Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis. Numerical analysis finds application in all fields of engineering and the physical sciences, but in the 21st century the life sciences, social sciences, medicine and the arts have adopted elements of scientific computations; the growth in computing power has revolutionized the use of realistic mathematical models in science and engineering, subtle numerical analysis is required to implement these detailed models of the world. For example, ordinary differential equations appear in celestial mechanics. Before the advent of modern computers, numerical methods depended on hand interpolation formulas applied to data from large printed tables. Since the mid 20th century, computers calculate the required functions instead, but many of the same formulas continue to be used as part of the software algorithms; the numerical point of view goes back to the earliest mathematical writings.

A tablet from the Yale Babylonian Collection, gives a sexagesimal numerical approximation of the square root of 2, the length of the diagonal in a unit square. Computing the sides of a triangle in terms of square roots is a basic practical problem, for example in astronomy and construction. Numerical analysis continues this long tradition: rather than exact symbolic answers, which can only be applied to real-world measurements by translation into digits, it gives approximate solutions within specified error bounds; the overall goal of the field of numerical analysis is the design and analysis of techniques to give approximate but accurate solutions to hard problems, the variety of, suggested by the following: Advanced numerical methods are essential in making numerical weather prediction feasible. Computing the trajectory of a spacecraft requires the accurate numerical solution of a system of ordinary differential equations. Car companies can improve the crash safety of their vehicles by using computer simulations of car crashes.

Such simulations consist of solving partial differential equations numerically. Hedge funds use tools from all fields of numerical analysis to attempt to calculate the value of stocks and derivatives more than other market participants. Airlines use sophisticated optimization algorithms to decide ticket prices and crew assignments and fuel needs; such algorithms were developed within the overlapping field of operations research. Insurance companies use numerical programs for actuarial analysis; the rest of this section outlines several important themes of numerical analysis. The field of numerical analysis predates the invention of modern computers by many centuries. Linear interpolation was in use more than 2000 years ago. Many great mathematicians of the past were preoccupied by numerical analysis, as is obvious from the names of important algorithms like Newton's method, Lagrange interpolation polynomial, Gaussian elimination, or Euler's method. To facilitate computations by hand, large books were produced with formulas and tables of data such as interpolation points and function coefficients.

Using these tables calculated out to 16 decimal places or more for some functions, one could look up values to plug into the formulas given and achieve good numerical estimates of some functions. The canonical work in the field is the NIST publication edited by Abramowitz and Stegun, a 1000-plus page book of a large number of used formulas and functions and their values at many points; the function values are no longer useful when a computer is available, but the large listing of formulas can still be handy. The mechanical calculator was developed as a tool for hand computation; these calculators evolved into electronic computers in the 1940s, it was found that these computers were useful for administrative purposes. But the invention of the computer influenced the field of numerical analysis, since now longer and more complicated calculations could be done. Direct methods compute the solution to a problem in a finite number of steps; these methods would give the precise answer. Examples include Gaussian elimination, the QR factorization method for solving systems of linear equations, the simplex method of linear programming.

In practice, finite precision is used and the result is an approximation of the true solution. In contrast to direct methods, iterative methods are not expected to terminate in a finite number of steps. Starting from an initial guess, iterative methods form successive approximations that converge to the exact solution only in the limit. A convergence test involving the residual, is specified in order to decide when a sufficiently accurate solution has been found. Using infinite precision arithmetic these methods would not reach the solution within a finite number of steps. Examples include Newton's method, the bisection method, Jacobi iteration. In computational matrix algebra, iterative methods are needed for large problems. Iterative methods are more common than direct methods in numerical analysis; some methods are direct in principle but are used as though they were not, e.g. GMRES and the conjugate gradient method. For these methods the number of steps needed to obtain the exact soluti

"Today" is a song by American alternative rock band The Smashing Pumpkins, written by lead vocalist and guitarist Billy Corgan. The song, though upbeat, contains dark lyrics. Corgan wrote the song about a day in which he was having suicidal thoughts, exemplified by the reference to depression and anger under the bridge; the contrast between the grim subject matter of the song and the soft instrumental part during the verses, coupled with use of irony in the lyrics, left many listeners unaware of the song's tale of depression and desperation. The song alternates between loud choruses with layered, distorted guitar. "Today" was released in September 1993 as the second single from the band's second album and major label debut, Siamese Dream. Although Corgan opted for "Cherub Rock", the lead single from the album, to be the opening track, "Today" and its follow-up "Disarm" are credited in AllMusic for popularizing the band and "sen into the stratosphere". "Today" has been well received by critics.

After the release and minor success of the band's debut album, The Smashing Pumpkins were being hyped as "the next Nirvana". However, the band was experiencing several difficulties at the time. Drummer Jimmy Chamberlin was undergoing an severe addiction to heroin. Corgan recalled that "after the first album, I became suicidal, it was an eight-month depression, give or take a month, I was pretty suicidal for about two or three months." Under the pressure and other complications, the Pumpkins entered the Triclops Sound Studios in Atlanta to record the follow-up to Gish."Today" was the first song Corgan wrote for Siamese Dream. Corgan said, "The day after I wrote'Today', my manager heard it and said,'It's a hit', I guess in a way, it was." Corgan played the self-recorded demo to producer Butch Vig and to the rest of his band, all of whom responded positively. "Today" had a chord progression and a melody, but Corgan felt there needed to be an opening riff to the song. One day, Corgan said. "When I added the opening riff, it changed the character of the song.

I had a song, starting out quiet and got loud." Soon afterwards, Virgin Records executives were sent to check up on the band after hearing about their problems, but were pleased with the demo. The reaction from the executives only served to put more stress on Corgan. Siamese Dream was completed after running four months after the deadline and exceeding its budget. Virgin Records executives saw "Today" as the ideal lead single, but Corgan wanted "Cherub Rock", one of the last songs written for the album, as the lead single. "Cherub Rock" was released first, but it was only a modest success. "Today" brought the band popularity through a successful music video. "Today" is played in standard tuning. While Corgan considered tuning the guitar down a half-step and playing the song as if it were in the key of E, he said, "There was something about the E♭ voicing that I liked. So it's intentionally in E♭, not E." "Today" is one of the Siamese Dream songs on which Corgan took over Iha and D'arcy's duties on guitar and bass to ensure quality.

The song, spanning three minutes and 21 seconds, begins with a one-measure opening guitar riff that alludes to the E♭ major-pentatonic scale. Corgan uses this riff to indicate shifts in the song. After the riff is played four times, the rest of the band enters backed with feedback-driven guitars. Ned Raggett of Allmusic commented on the song's ability to "alternate between calmer Cure-like sections and the louder crunches, soon-to-be-trademark guitar style taking My Bloody Valentine's own hypnotic riffing to more accessible results"; the chorus, consisting of multiple guitar tracks playing barre chords, is an example of the band's tendency to overdub several rhythm guitar tracks. While the song does not include a standard guitar solo, a short melodic guitar part appears during the bridge. Corgan explained: "The little guitar break over the C chord is a forwards sample that we laid in backwards. There's a weird vocal effect that moves across the speakers, a'yan-yan-yan-yan' thing, generated by using a Roland Space Echo to regenerate on the last word of the vocal line,'I wanna turn you on.'"The dark, ironic lyrics of "Today", describing a day when Corgan was feeling depressed and suicidal, contrast with the instrumentation.

Michael Snyder of the San Francisco Chronicle said that the song is "downright pretty as rock ballads go" but that "Corgan manages to convey the exhilaration and tragic release he seeks." Corgan told Rolling Stone that "I was suicidal... I just thought it was funny to write a song that said today is the greatest day of your life because it can't get any worse." Corgan compared writing the lyrics of "Today" and "Disarm" to "ripping guts out". They played the song live for the first time on September 20, 1993, on Late Night with David

Petroleum licensing or exploration license is the act of giving licenses to a company or a joint venture allowing them to search for commercially feasible deposits for the. Each sovereign country distributes licenses in what is called a licensing round; the procedure for such, can vary from country to country. The largest change is if it is a bid system or a grant system. In the bid system, each company or joint venture will offer a bid to gain the rights for the petroleum exploration at the license for a limited period of time; the highest bid, will obtain the rights. In the grant system, the license will be granted to the company or joint venture that shows the highest interest and ability for the exploration of the license; this can be shown by the company experience, projected plan for the exploration and or longest presence on the country as a petroleum exploration company. Malaysia - The Malaysia Petroleum Management is pleased to announce the launch of their ‘" 2018 Exploration Bidding Round".

United Kingdom - The Department of Energy and Climate Change grants licences. The UKCS is divided into quadrants of one degree longitude; each quadrant is divided into 30 blocks measuring 10 minutes of latitude and 12 minutes of longitude. Some blocks are divided further into part blocks where some areas are relinquished by previous licensees. For example, block 13/24a is the'a' part block; the UK government has traditionally issued licences via periodic licensing rounds. Blocks are awarded on the basis of the work programme bid by the participants; the UK government has solicited new entrants to the UKCS via "promote" licensing rounds with less demanding terms and the fallow acreage initiative, where non-active licences have to be relinquished. Norway - The Norwegian Petroleum Directorate grants licences; the NCS is divided into quads of 1 degree by 1 degree. Norwegian licence blocks are larger than British blocks, being 15 minutes of latitude by 20 minutes of longitude. Like in Britain, there are numerous part blocks formed by re-licensing relinquished areas.

One of the attractiveness of the license blocks in Norway are the economical benefits given by its taxation policy. The Norwegian tax system refunds 78% of all directly related exploration costs, which are paid the year after the costs occurred. Denmark - The Danish Energy Authority administers the Danish sector; the Danes divide their sector of the North Sea into 1 degree by 1 degree quadrants. Their blocks, are 10 minutes latitude by 15 minutes longitude. Part blocks exist. Germany - Germany and the Netherlands share a quadrant and block grid - quadrants are given letters rather than numbers; the blocks are 10 minutes latitude by 20 minutes longitude. Germany has the smallest sectors in the North Sea. Brazil - The National Petroleum Agency grants licenses; the license blocks are named followed by the state code and a number. BT means. While BM means the block is offshore, at sea. United States - Oil and gas rights in the onshore US tracts may be owned by private individuals or corporations, states, or the federal government.

Rights to explore for oil on owned mineral rights tracts are obtained by leases negotiated between the parties. The rules for obtaining licenses to explore on state mineral rights vary from state to state. Federal onshore exploration licenses are obtained through oral competitive bidding. Oil and gas rights in the offshore US are divided between the state and federal governments. Rights to explore on US federal offshore areas are obtained through competitive sealed bids. Individual tracts are 9 square miles. Current leases being offered in the Gulf of Mexico have 5-year terms for tracts in water depths of less than 400 m, 8 years for tracts in water greater than 400 m. Royalty rates are 18.75%. When a petroleum license is granted, the company or join venture is given a limited time for the exploration of the license. If after the limited time, the company has not discovered or performed its minimal obligations the license will be relinquished. Relinquished licenses can be re-attributed in next licensing rounds, or kept as relinquished if the government believes there is little interest on that area for the petroleum exploration.

Geology of the United Kingdom Geography of the United Kingdom Hydrocarbon exploration

Stingray CMusic is a British television music channel dedicated to classical and soundtrack music videos owned by Stingray Digital in Canada. The C in the channel's name stands for the genres of the channel's playlist which covers classical music, crossover music and cinema music. C Music Entertainment Ltd. which operates the channel, was founded in 2007 by Julian Rigamonti. In May 2017, Canadian company Stingray Digital acquired C Music Entertainment Ltd.. The channel was renamed as Stingray CMusic on 1 December 2018; the channel broadcasts short form music videos, avoids long form concerts and operas traditionally associated with classical music on television. Thema handles the worldwide distribution of the channel; the channel is distributed to host cable, DTH satellite, IPTV and mobile phone television providers via eight contribution satellites. To Europe on the Eurobird 9A satellite and on Eutelsat 10A, to North Africa on the Eutelsat 5 West A satellite, to South Africa on Eutelsat 36 B, to Portugal on Hispasat 1E and on SES 5, to the Middle East on Badr 5 and on Intelsat 20.

A high definition simulcast version was launched in 2013. In June 2008 C Music TV launched a subscription service on their website cmusic.tv, but it was suspended by the time the channel was rebranded as Stingray CMusic in December 2018. As C Music TV, the channel won the Award for'Best Music TV Channel 2008' at the Hot Bird European TV Awards which were announced on 14 November 2008 at a ceremony held at the “Scuola Grande di San Giovanni Evangelista” in Venice, Italy, it received the Hot Bird Jury's "Special Mention" Trophy in 2009 and in November 2012 it won the title of'Best Music Channel 2012' at the Eutelsat TV Awards 2012. Official website C Music Entertainment Limited at Companies House

The Duke of Silesia was the sons and descendants of the Polish Duke Bolesław III Wrymouth. In accordance with the last will and testament of Bolesław, upon his death his lands were divided into four or five hereditary provinces distributed among his sons, a royal province of Kraków reserved for the eldest, to be High Duke of all Poland; this was known as the fragmentation of Poland. Subsequent developments lead to further splintering of the duchies. At the beginning of the 14th century, fourteen independent Duchies existed in Silesia: Brzeg, Wrocław, Świdnica, Jawor, Ziębice, Głogów, Ścinawa, Żagan and Oleśnica in Lower Silesia. Between 1327 and 1329 most dukes accepted the overlordship of Bohemian king John of Bohemia, who acquired the right of succession for all of these duchies. In the coming centuries all branches of the Silesian Piasts died out, with the death of George William, Duke of Liegnitz the dynasty ceased to exist; the Duchy of Silesia, one of the hereditary provinces of Poland, was granted to Bolesław III's eldest son, Władysław II the Exile, was subsequently divided among his sons Bolesław I the Tall, Mieszko I Tanglefoot and Konrad Spindleshanks.

After Konrad's death Głogów was again united with the Duchy of Wrocław/Lower Silesia. In 1173 Bolesław returned and he agreed to let Mieszko and Bolesław rule in their own Duchies, separated from the Duchy of Silesia; this led to the creation of the Duchy of Racibórz for Mieszko I and the Duchy of Opole for Jarosław, beginning the fragmentation of the Duchy of Silesia. The territories controlled by Mieszko I and Jarosław corresponded to what is known as Upper Silesia, while the territories remaining with Bolesław I corresponded to Lower Silesia. Duchy of Lower Silesia was a direct continuation of the Duchy of Silesia, but without the territories corresponding to Upper Silesia; some sources refer to it as the Duchy of Silesia. Wrocław was the capital of the Duchy of Silesia, yet this early Duchy of Silesia should not be confused with the smaller Duchy of Wrocław, created with further fragmentation in 1248; the Duchy went through various border changes in the coming years, sometimes losing and sometimes gaining territory.

In 1248 Lower Silesia was divided when Bolesław II had to cede the Duchy of Wrocław to his younger brother Henry III. Upper Silesia was divided into the Duchies of Cieszyn, Opole-Racibórz. In 1340 the Duchy of Racibórz was united with a Bohemian fief. Below follows a simplified table of Silesia's partitions: A quick reminder avoiding confusion: Established in 1290 by High Duke Henry IV Probus, held by the Bishops of Wrocław 1302–1319 Henry of Wiebrzno 1326–1341 Nankier 1342–1376 Przecław of Pogarell 1382–1417 Wenceslaus II of Legnica 1417–1447 Konrad IV of Oleśnica 1447–1456 Peter II Nowak 1456–1467 Jošt of Rožmberk 1468–1482 Rudolf of Rüdesheim 1482–1506 Jan IV Roth 1506–1520 Jan V Thurzo 1520–1539 Jacob of Salza 1539–1562 Balthazar of Promnitz 1562–1574 Caspar of Logau 1574–1585 Martin Gerstmann 1585–1596 Andreas Jerin 1596–1599 Bonaventura Hahn 1599–1600 Paul Albert of Radolfzell 1600–1608 Jan VI of Sitsch 1608–1624 Charles of Austria, son of Charles II, Archduke of Austria 1625–1655 Karol Ferdynand Vasa, Duke of Opole from 1648 1656–1662 Leopold Wilhelm of Habsburg 1663–1664 Charles Joseph of Habsburg Grand Master of the Teutonic Order from 1662 1665–1671 Sebastian von Rostock 1671–1682 Frederick of Hesse-Darmstadt 1683–1732 Franz Ludwig von Pfalz-Neuburg 1732–1747 Philipp Ludwig von SinzendorfMajor part annexed by the Kingdom of Prussia after the First Silesian War in 1742.

1747–1795 Philipp Gotthard von Schaffgotsch 1795–1817 Joseph Christian Franz zu Hohenlohe-Waldenburg-BartensteinPrussian part secularised in 1810. 1823–1832 Emanuel von Schimonsky 1835–1840 Leopold von Sedlnitzky 1843–1844 Joseph Knauer 1845–1850 Melchior von DiepenbrockTheocracy abolished in 1850. List of Polish rulers Piast dynasty Dukes of Masovia Dukes of Greater Poland Dukes of Little Poland Dukes of Cuiavia Dukes of Sieradz-Łęczyca Neue deutsche Biographie, Berlin 2001, Bd.: 20, p. 403-407 Meyers Großes Konversations-Lexikon, Leipzig 1905–1909, Bd.: 17, p. 845-847 http://www.tacitus.nu/historical-atlas/regents/poland/silesia.htm

Wakamatsu-ku is a ward of Kitakyūshū, Japan. It has an area of 67.86 km2 and a population of about 88,594. Important facilities include Green Park, Ecotown recycling area, Gakuen Toshi academic area, Hibiki container terminal. An underwater tunnel was being constructed in 2005–2006 between Tobata and Wakamatsu wards to link the Hibiki container terminal with Kokura. Green Park is a large green area which includes a tropical house with rare plants and birds, it is not well known outside Kitakyushu. Sometimes indoor and open-air concerts are staged. There is an observation tower. There is a wind farm on harnessing wind power for electricity. There is no particular controversy about the sightliness or otherwise of this wind farm, as in some other countries, it is on windy reclaimed land. The beaches are pleasant. A larger one is beyond the city limits. Swimming is limited to the summer months; the writer Hino Ashihei was born in Wakamatsu and his birthhouse can be visited. Media related to Wakamatsu-ku, Kitakyūshū at Wikimedia Commons Wakamatsu ward office Green Park- Official Page Kyushu Institute of Technology Wakamatsu campus is in the Gakuen Toshi Academic zone.

Hino Ashihei