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Analytic function

In mathematics, an analytic function is a function, locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not hold for real analytic functions. A function is analytic if and only if its Taylor series about x0 converges to the function in some neighborhood for every x0 in its domain. Formally, a function f is real analytic on an open set D in the real line if for any x 0 ∈ D one can write f = ∑ n = 0 ∞ a n n = a 0 + a 1 + a 2 2 + a 3 3 + ⋯ in which the coefficients a 0, a 1, … are real numbers and the series is convergent to f for x in a neighborhood of x 0. Alternatively, an analytic function is an infinitely differentiable function such that the Taylor series at any point x 0 in its domain T = ∑ n = 0 ∞ f n! N converges to f for x in a neighborhood of x 0 pointwise.

The set of all real analytic functions on a given set D is denoted by C ω. A function f defined on some subset of the real line is said to be real analytic at a point x if there is a neighborhood D of x on which f is real analytic; the definition of a complex analytic function is obtained by replacing, in the definitions above, "real" with "complex" and "real line" with "complex plane". A function is complex analytic. For this reason the terms "holomorphic" and "analytic" are used interchangeably for such functions. Typical examples of analytic functions are: All elementary functions: All polynomials: if a polynomial has degree n, any terms of degree larger than n in its Taylor series expansion must vanish to 0, so this series will be trivially convergent. Furthermore, every polynomial is its own Maclaurin series; the exponential function is analytic. Any Taylor series for this function converges not only for x close enough to x0 but for all values of x; the trigonometric functions and the power functions are analytic on any open set of their domain.

Most special functions: hypergeometric functions Bessel functions gamma functionsTypical examples of functions that are not analytic are: The absolute value function when defined on the set of real numbers or complex numbers is not everywhere analytic because it is not differentiable at 0. Piecewise defined functions are not analytic where the pieces meet; the complex conjugate function z → z* is not complex analytic, although its restriction to the real line is the identity function and therefore real analytic, it is real analytic as a function from R 2 to R 2. Other non-analytic smooth functions; the following conditions are equivalent: 1. F is real analytic on an open set D. 2. There is a complex analytic extension of f to an open set G ⊂ C which contains D. 3. F is real smooth and for every compact set K ⊂ D there exists a constant C such that for every x ∈ K and every non-negative integer k the following bound holds | d

Sergio Aguza

Sergio Aguza Santiago is a Spanish footballer who plays for UD Almería as a midfielder. Born in Sant Boi de Llobregat, Catalonia, after playing youth football with EF Gavà and UD Cornellà, Aguza joined Real Madrid's youth facilities in 2008, aged 16. In 2011 -- 12 season Aguza made his senior debut. In July 2013, he was promoted to the reserves in the Segunda División. On 8 September Aguza made his professional debut, starting in a 0–1 home loss against CD Mirandés, he scored his first goal on 15 February of the following year, netting his side's second in a 3–1 home success against FC Barcelona B. On 31 July 2015, Aguza signed for English Football League Championship side Milton Keynes Dons on a two-year deal. Aguza made his debut for the club on 11 August 2015 in the 2–1 Football League Cup win over Leyton Orient. On 16 January 2016, following limited appearances for the club, Milton Keynes Dons announced that Aguza had left the club via mutual consent. On 18 January 2016, Aguza signed a two-and-a-half-year contract with SD Ponferradina, returning to Spain and its second level.

He scored his first goal for the club on 20 March, in a 2–1 home win against SD Huesca, added two more match-winners against CD Lugo and Albacete Balompié. On 7 July 2016, Aguza signed a one-year deal with AD Alcorcón in the second level; the following 31 January he rescinded his contract, signed a six-month deal with fellow league team Córdoba CF just hours later. On 13 July 2018, Aguza signed a two-year contract with UD Almería, still in the second division. Sergio Aguza at BDFutbol Sergio Aguza at La Preferente Sergio Aguza at Soccerway

1950 NCAA Men's Ice Hockey Tournament

The 1950 NCAA Men's Ice Hockey Tournament was the culmination of the 1949–50 NCAA men's ice hockey season, the 3rd such tournament in NCAA history. It was held between March 16 and 18, 1950, concluded with Colorado College defeating Boston University 13-4. All games were played at the Broadmoor Ice Palace in Colorado; the 13 goals Colorado College scored in the championship game is a record that has only been equaled once. Additionally, the 23 goals scored in their two games is a record for one team in a Frozen Four. CC's goal differential is a record for an NCAA tournament, matched by Michigan in 1953 and Wisconsin in 1983. Four teams qualified for two each from the eastern and western regions; the teams were selected by a committee based upon both their overall record and the strength of their opponents. The eastern and western teams judged as better were seeded as the top regional teams; the second eastern seed was slotted to play the top western vice versa. All games were played at the Broadmoor Ice Palace.

All matches were Single-game eliminations with the semifinal winners advancing to the national championship game and the losers playing in a consolation game. Note: * denotes overtime period


WTIX-FM is an oldies outlet serving the New Orleans area. The station is owned by Michael A. "Michael In The Morning" Costello and operates at 94.3 MHz with an ERP of 100 kW. Its city of license is Louisiana, its studios are in Metairie and its transmitter is in St. Bernard Parish, Louisiana. Costello was program director at WRNO-FM from 1976 to 1991 and hosted the morning show as "Michael In The Morning". Additionally, he hosted the Sunday night edition of The Rock'N'Roll Hall Of Fame, he acquired the former KLEB-FM from the Harold Callais family, a country music outlet in 1995 and flipped it to its current oldies format. The station is modeled after the original Top 40 AM station WTIX, a successful Top 40 outlet in New Orleans from the 1950s through the 1970s, he increased the transmitted power and moved the antenna from its original location to its present location in 1998 to better serve the New Orleans metro area. Mike Costello handles GM, PD, hosts the morning show. Many of the jingles used on WTIX-FM are the originals used by WTIX/690 in the 1970s.

WTIX-FM adapts a time-honored feature from the old WTIX: the "Chime Time", wherein a chime rang whenever jocks gave the time on the air. WTIX-FM's format, Oldies, is patterned after half of WTIX's late-1980s hybrid Talk/Oldies format. Three of the WTIX-FM disc jockeys, "Hot" Rod Glenn, Bobby Reno, Terry Knight had minor and short-term ties to the original WTIX. Terry Knight left the station several years ago. Sgt. T-Ben's daily segment, "The News You Need Now!," which debuted in June 2004, is a tongue-in-cheek mock news report concerning local politicians, national celebrities and various topical news stories, concluding with his signature catchphrase, "Nobody cares!" And "JD The DJ," J. Douglas, veteran DJ at such stations as WPTR in Albany, KIRL in St. Louis, WIXO-FM, WNOE and WRNO-FM in New Orleans, hosts the "Rock and Roll Flight to Midnight". WTIX-FM 17 Choice Oldies, an album released on LP by The Mighty 690 in 1967, had been reissued on compact disc in 2005. Another CD, WTIX At The Beach, a compilation of summertime-themed oldies, was released in 2008, there is a new disc, WTIX Back To The Beach, released in July 2011.

Its current slogan is "New Orleans' Oldies Station, TIX-FM." Michael Costello Ben Walsh "Hot" Rod Glenn "Your Pal" Al Nassar J. Douglas Terry Knight - a DJ on the original WTIX. Bruce Charles - a DJ on the original WTIX. Bobby Reno WTIX official website Query the FCC's FM station database for WTIX Radio-Locator information on WTIX Query Nielsen Audio's FM station database for WTIXJD the DJ - J. Douglas' official site Benny The Jokeman - Home Page - Sgt. T-Ben Boudreaux's official site

Leszek Balcerowicz

Leszek Henryk Balcerowicz is a Polish economist and Professor at Warsaw School of Economics. He served as Chairman of the National Bank of Poland and twice as Deputy Prime Minister of Poland. In 1989, he became Minister of Finance in Tadeusz Mazowiecki's first non-communist government and led the free-market economic reforms, which have transformed Poland into one of Europe's most successful economies. In 2007, he became the Chairman of its Council. In 1970 he graduated with distinction from the Foreign Trade faculty of the Central School of Planning and Statistics in Warsaw. Balcerowicz received his MBA from St. John's University in New York, in 1974 and doctorate from the Central School of Planning and Statistics in 1975, he was a member of the Polish communist party from 1969 until the declaration of martial law in Poland, in 1981. In the late 1970s he participated in an economic-advisory team associated with the prime minister of People's Republic of Poland. In 1978–1980 he worked at the Institute of Marxism-Leninism.

He became an economics expert in the pro-democracy independent trade union Solidarity. From 1989 to 1991 and between 1997 and 2000 he was the Deputy Prime Minister and Finance Minister of Poland. Between 1995 and 2000 he was the chairman of a centrist free-market political party. On 22 December 2000 he became the Chairman of the National Bank of Poland, he was a columnist for Wprost, a Polish news magazine. On 11 November 2005, the President of Poland, Aleksander Kwaśniewski, awarded L. Balcerowicz with the Order of the White Eagle for his "contribution to Poland's economic transformation". In 2006 he was elected member of Galeria Chwały Polskiej Ekonomii, a hall of fame for "outstanding Polish economists". Balcerowicz is a member of the Commission on Legal Empowerment of the Poor, an independent initiative hosted by the UNDP and the first global initiative to focus on the link between exclusion and the law, he is a member of the influential Washington-based financial advisory body, the Group of Thirty, is a Board member of renowned Washington, D.

C. think-tank the Peterson Institute. Fellow of Collegium Invisibile. Since 11 June 2008 Balcerowicz has been a member of the board of Bruegel, the Brussels-based think tank on international economics. Since 2007, he has led the Civil Development Forum, a think-tank with a mission of "increasing active support of the society for a wide range of individual freedoms, - what goes with it - for strengthening the rule of law in the country". In 2016 he was appointed as representative of the Ukrainian President in the Cabinet of ministers; the Balcerowicz Plan was a series of reforms, which brought the to end hyperinflation, dismantled inefficient economic structures, balanced the national budget. The prices of most consumer goods were freed and caps for annual increases established in state-sector employees' wages. Poland's currency, the złoty, was made convertible within the country's borders; this resulted in a substantial increase in prices and had forced state-owned companies, while making them economically competitive.

This amounted to a two-year shock to the Polish economy. Among other actions included in the plan was the negotiation of a significant reduction of the debts inherited from the Polish People's Republic; the severity of the reforms was controversial and made Balcerowicz an object of criticism by some politicians in Poland. On the other hand, many economists and experts such as Krzysztof Sobczak, Jeffrey Sachs, Jacek Rostowski agree that without introducing such radical changes, Poland's economic success and steady economic growth would not have been possible. Since 1991, Poland's annual growth rate was one of the highest of all post-Communist economies, while the country has experienced uninterrupted growth for over 28 years - the longest in the world, along Australia. High unemployment remained a problem for some two decades after the implementation of the reforms, leaving certain poverty-stricken regions with structural unemployment. Reducing unprofitability of the state-owned companies required significant layoffs.

Though over 2 million Poles emigrated from Poland since its entry into the EU, until 2010s, the unemployment level remained at 13%. Populist politician Andrzej Lepper, the leader of the populist Self-Defense party, created the slogan: "Balcerowicz must go", echoing the disgruntlement felt by some Poles with Balcerowicz's plan. However, since 2013, the unemployment rate has not exceeded 10% and in 2019 has reached the record low of 3.8%. During the Eurozone crisis Balcerowicz has been an outspoken supporter for fiscal discipline and has been dubbed the anti-Bernanke for his scorn of distortionary fiscal stimulus. In various articles he has developed a comparison between the fiscally-profligate PIGS and the fiscally-disciplined BELLs. Responsible fiscal policy brings about better growth outcomes, claims Leszek Balcerowicz, he has many followers among East European economists, most prominently Simeon Djankov, Deputy prime Minister and Minister of Finance of Bulgaria between 2009 and 2013. Balcerowicz was a competetive athlete in his youth.

In 1966, he became Poland's youth champion in cross country at the distance of 1500 meters. Since 1977, he has been married to an economist, he has three children. 1993 University

Samuel Sharpe

Samuel Sharpe, or Sharp known as Sam Sharpe, was an enslaved Jamaican man, the leader of the widespread 1832 Baptist War slave rebellion in Jamaica. He was proclaimed a National Hero of Jamaica in 1975 and his image is on the $50 Jamaican banknote. Samuel Sharpe was born into enslavement in the parish of St James, Jamaica, on a plantation owned by Samuel and Jane Sharpe around 1805; the Slave Return of 1832 announcing his death gave his name as Archer aka Samuel Sharpe, the son of Eve, he was only 31 years old when he died. The Slave Return of Samuel and Jane Sharpe 1817 showed a young 12-year-old Archer on the plantation with his mother Juda Bligom and siblings Joe and Eliza, he was allowed to become educated. Because of his education, he was respected by other slaves. Sharpe became a well-known preacher and leader in the Baptist Church, which had long welcomed slaves as members and recognized them as preachers, he was a deacon at the Burchell Baptist Church in Montego Bay, whose pastor was Rev. Thomas Burchell, a missionary from England.

Sharpe spent most of his time travelling to different parishes in Jamaica, educating the slaves about Christianity, which he believed promised freedom. Slaves learned. In the mistaken belief that emancipation had been granted by the British Parliament, Sharpe organised a peaceful general strike across many estates in western Jamaica to protest working conditions; as this was the harvest of the sugar cane, it was a critical time for the plantation owners: the workforce had to work overtime to process the cane at its peak. The Christmas Rebellion began on 27 December 1831 at the Kensington Estate. Reprisals by the plantation owners led to the rebels' burning the crops. Sharpe's peaceful protest turned into Jamaica's largest slave rebellion; the uprising lasted for 10 days and spread throughout the entire island, mobilizing as many as 60,000 of Jamaica's enslaved population. The colonial government used the armed Jamaican military forces and warriors from the towns of the Jamaican Maroons to put down the rebellion, suppressing it within two weeks.

Some 14 whites were killed by armed slave battalions, but more than 200 slaves were killed by troops. Afterwards, more reprisals followed; the government tried and hanged many of the ringleaders, including Sharpe, in 1832. A total of 310 to 340 were executed through the judicial process, including many for minor offenses such as theft of livestock. In the months leading up to his execution, while in jail, Sharpe had several meetings with Rev. Henry Bleby, a missionary, who reported that Sharpe told him: "I would rather die upon yonders gallows than live my life in slavery, than live a slave." The rebellion and government response provoked two detailed Parliamentary Inquiries. The Jamaican government's severe reprisals in the aftermath of the rebellion are believed to have contributed to passage by Parliament of the 1833 Slavery Abolition Act and final abolition of slavery across the British Empire in 1838. In 1975, the government of independent Jamaica proclaimed Sharpe a National Hero, with the posthumous title of Rt.

Excellent Samuel Sharpe. In 1975, Sam Sharpe Teachers' College was founded and named in his honor in Granville, a suburb of Montego Bay. Sharpe's image is used on the modern Jamaican $50 bill; the British jazz saxophonist Courtney Pine included an instrumental composition'Samuel Sharpe' as a tribute on his 2012 album House of Legends. Sam Sharpe is referenced by Vybz Kartel in his 2019 song "Stand Strong". History of Jamaica Slavery in the British and French Caribbean Thomas Burchell Rodriguez, Junius P. Encyclopedia of Slave Resistance and Rebellion. Westport, Connecticut: Greenwood, 2006. Reid-Salmon, Burning for Freedom: A Theology of the Black Atlantic Struggle for Liberation. Jamaica: Ian Randle Publishers, 2012. Jamaican History Reid-Salmon, Burning for Freedom: A Theology of the Black Atlantic Struggle for Liberation