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f090849b5ae56c3025dea23b2dd6ff17b74a18e3 | on march 5, 2020, riat was signed to | on march 5, 2020, riat was signed to a two-year entry-level contract by the washington capitals of the national hockey league (nhl). on august 4, 2020, the washington capitals agreed to return riat on loan to former club genève-servette hc for the beginning of the 2020–21 season due to the covid-19 pandemic delaying th... | wikipedia |
0f108938df7eeedb04a6145f9db225beab8df3a1 | riat made his professional debut with genève-servette hc | riat made his professional debut with genève-servette hc in the 2015–16 season, appearing in 45 national league games this season and putting up 21 points (9 goals). he went on to play three more seasons with geneva, playing 139 regular season games (59 points) and 13 playoffs games (4 points). | wikipedia |
f600fc854a5a4ebb65051837dc72dde4788351b7 | damien riat (born 26 february 1997) is a | damien riat (born 26 february 1997) is a swiss professional ice hockey winger who is currently playing with lausanne hc of the national league (nl). riat was drafted 117th overall by the washington capitals in the 2016 nhl entry draft. | wikipedia |
a798b352741c88e802d57359c4cc39951655c491 | electron nuclear double resonance (endor) is a magnetic | electron nuclear double resonance (endor) is a magnetic resonance technique for elucidating the molecular and electronic structure of paramagnetic species. the technique was first introduced to resolve interactions in electron paramagnetic resonance (epr) spectra. it is currently practiced in a variety of modalities, m... | wikipedia |
a23521703710fb82f03f0138ea3909efbab2dd6f | hf pulsed endor is generally applied to biological | hf pulsed endor is generally applied to biological and related model systems. applications have been primarily to biology with a heavy focus on photosynthesis related radicals or paramagnetic metal ions centers in matalloenzymes or metalloproteins. additional applications have been to magnetic resonance imaging contras... | wikipedia |
6a757df0b5b7a78c5ef3df46dcee4f4cabd072ec | endor technique has been used to characterize of | endor technique has been used to characterize of spatial and electronic structure of metal-containing sites. paramagnetic metal ions/complexes introduced for catalysis; metal clusters producing magnetic materials; trapped radicals introduced as probes for disclosing the surface acid/base properties; color centers and d... | wikipedia |
937b2a2b108686858446897cb5d27f674f544e63 | at θ = 0˚ the endor spectra contain | at θ = 0˚ the endor spectra contain only the component of hyperfine coupling that is parallel to the axial protons and perpendicular to the equatorial protons. at θ = 90˚ endor spectra contain only the component of hyperfine coupling that is perpendicular to the axial protons and parallel to the equatorial protons. the... | wikipedia |
f603f268095d095d5df2147fd6d5bbe7dd845b0a | the traditional convention of magnetic resonance envisions the | the traditional convention of magnetic resonance envisions the paramagnets aligning with the external magnetic field; however, in practice it is simpler to treat the paramagnets as fixed and the external magnetic field as a vector. specifying positional relationships requires three separate but related pieces of inform... | wikipedia |
d700f210225f4942cfdcbf00dc12fbc596ef2092 | in polycrystalline media or frozen solution, endor can | in polycrystalline media or frozen solution, endor can provide spatial relationships between the coupled nuclei and electron spins. this is possible in solid phases where the epr spectrum arises from the observance of all orientations of paramagnetic species; as such the epr spectrum is dominated by large anisotropic i... | wikipedia |
f9ad7d10b409a1050c766b119b8e0525b70db15b | polarization modulated endor (pm-endor) uses two perpendicular rf | polarization modulated endor (pm-endor) uses two perpendicular rf fields with similar phase control units to cp-endor. however, a linearly polarized rf field which rotates in the xy-plane at a frequency less than the modulation frequency of the rf carrier is used. | wikipedia |
4aadd4df20050bc3e2e56460be9cb9e017621d6d | the cp-endor technique makes use of circularly polarized | the cp-endor technique makes use of circularly polarized rf fields. two linearly polarized fields are generated by rf currents in two wires which are oriented parallel to the magnetic field. the wires are then connected into half loops which then cross at a 90 degree angle. this technique was developed by schweiger and... | wikipedia |
de6f2ff41487b418dc7ad625a4cd46769b7c1ff4 | double electron-nuclear-double resonance (double endor) requires the application | double electron-nuclear-double resonance (double endor) requires the application of two rf (rf1 and rf2) fields to the sample. the change in signal intensity of rf1 is observed while rf2 is swept through the spectrum. the two fields are perpendicularly oriented and are controlled by two tunable resonance circuits which... | wikipedia |
8da9b799c2651424fd046712502c182200e6f6b1 | this technique was established by hyde and is | this technique was established by hyde and is especially useful for separating overlapping epr signals which result from different radicals, molecular conformations or magnetic sites. ei-epr spectra monitor changes in the amplitude of an endor line of the paramagnetic sample, displayed as a function of the magnetic fie... | wikipedia |
ad85ce2a5b241194ce1812c702a21c116004ee73 | endor-induced epr (ei-epr) displays endor transitions as a | endor-induced epr (ei-epr) displays endor transitions as a function of the magnetic field. while the magnetic field is swept through the epr spectrum, the frequency follows the zeeman frequency of the nucleus. the ei-epr spectra can be collected in two ways: (1) difference spectra (2) frequency modulated rf field witho... | wikipedia |
a8b7e4dce1d9189121c8dc3a078d4747176ccc30 | where γ e {\displaystyle \gamma _{\mathrm {e} }} | where γ e {\displaystyle \gamma _{\mathrm {e} }} and γ n {\displaystyle \gamma _{\mathrm {n} }} are the gyromagnetic ratio of the electron and the nucleus respectively. b 1 {\displaystyle b_{1}} is the magnetic field of the observer which is microwave radiation while b 2 {\displaystyle b_{2}} is the magnetic field of t... | wikipedia |
90a4d6921f96ae46856e4013d5f60a4dbd1d16c9 | the right figure illustrates the energy diagram of | the right figure illustrates the energy diagram of the simplest spin system where a is the isotropic hyperfine coupling constant in hertz (hz). this diagram indicates the electron zeeman, nuclear zeeman and hyperfine splittings. in a steady state endor experiment, an epr transition (a, d), called the observer, is partl... | wikipedia |
d5fa74f9fd8bc7d0313dccaa6903ee77542e89b5 | the electron zeeman interaction describes the interaction between | the electron zeeman interaction describes the interaction between an electron spin and the applied magnetic field. the nuclear zeeman interaction is the interaction of the magnetic moment of the proton with an applied magnetic field. the hyperfine interaction is the coupling between the electron spin and the proton's n... | wikipedia |
9fb639b52d7b18c8417a04e76591d1bcbaca07bf | in the standard continuous wave (cwendor) experiment, a | in the standard continuous wave (cwendor) experiment, a sample is placed in a magnetic field and irradiated sequentially with a microwave followed by radio frequency. the changes are then detected by monitoring variations in the polarization of the saturated electron paramagnetic resonance (epr) transition. | wikipedia |
a798b352741c88e802d57359c4cc39951655c491 | electron nuclear double resonance (endor) is a magnetic | electron nuclear double resonance (endor) is a magnetic resonance technique for elucidating the molecular and electronic structure of paramagnetic species. the technique was first introduced to resolve interactions in electron paramagnetic resonance (epr) spectra. it is currently practiced in a variety of modalities, m... | wikipedia |
0d5ad47d3037837ec029f5ed5f2fcb1100dc036f | the championships were part of the uslta circuit | the championships were part of the uslta circuit from 1887 until 1923. it became of part of the iltf circuit following the united states joining the international lawn tennis federation in 1925 until 1969 in 1970 the tournament became part of the iltf grand prix circuit. in 1955 and from 1969 until 1973 the women's eve... | wikipedia |
49e4f1dba456acb6804ece4d740f91745510ff96 | on july 4, 1887, the first championships of | on july 4, 1887, the first championships of the western states were inaugurated at the scarlett ribbon club in chicago, illinois, united states, and the first men's champion was charles amherst chase. in 1895 a women's championship event was added to the schedule which was won by marion capwell. | wikipedia |
16ea10399c6b8db1630d73fecb798ad62358311f | edward i. kasid was a polish american professional | edward i. kasid was a polish american professional basketball player. he spent one season in the basketball association of america (baa) as a member of the toronto huskies during the 1946–47 season. he was born in minnesota to joseph and josephine kasid. kasid attended minneapolis vocational high school. | wikipedia |
387a81158366e407be9dfc5ef1ec4d23519c523b | edward i. kasid (august 13, 1923 – november | edward i. kasid (august 13, 1923 – november 3, 1989) was a polish american professional basketball player. he spent one season in the basketball association of america (baa) as a member of the toronto huskies during the 1946–47 season. he was born in minnesota to joseph and josephine kasid. kasid attended minneapolis v... | wikipedia |
ff728f6f76232afb79208426a3f47b5dff004c29 | grabarkewitz was part of a core of young | grabarkewitz was part of a core of young dodgers prospects that became known as "the mod squad" after the popular tv series of the same name, and appeared on the cover of the may 19, 1969 edition of sports illustrated, along with his fellow mod squad members. in 1977, he became an independent life insurance agent in sa... | wikipedia |
5c9acfcc2dab7a097ff71b8d960d76ac8669e831 | grabarkewitz was released during spring training 1975, and | grabarkewitz was released during spring training 1975, and joined his friend williams with oakland two days before the season started. he spent most of the season with oakland, but appeared in only his sixth game before breaking his ribs and shoulder from a collision with the kansas city royals ' hal mcrae. he was assi... | wikipedia |
8d2990f243023979c95a50cafaebc7989e5e413f | the cubs were battling injuries, and had something | the cubs were battling injuries, and had something of a revolving door at second base until grabarkewitz's arrival. he took over the job, and batted a respectable.248 over the rest of the season while also playing some short. his only home run as a cub came against the phillies on july 26. once again grabarkewitz saw h... | wikipedia |
f4e40fec1bba7fe14683325f3e17eb78267f8808 | grabarkewitz's contract was sold by the angels to | grabarkewitz's contract was sold by the angels to the philadelphia phillies on august 14, 1973 in a transaction that was completed four months later at the winter meetings on december 6 when denny doyle was sent to california for aurelio monteagudo and chris coletta. returning to the national league seemed to rejuvenat... | wikipedia |
34fc0d7d26dc6aefe779e5888332f0331aca4525 | the angels used grabarkewitz at second, third and | the angels used grabarkewitz at second, third and short, and even tried him in the outfield and the newly formed designated hitter position, however, the shoulder and ankle problems continued, as grabarkewitz's average never climbed above.200 after april. | wikipedia |
cff349b43d813f32281677959e67a185fca08e79 | during the off season, the dodgers dealt second | during the off season, the dodgers dealt second baseman ted sizemore and minor league catcher bob stinson to the st. louis cardinals for third baseman dick allen with the intention of shifting grabarkewitz back to second. unfortunately, a spring training arm injury limited grabarkewitz's services to start the season, a... | wikipedia |
b77b83a1df5de2a33958355cc2cad67b02120150 | grabarkewitz's batting average was.341 at the time of | grabarkewitz's batting average was.341 at the time of the all-star game, and he was embroiled in a battle for the nl batting crown with rico carty and roberto clemente. however, he batted only.232 over the remainder of the season, and ended the season with a.289 average. regardless, he led the dodgers with a.399 on-bas... | wikipedia |
19df81e367a6d3332102844099b60cfe2158b236 | grabarkewitz began the 1970 season at second base, | grabarkewitz began the 1970 season at second base, but shifted over to third when steve garvey was demoted to spokane. following a 2-for-3 performance against the san francisco giants on may 17, grabarkewitz's batting average peaked at a season high of.420, and earned him serious consideration as a write-in candidate f... | wikipedia |
d0b88da1b10e0ae4f3b8d5c5d1709f50c2eeebc7 | all told, grabarkewitz batted.283 with 48 home runs, | all told, grabarkewitz batted.283 with 48 home runs, 157 runs batted in and 287 runs scored over his minor league career when he made his major league debut with the dodgers on april 22, 1969. grabarkewitz was given the opportunity to earn the dodgers' starting shortstop job, but struggled in that role. he was batting.... | wikipedia |
d7cef1ae088be56e5b3469fcbdc03be7c8231530 | grabarkewitz immediately impressed in his first professional season. | grabarkewitz immediately impressed in his first professional season. the second baseman clubbed eleven home runs and scored a northwest league leading 62 runs for the duke snider led tri-city atoms to lead the team to its second straight title. in 1967, grabarkewitz was shifted to shortstop with the california league '... | wikipedia |
9c2b7f7e5046b85e59a7cd1df37b0f2cb6154ed8 | grabarkewitz lettered in baseball, basketball, football, golf and | grabarkewitz lettered in baseball, basketball, football, golf and track at alamo heights high school, and in 1963, he played for the now defunct sagamore clouters of the cape cod baseball league. after high school, he attended st. mary's university in nearby san antonio, texas for two years before being selected by the... | wikipedia |
16cff4b78aa6c224f9a0ed7bad6e36907836e054 | when he died in 1909 at woolton woods | when he died in 1909 at woolton woods his probate was almost £500,000. he was buried in the churchyard of cairo street chapel, warrington. the estate of woolton woods passed to his sons who sold it to col. james p. reynolds, who in turn sold it to liverpool city council. | wikipedia |
4f8895024a47b5673b8ef8c3d34c3ad3599de5c9 | with his accumulated wealth holbrook gaskell moved to | with his accumulated wealth holbrook gaskell moved to woolton woods in much woolton. he became a renowned collector of orchids. frederick sander, an orchid dealer, received a new cattleya species in 1883 from his collector seidl and named it cattleya gaskelliana after holbrook gaskell in recognition of a good customer ... | wikipedia |
386dab1ae0c3637ec5196fd02f10129b96d250cb | gaskell served as a magistrate in widnes, was | gaskell served as a magistrate in widnes, was an active liberal and a member of the liverpool reform club, supporting causes including the extension of the franchise. he endowed a chair of botany and provided chemistry laboratories at university college, liverpool. he paid for public baths in widnes and supported conva... | wikipedia |
f210b45d02b0bea452af359669cf8b3911d628f6 | in 1860 when the governments of britain and | in 1860 when the governments of britain and france formed a treaty to raise duties on materials made from salt, holbrook gaskell went with edmund knowles muspratt to paris to negotiate terms for the manufacturers. gaskell remained a director of the company until 1890 when it became part of the united alkali company. he... | wikipedia |
956fb64e5452af0f1982c5de6060d33d532adfd7 | deacon's plant in widnes was set up to | deacon's plant in widnes was set up to develop the ammonia-soda process that deacon believed he could make successful. however, after various setbacks gaskell could not see this making money and he forced deacon to abandon the venture. instead they established one of the largest and most successful leblanc factories in... | wikipedia |
1227674fe675725fb3acd8e3e62bcd699614897e | in 1855 gaskell was well enough to enter | in 1855 gaskell was well enough to enter into a second partnership with the industrial chemist henry deacon, who had worked with him in nasmyth, gaskell & co. nasmyth wrote "in course of time the alarming symptoms departed, and he recovered his former health. he then embarked in an extensive soda manufactory, in conjun... | wikipedia |
246741d74feefe072e66f1521489a168d408b676 | gaskell married frances ann bellhouse in 1841, who | gaskell married frances ann bellhouse in 1841, who was the daughter of henry bellhouse of manchester and niece of david bellhouse, the manchester builder nasmyth and gaskell had contracted during the initial building of the patricroft site. over the next 14 years they had nine children, six daughters and three sons. | wikipedia |
2e154b8ba983f48957c664360658448aac67c79f | in 1836 he formed a partnership with james | in 1836 he formed a partnership with james nasmyth which led to the creation of nasmyth, gaskell and company and the building of the bridgewater foundry at patricroft near manchester. nasmyth recalls this in his biography "he had served his time at yates and cox, iron merchants, of liverpool. having obtained considerab... | wikipedia |
e8bcb970ab756c4889cd472a08fd8175aff94d1f | gaskell was born in wavertree, liverpool. he was | gaskell was born in wavertree, liverpool. he was the eldest son of roger gaskell, a sailcloth manufacturer, from his marriage to his cousin anne hunter. he was baptised at the paradise street unitarian chapel in liverpool. he was the cousin of the unitarian minister william gaskell, (husband of the novelist mrs gaskell... | wikipedia |
d961e0fbcc4932f32f407b406d3702e38225d657 | he was born in přelouč and died in | he was born in přelouč and died in pardubice. brebera spent his career in czechoslovakian state-hold research institute for chemical industry, currently explosia pardubice. he is best known for the developing the plastic explosive known as semtex in the late 1950s, which was put on the market in 1964. | wikipedia |
2ac1d74650b63c923964d39a9f298de1d1416f10 | brebera spent his career in czechoslovakian state-hold research | brebera spent his career in czechoslovakian state-hold research institute for chemical industry, currently explosia pardubice. he is best known for the developing the plastic explosive known as semtex in the late 1950s, which was put on the market in 1964. | wikipedia |
78aba7c8b49824b3c3be6cf6c60c398c3f36ac5a | edward t. peterson was an american professional basketball | edward t. peterson was an american professional basketball player. a 6'9" center from cornell university, peterson played three seasons in the nbl and nba as a member of the syracuse nationals and tri-cities blackhawks. he was elected into the sphinx head society during his senior year at cornell university. | wikipedia |
82c060ede89e2d13f39192df629c64c86e18155e | juan carlos valerón santana is a spanish former | juan carlos valerón santana is a spanish former professional footballer who played as an attacking midfielder. over 15 seasons, he amassed la liga totals of 390 matches and 29 goals with mallorca, atlético madrid and deportivo, spending 13 years with the latter club. he started and finished his 22-year senior career wi... | wikipedia |
521d8fe92f4d13f17a6ce023e2259eaaea79cba7 | valerón was a close friend of fellow canarian | valerón was a close friend of fellow canarian manuel pablo, who also played for las palmas, deportivo and spain. he was also known for his devotion to god, but admitted that he and his family did not follow any religion. | wikipedia |
befda7313ef591a0b390b0e594bb45a4457fed20 | valerón's older brother, miguel Ángel, was also a | valerón's older brother, miguel Ángel, was also a footballer and a midfielder; having represented las palmas and mallorca b, his career was also curtailed by injuries. later, together with another sibling, pedro, they created a football club/school named abrisajac, from biblical characters abraham, isaac and jacob, and... | wikipedia |
c27433d1424702994f295affa61d0955cdc214f0 | miguel Ángel ramírez, las palmas president, said in | miguel Ángel ramírez, las palmas president, said in 2015 he was trying to persuade valerón to play another season so that he would be able to say goodbye to all the stadia in spain where he was consistently cheered – this was exemplified by his last game at the camp nou, with former national teammate luis enrique and c... | wikipedia |
907e085b29d48ea41c5b76157809ece280561557 | valerón was widely regarded as one of the | valerón was widely regarded as one of the most respected players in spain, and as an important figure for deportivo. andrés iniesta said he would pay to watch him play, whilst manager juan antonio anquela called him a reference for spanish football and fellow coach vicente del bosque admitted that he would always fit i... | wikipedia |
12f219168b2f19ad069cf9813dcd08522aedde07 | on 24 january 2018, valerón agreed to join | on 24 january 2018, valerón agreed to join juan manuel rodríguez 's staff at las palmas b, the reserve team. he resigned from his post in june for personal reasons, but the following january resumed his work with the youth sides. | wikipedia |
616e5fa0df08ca4ff54a4954a1c127c999c513b7 | shortly after retiring, valerón began working as a | shortly after retiring, valerón began working as a manager after completing the uefa b course. he was appointed youth coach at las palmas and, ahead of the 2017–18 season, was named assistant to manolo márquez in the first team; the latter left his position three months later, however, and the former returned to the ac... | wikipedia |
6bf02a5acb7b92e7af432fcf9c27afa766f1729c | he made his last appearance for his country | he made his last appearance for his country on 26 march 2005, a 3–0 defeat of china. in late july 2019, he was hired by the canarian football federation to be responsible for the several teams in the region that took part in the corresponding competitions. | wikipedia |
2bac1702db55859aa3ab7b2a922519a7b562fbda | a spain international since 18 november 1998 in | a spain international since 18 november 1998 in a 2–2 friendly draw with italy in salerno, valerón appeared with the national side at uefa euro 2000, the 2002 fifa world cup (where he scored in a 3–1 win against slovenia) and euro 2004, netting immediately after coming from the bench in a 1–0 victory over russia in the... | wikipedia |
0ab0f430592035a151976f22e692b2c119062de2 | in summer 2015, valerón renewed his contract with | in summer 2015, valerón renewed his contract with the amarillos for a further year. on 26 september, he first appeared with the club in the top flight, featuring 22 minutes in a 1–2 away loss to barcelona in what was his first game in the competition in 847 days, and becoming the fifth oldest player to play there at th... | wikipedia |
89c097665fafc3d3a32fa0f19b103bfcdb4f0d5c | on 14 july 2013, following another deportivo relegation, | on 14 july 2013, following another deportivo relegation, valerón returned to his first club las palmas after 16 years, signing a one-year contract with an option for a second. he continued to be an important first-team member during his tenure, achieving promotion to the top flight in 2015. | wikipedia |
05714ba1cf50712d9e1c79a61c2aa22c43306702 | the 36-year-old valerón was an undisputed starter for | the 36-year-old valerón was an undisputed starter for depor in the 2011–12 season, scoring a career-best five goals in nearly 3,000 minutes of action in segunda división as his team returned to the top flight after one year out, as champions. even though he had a contract until 2015, he chose to leave in june 2013, hav... | wikipedia |
8dd25fbe0cd78011664d6cd365935eca37bd53f6 | on 27 january 2008, valerón returned to the | on 27 january 2008, valerón returned to the bench in deportivo's 3–1 home win against real valladolid, coming on as a substitute for andrés guardado for the final 15 minutes – his first match for over a year. in 2008–09, aged 33, he eventually became an important first-team fixture again, both on domestic and european ... | wikipedia |
3db49465832cdb5fd7c95b6d72477997348c7f15 | in january 2006, however, valerón started a bad | in january 2006, however, valerón started a bad run with injuries. he suffered a knee injury which relapsed in july and early 2007 (with him only managing two league appearances in the process), leading to another surgery. consequently, he did not reappear until midway through the 2007–08 campaign. | wikipedia |
6d8ade2e1c39e8d0bd339b288c363dc34e87bde5 | in the next two years, valerón played for | in the next two years, valerón played for atlético madrid where he was an undisputed starter but, following the side's relegation in 2000, he joined deportivo de la coruña, sharing club and position with equally talented brazilian djalminha. he gradually would become first choice, signing a contract to eventually see o... | wikipedia |
2db5e7fab4ec3a0257b7d1467bcc8a2f0fb1d7d1 | he was an instrumental figure in the club's | he was an instrumental figure in the club's qualification for the uefa cup winners' cup, with the team also finishing fifth in the league and reaching the final of the copa del rey, lost to eventual champions barcelona in a penalty shoot-out. | wikipedia |
bbe7a3565d3949baac5690b6e8413798dcb717ac | born in arguineguín, gran canaria, valerón started playing | born in arguineguín, gran canaria, valerón started playing with hometown's las palmas, but switched to the balearic islands in the 1997–98 season, representing mallorca and making his la liga debut on 31 august 1997 by playing ten minutes in a 2–1 home win over valencia. | wikipedia |
b1418e084642473387c4c894d52d9e1f6c6d75b3 | over 15 seasons, he amassed la liga totals | over 15 seasons, he amassed la liga totals of 390 matches and 29 goals with mallorca, atlético madrid and deportivo, spending 13 years with the latter club. he started and finished his 22-year senior career with las palmas. | wikipedia |
193ec2bb49297f20acd886e30a4d2c1601725b97 | in mathematics, the abel–plana formula is a summation | in mathematics, the abel–plana formula is a summation formula discovered independently by niels henrik abel (1823) and giovanni antonio amedeo plana (1820). it states that ∑ n = 0 ∞ f ( a + n ) = ∫ a ∞ f ( x ) d x + f ( a ) 2 + ∫ 0 ∞ f ( a − i x ) − f ( a + i x ) i ( e 2 π x − 1 ) d x for the case a = 0 we have it hold... | wikipedia |
adcec0343c128a873932440e65416f545ab98e66 | the case ƒ (0) ≠ 0 is obtained | the case ƒ (0) ≠ 0 is obtained similarly, replacing ∫ a − 1 ∞ a ∞ f (z) e − 2 i π z − 1 d z {\textstyle \int _{a^{-1}\infty }^{a\infty }{\frac {f(z)}{e^{-2i\pi z}-1}}\,dz} by two integrals following the same curves with a small indentation on the left and right of 0. | wikipedia |
fc346c76fe2e33a3443f07a3401ae82620ec0c7e | this identity stays true by analytic continuation everywhere | this identity stays true by analytic continuation everywhere the integral converges, letting a → i {\displaystyle a\to i} we obtain the abel–plana formula ∑ n = 0 ∞ f (n) = ∫ 0 ∞ (f (t) + i f (i t) − i f (− i t) e 2 π t − 1) d t. {\displaystyle \sum _{n=0}^{\infty }f(n)=\int _{0}^{\infty }\left(f(t)+{\frac {i\,f(it)-i\... | wikipedia |
969cd04b3560ed7038fdfcbdf951fec8831a14db | using the cauchy integral theorem for the last | using the cauchy integral theorem for the last one. ∫ 0 a ∞ f (z) e − 2 i π z − 1 d z = ∫ 0 ∞ f (a t) e − 2 i π a t − 1 d (a t), {\displaystyle \int _{0}^{a\infty }{\frac {f(z)}{e^{-2i\pi z}-1}}\,dz=\int _{0}^{\infty }{\frac {f(at)}{e^{-2i\pi at}-1}}\,d(at),} thus obtaining ∑ n = 0 ∞ f (n) = ∫ 0 ∞ (f (t) + a f (a t) e ... | wikipedia |
3663b65c37b25af413beea903ed17c3540e1d25e | then ∫ a − 1 ∞ 0 f | then ∫ a − 1 ∞ 0 f (z) e − 2 i π z − 1 d z = − ∫ 0 a − 1 ∞ f (z) e − 2 i π z − 1 d z = ∫ 0 a − 1 ∞ f (z) e 2 i π z − 1 d z + ∫ 0 a − 1 ∞ f (z) d z = ∫ 0 ∞ f (a − 1 t) e 2 i π a − 1 t − 1 d (a − 1 t) + ∫ 0 ∞ f (t) d t. {\displaystyle {\begin{aligned}\int _{a^{-1}\infty }^{0}{\frac {f(z)}{e^{-2i\pi z}-1}}\,dz&=-\int _{0}... | wikipedia |
cd8dd53dfdec8dd6a57ccc31294852d3218f5197 | let f {\displaystyle f} be holomorphic on ℜ | let f {\displaystyle f} be holomorphic on ℜ (z) ≥ 0 {\displaystyle \re (z)\geq 0}, such that f (0) = 0 {\displaystyle f(0)=0}, f (z) = o (| z | k) {\displaystyle f(z)=o(|z|^{k})} and for arg (z) ∈ (− β, β) {\displaystyle \operatorname {arg} (z)\in (-\beta,\beta)}, f (z) = o (| z | − 1 − δ) {\displaystyle f(z)=o(|z|^{-1... | wikipedia |
e7b69450271e98b7ba58ef335ab8d6ddb703f726 | Γ (x + 1) = li − x | Γ (x + 1) = li − x (e − 1) + θ (x) {\displaystyle \gamma (x+1)=\operatorname {li} _{-x}\left(e^{-1}\right)+\theta (x)} where Γ (x) {\displaystyle \gamma (x)} is the gamma function, li s (z) {\displaystyle \operatorname {li} _{s}\left(z\right)} is the polylogarithm and θ (x) = ∫ 0 ∞ 2 t x e 2 π t − 1 sin (π x 2 − t) d t... | wikipedia |
f615e13298cd01d500cb72ffe62acbd9accba3a9 | which holds for all s ∈ c {\displaystyle | which holds for all s ∈ c {\displaystyle s\in \mathbb {c} }, s ≠ 1. another powerful example is applying the formula to the function e − n n x {\displaystyle e^{-n}n^{x}}: we obtain | wikipedia |
20a6b04fd9ba4da515e0f89008d61954d6dfbfd9 | it holds for functions ƒ that are holomorphic | it holds for functions ƒ that are holomorphic in the region re(z) ≥ 0, and satisfy a suitable growth condition in this region; for example it is enough to assume that | ƒ | is bounded by c /| z | in this region for some constants c, ε > 0, though the formula also holds under much weaker bounds. (olver 1997, p.290). | wikipedia |
dcd9b54fad16ba9a6194ec5c155eb1cc6bad39d6 | ∑ n = 0 ∞ f (a + | ∑ n = 0 ∞ f (a + n) = ∫ a ∞ f (x) d x + f (a) 2 + ∫ 0 ∞ f (a − i x) − f (a + i x) i (e 2 π x − 1) d x {\displaystyle \sum _{n=0}^{\infty }f\left(a+n\right)=\int _{a}^{\infty }f\left(x\right)dx+{\frac {f\left(a\right)}{2}}+\int _{0}^{\infty }{\frac {f\left(a-ix\right)-f\left(a+ix\right)}{i\left(e^{2\pi x}-1\right)}}dx} | wikipedia |
dbbb25345640d66c2fe3937b8f40f79158354997 | carathéodory's theorem is a theorem in convex geometry. | carathéodory's theorem is a theorem in convex geometry. it states that if a point x lies in the convex hull c o n v of a set p ⊂ r d , then x lies in some d -dimensional simplex with vertices in p . equivalently, x can be written as the convex combination of at most d + 1 points in p . additionally, x can be written as... | wikipedia |
c56a93c9dc7d7087a204e744a1c24f47cc46db6d | this theorem has a variant in which the | this theorem has a variant in which the convex hull is replaced by the conical hull. let x,..., x be sets in r and let x be a point contained in the intersection of the conical hulls of all these d sets. then there is a set t = { x,..., x }, where x ∈ x,..., x ∈ x, such that the conical hull of t contains the point x. | wikipedia |
31fe38880480c84d28e1edcc86cee34129044f26 | by viewing the sets x,..., x as different | by viewing the sets x,..., x as different colors, the set t is made by points of all colors, hence the "colorful" in the theorem's name. the set t is also called a rainbow simplex, since it is a d -dimensional simplex in which each corner has a different color. | wikipedia |
e0ec57263f56967d767d24cde8da082f6cb4c6bc | carathéodory's theorem simply states that any nonempty subset | carathéodory's theorem simply states that any nonempty subset of r d {\displaystyle \mathbb {r} ^{d}} has carathéodory's number ≤ d + 1 {\displaystyle \leq d+1}. this upper bound is not necessarily reached. for example, the unit sphere in r d {\displaystyle \mathbb {r} ^{d}} has carathéodory's number equal to 2, since ... | wikipedia |
4c609ad0ae7cec486983f5fdcbdb89373519c389 | for any nonempty p ⊂ r d {\displaystyle | for any nonempty p ⊂ r d {\displaystyle p\subset \mathbb {r} ^{d}}, define its carathéodory's number to be the smallest integer r {\displaystyle r}, such that for any x ∈ c o n v (p) {\displaystyle x\in \mathrm {conv} (p)}, there exists a representation of x {\displaystyle x} as a convex sum of up to r {\displaystyle r... | wikipedia |
b4600041fe4efd17c031785f70567acd2a8344d9 | if w n + w d + 1 | if w n + w d + 1 u n ≥ 0 {\displaystyle w_{n}+w_{d+1}u_{n}\geq 0} for all n = 1,..., d {\displaystyle n=1,...,d}, then set θ = 1 {\displaystyle \theta =1}. otherwise, let θ {\displaystyle \theta } be the smallest θ {\displaystyle \theta } such that one of w n + θ w d + 1 u n = 0 {\displaystyle w_{n}+\theta w_{d+1}u_{n}... | wikipedia |
170fab2935f72db15b8d3dfe808a02a70cf323d0 | x = ∑ n = 1 d (w | x = ∑ n = 1 d (w n + θ w d + 1 u n) q n + (1 − θ) w d + 1 q d + 1; θ ∈ {\displaystyle x=\sum _{n=1}^{d}(w_{n}+\theta w_{d+1}u_{n})q_{n}+(1-\theta)w_{d+1}q_{d+1};\quad \theta \in } | wikipedia |
6f38ae46d2ffb6f2d1536b835a31ac2270cc3069 | else, there exists (u 1,..., u d) ∈ | else, there exists (u 1,..., u d) ∈ r d {\displaystyle (u_{1},...,u_{d})\in \mathbb {r} ^{d}}, such that ∑ n = 1 d u n q n = q d + 1 {\displaystyle \sum _{n=1}^{d}u_{n}q_{n}=q_{d+1}}. then we can interpolate a full interval of representations: | wikipedia |
a2fe569fc70ef8b910cbadef160eef719750678a | if { q 1,..., q d } {\displaystyle | if { q 1,..., q d } {\displaystyle \{q_{1},...,q_{d}\}} is linearly dependent, then we can use induction on its linear span to eliminate one nonzero term in ∑ n = 1 d w n w 1 + ⋯ + w d q n {\displaystyle \sum _{n=1}^{d}{\frac {w_{n}}{w_{1}+\cdots +w_{d}}}q_{n}}, and thus eliminate it in x = ∑ n = 1 d + 1 w n q n {\disp... | wikipedia |
b08473502bcc453090baae0e97c4ed2e5111cd69 | induction case. represent x = ∑ n = | induction case. represent x = ∑ n = 1 d + 1 w n q n {\displaystyle x=\sum _{n=1}^{d+1}w_{n}q_{n}}. if some w n = 0 {\displaystyle w_{n}=0}, then the proof is finished. so assume all w n > 0 {\displaystyle w_{n}>0} | wikipedia |
4d544da15fde24152a658dbf6cc86076a954958b | this is trivial when n ≤ d {\displaystyle | this is trivial when n ≤ d {\displaystyle n\leq d}. if we can prove it for all n = d + 1 {\displaystyle n=d+1}, then by induction we have proved it for all n ≥ d + 1 {\displaystyle n\geq d+1}. thus it remains to prove it for n = d + 1 {\displaystyle n=d+1}. this we prove by induction on d {\displaystyle d}. | wikipedia |
22a2bb9313720eb224f6e5f62f073e8389f0ba88 | any x ∈ s {\displaystyle x\in s}, by | any x ∈ s {\displaystyle x\in s}, by first part, has a "lifted" representation (x, 1) = ∑ n = 1 n w n (q n, 1) {\displaystyle (x,1)=\sum _{n=1}^{n}w_{n}(q_{n},1)} where at most d + 1 {\displaystyle d+1} of w n {\displaystyle w_{n}} are nonzero. that is, x = ∑ n = 1 n w n q n {\displaystyle x=\sum _{n=1}^{n}w_{n}q_{n}},... | wikipedia |
462a2edaae01bfc4066003bd32bbf024ce3a4123 | explicitly, let s ⊂ r d {\displaystyle s\subset | explicitly, let s ⊂ r d {\displaystyle s\subset \mathbb {r} ^{d}}, then identify r d {\displaystyle \mathbb {r} ^{d}} with the subset { w ∈ r d + 1: w d + 1 = 1 } {\displaystyle \{w\in \mathbb {r} ^{d+1}:w_{d+1}=1\}}. this induces an embedding of s {\displaystyle s} into s × { 1 } ⊂ r d + 1 {\displaystyle s\times \{1\}... | wikipedia |
522a5b755f5aedbf5efa2a390910ef39de2c78e3 | for any x ∈ c o n v | for any x ∈ c o n v (s) {\displaystyle x\in \mathrm {conv} (s)}, represent x = ∑ n = 1 n w n q n {\displaystyle x=\sum _{n=1}^{n}w_{n}q_{n}} for some q 1,..., q n ∈ s {\displaystyle q_{1},...,q_{n}\in s}, then x ∈ c o n v ({ q 1,..., q n }) {\displaystyle x\in \mathrm {conv} (\{q_{1},...,q_{n}\})}, and we use the lemma... | wikipedia |
07bca389f3661d550021bed4e15e58dbc5790d76 | lemma — if q 1,..., q n ∈ | lemma — if q 1,..., q n ∈ r d {\displaystyle q_{1},...,q_{n}\in \mathbb {r} ^{d}} then ∀ x ∈ c o n v ({ q 1,..., q n }) {\displaystyle \forall x\in \mathrm {conv} (\{q_{1},...,q_{n}\})}, exists w 1,..., w n ≥ 0 {\displaystyle w_{1},...,w_{n}\geq 0} such that x = ∑ n w n q n {\displaystyle x=\sum _{n}w_{n}q_{n}}, and at... | wikipedia |
c5e42efbad5b9d65d1108499d32383290d8872e6 | the essence of carathéodory's theorem is in the | the essence of carathéodory's theorem is in the finite case. this reduction to the finite case is possible because c o n v (s) {\displaystyle \mathrm {conv} (s)} is the set of finite convex combination of elements of s {\displaystyle s} (see the convex hull page for details). | wikipedia |
b9d4a049a8fd1e12c4faddb09225a6348a888381 | if x ∈ c o n v (s) | if x ∈ c o n v (s) ⊂ r d {\displaystyle x\in \mathrm {conv} (s)\subset \mathbb {r} ^{d}}, then x {\displaystyle x} is the convex sum of at most d + 1 {\displaystyle d+1} points of s {\displaystyle s}. | wikipedia |
1a2fb49f636ca0b4c69ac0ec8791ac9e9adb3f94 | carathéodory's theorem — if x ∈ c o | carathéodory's theorem — if x ∈ c o n e (s) ⊂ r d {\displaystyle x\in \mathrm {cone} (s)\subset \mathbb {r} ^{d}}, then x {\displaystyle x} is the nonnegative sum of at most d {\displaystyle d} points of s {\displaystyle s}. | wikipedia |
dd38612bc979e66a26c9d27871b15da0224696b8 | note: we will only use the fact that | note: we will only use the fact that r {\displaystyle \mathbb {r} } is an ordered field, so the theorem and proof works even when r {\displaystyle \mathbb {r} } is replaced by any field f {\displaystyle \mathbb {f} }, together with a total order. | wikipedia |
ff057b0f66d69609422b18e8ac30ea95fc93e353 | for example, let p = {(0,0), (0,1), (1,0), | for example, let p = {(0,0), (0,1), (1,0), (1,1)}. the convex hull of this set is a square. let x = (1/4, 1/4) in the convex hull of p. we can then construct a set {(0,0),(0,1),(1,0)} = p ′, the convex hull of which is a triangle and encloses x. | wikipedia |
9986c23e5c4d249abff5dd247cc2cd420b1aa209 | the result is named for constantin carathéodory, who | the result is named for constantin carathéodory, who proved the theorem in 1911 for the case when p {\displaystyle p} is compact. in 1914 ernst steinitz expanded carathéodory's theorem for arbitrary sets. | wikipedia |
1bd549696d14ac443ac0ae3d77e9584f29734fb8 | an equivalent theorem for conical combinations states that | an equivalent theorem for conical combinations states that if a point x {\displaystyle x} lies in the conical hull c o n e (p) {\displaystyle \mathrm {cone} (p)} of a set p ⊂ r d {\displaystyle p\subset \mathbb {r} ^{d}}, then x {\displaystyle x} can be written as the conical combination of at most d {\displaystyle d} ... | wikipedia |
c7de85e2cc37f45f85787775c614738f78d73542 | carathéodory's theorem is a theorem in convex geometry. | carathéodory's theorem is a theorem in convex geometry. it states that if a point x {\displaystyle x} lies in the convex hull c o n v (p) {\displaystyle \mathrm {conv} (p)} of a set p ⊂ r d {\displaystyle p\subset \mathbb {r} ^{d}}, then x {\displaystyle x} lies in some d {\displaystyle d} -dimensional simplex with ver... | wikipedia |
c134766fd3c81ed4e19a289545e2c96b756a98a2 | john murray anderson is a canadian former ice | john murray anderson is a canadian former ice hockey right winger. he was the head coach of the chicago wolves of the international hockey league (ihl) and american hockey league (ahl) from 1997 to 2008 ,again from 2013 to 2016. anderson also serves as interim head coach for the wolves in 2023. in the national hockey l... | wikipedia |
8eb62f45300d0f458d74c4f2fccbdb76dd975b93 | anderson also helped establish john anderson's, a diner | anderson also helped establish john anderson's, a diner best known for its "banquet burger", as well as its $4 breakfast special. the original restaurant is located at victoria park ave. and van horne ave. in toronto, ontario. | wikipedia |
21b6c7bd31534e97b1373592f9d91bc04e9ffddb | on july 12, 2011, anderson became an assistant | on july 12, 2011, anderson became an assistant coach for the phoenix coyotes. on july 10, 2013, anderson was rehired as the head coach of the chicago wolves. after leaving the wolves in 2016, he joined the minnesota wild as an assistant head coach until 2018. in february 2022, he agreed to join the bakersfield condors ... | wikipedia |
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