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62bd4e2f6a0290ccca6586e3089262afd1a2b058 | a potential difference (voltage) between two points of | a potential difference (voltage) between two points of a conductor creates an electric field that accelerates charge carriers in the direction of the electric field, giving them kinetic energy. when the charged particles collide with the quasi-particles in the conductor (i.e. the canonically quantized, ionic lattice os... | wikipedia |
514e63d7a8261d1a49c3c566cd47512da4859e2d | in 1841 and 1842, subsequent experiments showed that | in 1841 and 1842, subsequent experiments showed that the amount of heat generated was proportional to the chemical energy used in the voltaic pile that generated the template. this led joule to reject the caloric theory (at that time the dominant theory) in favor of the mechanical theory of heat (according to which hea... | wikipedia |
8f7761026ec91ca46af2de9a7fe18689e8397216 | james prescott joule first published in december 1840, | james prescott joule first published in december 1840, an abstract in the proceedings of the royal society, suggesting that heat could be generated by an electrical current. joule immersed a length of wire in a fixed mass of water and measured the temperature rise due to a known current flowing through the wire for a 3... | wikipedia |
30c96d5510d5af7bc0d7a9e355a9f032531da5eb | joule's first law (also just joule's law), also | joule's first law (also just joule's law), also known in countries of the former ussr as the joule–lenz law, states that the power of heating generated by an electrical conductor equals the product of its resistance and the square of the current. joule heating affects the whole electric conductor, unlike the peltier ef... | wikipedia |
d4cf0fa4a3aa94124eb755ec2dd1aa0015de7504 | adamo giuliano fantilli is a canadian professional ice | adamo giuliano fantilli is a canadian professional ice hockey centre for the columbus blue jackets of the national hockey league (nhl). he was drafted third overall by the blue jackets in the 2023 nhl entry draft. during his freshman season at the university of michigan he was the leading goal and point-scorer and won ... | wikipedia |
a2a4a6eef26a447889bf501afb6421f2dd2e6379 | after missing the final months of the 2023–24 | after missing the final months of the 2023–24 season with injury, fantilli rejoined the senior national team for the 2024 iihf world championship. citing his shortened professional season as a motive for competing, he said: "i have to get my game going. that's my concern." however, fantilli was subsequently dropped fro... | wikipedia |
99ddb9ffe57a25e7ee313366208c0fb92e734034 | on may 5, 2023, fantilli was named to | on may 5, 2023, fantilli was named to canada's national senior team at the 2023 iihf world championship. he scored the game-winning goal in the semi-final against latvia, his first of the tournament. he finished the tournament with one goal and two assists and won a gold medal. he became the second canadian to win gold... | wikipedia |
1c5e1e1c32f53b7580b010aa1ad03cd50cd871f0 | in 2022, fantilli was named to canada's national | in 2022, fantilli was named to canada's national under-18 team at the 2022 iihf world u18 championships, where he again served as alternate captain. he recorded one goal and five assists in four games before the team was knocked out in the quarterfinal stage. on december 12, 2022, he was named to canada's national juni... | wikipedia |
64d33769c88d220599c14097e11520030109d41e | fantilli made his nhl debut on october 12, | fantilli made his nhl debut on october 12, 2023, his 19th birthday, against the philadelphia flyers. he scored his first career nhl goal on october 21, 2023, as part of a 5–4 overtime win against the minnesota wild. fantilli played 49 games before his debut season was cut short by a calf laceration in a january 28 game... | wikipedia |
c6d665fa32bd61badb35d7dd89a442429102f566 | fantilli was drafted third overall by the columbus | fantilli was drafted third overall by the columbus blue jackets in the 2023 nhl entry draft. on july 1, 2023, fantilli signed a three-year entry-level contract with the blue jackets. he will forego the rest of his college career to go play in the nhl. | wikipedia |
e46a5979567a1ae4dd48b7a109c17d1b2975dd92 | during the 2023 big ten men's ice hockey | during the 2023 big ten men's ice hockey tournament he set the big ten tournament record for goals in a single tournament with seven and in points with 11. he was subsequently named the big ten tournament mvp. he finished his freshman year with 30 goals and 35 assists in 36 games. he led the nation in points, and goals... | wikipedia |
310328248efc4bec1e1f09c934b9d66ebb51d6b3 | in august 2021, fantilli verbally committed to the | in august 2021, fantilli verbally committed to the university of michigan. he began his collegiate career for the michigan wolverines during the 2022–23 season. he made his debut for michigan on october 7, 2022, in a game against lindenwood, where he won 15 of 19 faceoffs and recorded two assists. the next day, he scor... | wikipedia |
7a1f12baead49c1f5207df32fe6b48cdb14c863f | in august 2021, fantilli verbally committed to the | regular season he recorded 11 goals and 19 assists in 19 conference games for a league-best 1.58 points per game. following an outstanding season, he was named to the all- big ten freshman team, the all-big ten first team and was named big ten freshman of the year. | wikipedia |
d8f1521d9bd8c70f4754223884e2c899dac5ec3f | fantilli attended kimball union academy where he recorded | fantilli attended kimball union academy where he recorded 18 goals and 18 assists in 25 games. on march 25, 2020, fantilli signed a ushl tender agreement with the chicago steel for the 2020–21 season. during the regular season he recorded 18 goals and 18 assists in 49 games for the steel and was named to the ushl all-r... | wikipedia |
f0a4ac45fe20941468b59a68706102b7ed918c4a | fantilli was drafted in the first round, 18th | fantilli was drafted in the first round, 18th overall, by the saginaw spirit in the 2020 ohl priority selection draft. he was considered the top canadian prospect in the draft, however, he had already committed to play for the chicago steel. on september 13, 2021, the spirit exercised their defected player rule and tra... | wikipedia |
03bede3f3d9f5dd05461a5b2e5ae091653820a14 | adamo giuliano fantilli (born october 12, 2004) is | adamo giuliano fantilli (born october 12, 2004) is a canadian professional ice hockey centre for the columbus blue jackets of the national hockey league (nhl). he was drafted third overall by the blue jackets in the 2023 nhl entry draft. during his freshman season at the university of michigan he was the leading goal a... | wikipedia |
42fa5afb04d5790c5fd40555c570af55f21ad7b4 | following three seasons in the khl with spartak | following three seasons in the khl with spartak moscow, nikishin was traded to powerhouse club ska saint petersburg in exchange for 5 contracted players and the rights to four further players and an additional 50 million rubles on 31 july 2022. with saint petersburg, nikishin's production skyrocketed, putting up 55 poi... | wikipedia |
413ccd4fca1817104049f417024e6224efc649d2 | nikishin made his khl debut for hc spartak | nikishin made his khl debut for hc spartak moskva during the 2019–20 season. he was drafted by the carolina hurricanes in the 3rd round of the 2020 nhl entry draft with the 69th overall pick. | wikipedia |
6728890e238e2df15781637f72b90d836336bfc9 | alexander alexandrovich nikishin (russian: Александр Александрович Никишин; born | alexander alexandrovich nikishin (russian: Александр Александрович Никишин; born 2 october 2001) is a russian professional ice hockey defenseman currently playing for ska saint petersburg in the kontinental hockey league (khl). | wikipedia |
3a53a3bf64ed1c97d0c62cf36c5ab2183ac857c0 | john francis "moe" murphy was an american professional | john francis "moe" murphy was an american professional basketball and baseball player. murphy was born in philadelphia, pennsylvania and attended simon gratz high school, where he led the philadelphia public league in points per game with 13.7 in 1942. he spent three seasons playing for the wilmington bombers of the am... | wikipedia |
ae22f06e0d5bfa3b88195970415ff33141ba41cd | john francis " moe " murphy (september 13, | john francis " moe " murphy (september 13, 1924 – january 29, 2003) was an american professional basketball and baseball player. murphy was born in philadelphia, pennsylvania and attended simon gratz high school, where he led the philadelphia public league in points per game with 13.7 in 1942. he spent three seasons pl... | wikipedia |
2e5c17f405d205bd8f24d3b4bc371a423522169f | the peugeot 8aa, or l112, is a water-cooled | the peugeot 8aa, or l112, is a water-cooled v8 aircraft engine that equipped the 1,123 voisin viii bombers and fighters built during ww1. the 8aa was the first engine designed by peugeot for airplane use and the only one that saw service in ww1. in publications the engine is often referred to as the peugeot 200 hp, or ... | wikipedia |
0769c1413cb36dc7ed0734009281e22a10f76cef | the peugeot 8aa was a remarkably fuel efficient | the peugeot 8aa was a remarkably fuel efficient engine for its time. at its most efficient point, the engine's specific fuel consumption was 0.252 kg/kwh (0.414 lb/(hp⋅h)) corresponding to a thermal efficiency of more than 30%. | wikipedia |
edb42f12bd1437d4ade23a64002a4ff2b12f895c | the peugeot 8aa shared many common parts with | the peugeot 8aa shared many common parts with the hispano-suiza 8 and its british derivative the wolseley viper. camshafts, piston rings and some bearings were interchangeable. peugeot was one of the main licensed producers of hispano-suiza 8 engines. | wikipedia |
159bae159334f8d584cfceeda45c145b0c25872d | 2 carburetors (one for each cylinder bank) are | 2 carburetors (one for each cylinder bank) are attached to the out-side inlet manifolds. carburetors were heated by the exhaust however this arrangement was ineffective at the cold temperatures found at high altitudes which contributed to the type's poor reliability. | wikipedia |
fbf0817470ead36342c7bb2e6ccbfc31cb002803 | the cylinders are cast from iron in blocks | the cylinders are cast from iron in blocks of four with integral water jackets. in each cylinder head there are four inclined valves guided in bronze bushings. the engine's four camshafts (two per cylinder bank) are enclosed in aluminum housings and operate the valves through tappets. | wikipedia |
225d388eed5d15dd4feac9dc23eeab7971273902 | a shaft, extending directly back through the vee, | a shaft, extending directly back through the vee, drives the camshaft spur gear train and the accessory units. the crankcase is of the barrel type with the four-throw crankshaft supported on three large double-row ball bearings. the crankshaft is made from two sections and is joined at the center bearing by a taper, ke... | wikipedia |
a388ab8907b93c56954a7f8ac5e29eb305a9a0b6 | the engine's eight cylinders are in a 90 | the engine's eight cylinders are in a 90 degree vee arrangement. reduction gears rotated the propeller at half the speed of the crankshaft. the propeller shaft extends far out in front of the crankcase and is mounted on ball bearings. | wikipedia |
9430b3fc23b9d51ba85b386ed31aba2fa7654700 | peugeot's design followed that of their successful racing | peugeot's design followed that of their successful racing car engines which had been in production since 1910. for its time period, the engine had advanced features including double overhead camshafts (dohc) and a dry sump design patented by peugeot. | wikipedia |
8b3a42f77e208aa8713aa35dd223c52fa22bdf67 | the peugeot 8aa was developed to meet a | the peugeot 8aa was developed to meet a french military requirement for 200 hp aero-engines to equip a new class of bomber aircraft designed to hit targets inside germany. to achieve this the engines needed to run for six hours with most of the journey taking place at altitudes around 4,400 m (14,400 ft). the requireme... | wikipedia |
417224344f967823dc937c91d6aa82f68166d71d | the peugeot 8aa, or l112, is a water-cooled | the peugeot 8aa, or l112, is a water-cooled v8 aircraft engine that equipped the 1,123 voisin viii bombers and fighters built during ww1. the 8aa was the first engine designed by peugeot for airplane use and the only one that saw service in ww1. in publications the engine is often referred to as the peugeot 200 hp, or ... | wikipedia |
26ac7c080c27bb906099ff4c00c432c3dd6dc9b1 | his work has been cited, together with that | his work has been cited, together with that of marcel riesz, for its importance to clifford algebras and mathematical physics in the proceedings of a workshop held in france in 1989 which had been dedicated to these two mathematicians. | wikipedia |
90223887269721e2dffe0daa053add446915c7c6 | he wrote a series of publications of 1957/1958 | he wrote a series of publications of 1957/1958 on geometric algebras that stand in relation to quantum physics and quantum field theory. he pointed out that those algebras can be described in terms of extensions of the commutative and the anti-commutative grassmann algebras which have the same structure as the boson al... | wikipedia |
92d59536310a0299e06ffc317e59d1d999a65064 | in the university of são paulo had schönberg | in the university of são paulo had schönberg interacted closely with david bohm during the final years of bohm's exile in brazil, and, in 1954, schönberg demonstrated a link among the quantized motion of the madelung fluid and the trajectories of the de broglie–bohm theory. | wikipedia |
1adf253285801d99ac9193ca3e8dfc7298acad24 | together with indian physicist subrahmanyan chandrasekhar (1910–1995), he | together with indian physicist subrahmanyan chandrasekhar (1910–1995), he discovered and published in 1942 the so-called schönberg–chandrasekhar limit, which is the maximum mass of the core of a star that can support the overlying layers against gravitational collapse, once the core hydrogen is exhausted. | wikipedia |
1a7d223ca701a0f67c1a324e4b35fbf85d496d9b | widely regarded as one of brazil's most important | widely regarded as one of brazil's most important theoretical physicists, schenberg is best remembered for his contributions to astrophysics, particularly the theory of nuclear processes in the formation of supernova stars. he provided the inspiration for the name of the so-called urca process, a cycle of nuclear react... | wikipedia |
a88e89be774c452180a643d8c314b01a2eb17290 | schenberg was born in recife, brazil as mayer | schenberg was born in recife, brazil as mayer schönberg. his parents were russian jews of german origin. from early on he showed remarkable ability for mathematics, enchanting himself with geometry, which had a strong influence on his works. he took the primary and secondary courses in recife. because of his family's f... | wikipedia |
8e6dc412abc8a93e22296281ad4146aee05ed030 | tan cheng hoe is a malaysian football manager | tan cheng hoe is a malaysian football manager and former football player who was recently the manager of thai league 1 club police tero. he was the assistant coach for the malaysia national team during k. rajagopal and nelo vingada era before he was chosen as the head coach of the national team. | wikipedia |
c75275ceef32fbb8e515f0fa359afc4b8f013f96 | on 29 february 2024, thai league 1 club | on 29 february 2024, thai league 1 club police tero announced the appointment of tan as the head coach and designated worrawoot srimaka as the technical director. he secured nine points from his games in charge, which were ultimately not enough to keep the team in the league, with the club's relegation confirmed on 13 ... | wikipedia |
5476394b1131798a7409caf1b619173d59720dff | on 24 september 2022, tan cheng hoe appointed | on 24 september 2022, tan cheng hoe appointed as selangor first team head coach. tan, who took the helm of selangor in september 2022, has been instrumental in the club's recent successes, notably leading them to runner-up positions in both the 2022 malaysia cup and the 2023 malaysia super league. such feats not only e... | wikipedia |
da98c1d75c0e07fd2740886c486d5f3992846e59 | tan continued to lead the malaysian team in | tan continued to lead the malaysian team in the 2022 fifa world cup qualification after he been reward by the football association of malaysia (fam) to manage the team for another two years until his contract ended in 2020. although the team under his management able to easily routed timor-leste in the first round by 1... | wikipedia |
abffbc5350625c1f67b0d810dabcd053a0f5b896 | this was continued in the finals first leg | this was continued in the finals first leg when the malaysian team under his led also managed to hold the tournament favourites of vietnam 2–2 at home in their second meet in the tournament despite the latter had already scoring two goals in the first half. however, in the second leg in hanoi, his team failed to repeat... | wikipedia |
63e519a2293412a6697c2b72a4251889a0fc374a | in the preparation for the 2018 aff championship, | in the preparation for the 2018 aff championship, tan had mentioned through an article published by the fox sports asia that malaysia are underdogs team and hope their country fans will not put their hope too much despite being drawn into a favourable group a consisting of vietnam, myanmar, cambodia and laos, adding th... | wikipedia |
51bd14e0eb94c183169af924184ee3a6a76a0a76 | tan began his 2nd spell with malaysia in | tan began his 2nd spell with malaysia in april 2017 as he took up the position of assistant manager under nelo vingada. tan was then made the head coach of malaysia after vingada stepped down following a string of poor results. | wikipedia |
8ed96b64a056401ec2709e6e1f429987a30deceb | a break from football lasting 5 years then | a break from football lasting 5 years then followed with tan's next role as manager at second-tier malaysia premier league club, kedah beginning in april 2014. in 2015, he guided kedah to win the 2015 malaysia premier league which also gain promotion to the 2016 malaysia super league. he was named the m-league's best c... | wikipedia |
ecf8d6ea99de51bde5b1a6043963e22468baa5af | tan was the assistant coach to k. rajagopal | tan was the assistant coach to k. rajagopal from 2004 to 2013. tan worked as an under 19s coach for malaysia from december 2004 until october 2009. tan continued his development as coach for harimau muda from october 2007 until october 2009. under rajagopal and tan, malaysia achieved many good results such as reaching ... | wikipedia |
a4b8f9596990eb9771276514056f4f5413e773e4 | tan made an appearance for malaysia selection in | tan made an appearance for malaysia selection in a match against aston villa on 20 may 1991. in september 1991, he was part of malaysia b team managed by m. karathu for taan cup, an invitational tournament in kathmandu nepal. malaysia b team won the tournament beating tiong bahru of singapore 2–0. | wikipedia |
525ef777ad84dd1c153844020b5ee18a4fa49cc3 | tan cheng hoe (chinese: 陈清和) is a malaysian | tan cheng hoe (chinese: 陈清和) is a malaysian football manager and former football player who was recently the manager of thai league 1 club police tero. he was the assistant coach for the malaysia national team during k. rajagopal and nelo vingada era before he was chosen as the head coach of the national team. | wikipedia |
8fe8d7fa5bdfe7cc0e41e476069615818df449ff | testosterone enanthate is an androgen and anabolic steroid | testosterone enanthate is an androgen and anabolic steroid (aas) medication which is used mainly in the treatment of low testosterone levels in men. it is also used in hormone therapy for transgender men. it is given by injection into muscle or subcutaneously usually once every one to four weeks. side effects of testos... | wikipedia |
301cfae91d5370a17f45e77f883253c3ee31feb7 | on october 1, 2018, the u.s. food and | on october 1, 2018, the u.s. food and drug administration (fda) announced the approval of xyosted. xyosted, a product of antares pharma, inc., is a single-use disposable auto-injector that dispenses testosterone enanthate. xyosted is the first fda-approved subcutaneous testosterone enanthate product for testosterone re... | wikipedia |
bdb86060c94a2749b26b4a2de9e531ae3c0b82bc | testosterone enanthate, or testosterone 17β-heptanoate, is a synthetic | testosterone enanthate, or testosterone 17β-heptanoate, is a synthetic androstane steroid and a derivative of testosterone. it is an androgen ester; specifically, it is the c17β enanthate (heptanoate) ester of testosterone. | wikipedia |
91b6a50fdfd95189baf2a85878166282a779537f | testosterone enanthate has an elimination half-life of 4.5 | testosterone enanthate has an elimination half-life of 4.5 days and a mean residence time of 8.5 days when used as a depot intramuscular injection. it requires frequent administration of approximately once per week, and large fluctuations in testosterone levels result with it, with levels initially being elevated and s... | wikipedia |
57f40eecde9f3aa232fa5d8718c4787e18fd776f | testosterone cypionate is converted by the body to | testosterone cypionate is converted by the body to testosterone that has both androgenic effects and anabolic effects; still, the relative potency of these effects can depend on various factors and is a topic of ongoing research. testosterone can either directly exert effects on target tissues or be metabolized by 5α-r... | wikipedia |
e37e203ef0f22791537862e83b9d30d881fa5899 | side effects of testosterone enanthate include virilization among | side effects of testosterone enanthate include virilization among others. approximately 10 percent of testosterone enanthate will be converted to dihydrotestosterone in normal men. dihydrotestosterone (dht) can promote masculine characteristics in both males and females. these masculine characteristics include: clitora... | wikipedia |
7eafe4bae78d2af3afc7c9c0010f2cf880052dfc | testosterone enanthate is used primarily in androgen replacement | testosterone enanthate is used primarily in androgen replacement therapy. it is the most widely used form of testosterone in androgen replacement therapy. the medication is specifically approved, in the united states, for the treatment of hypogonadism in men, delayed puberty in boys, and breast cancer in women. it is a... | wikipedia |
3d2d037328206fd5e4353f68324adacf87011cb1 | testosterone enanthate was introduced for medical use in | testosterone enanthate was introduced for medical use in 1954. along with testosterone cypionate, testosterone undecanoate, and testosterone propionate, it is one of the most widely used testosterone esters. in addition to its medical use, testosterone enanthate is used to improve physique and performance. the drug is ... | wikipedia |
b235d84a542f693cc56a2d331b10c24766212a98 | side effects of testosterone enanthate include symptoms of | side effects of testosterone enanthate include symptoms of masculinization like acne, increased hair growth, voice changes, and increased sexual desire. the drug is a synthetic androgen and anabolic steroid and hence is an agonist of the androgen receptor (ar), the biological target of androgens like testosterone and d... | wikipedia |
d5e812bbd7553a78659a5b1799c8414e74e630be | testosterone enanthate is an androgen and anabolic steroid | testosterone enanthate is an androgen and anabolic steroid (aas) medication which is used mainly in the treatment of low testosterone levels in men. it is also used in hormone therapy for transgender men. it is given by injection into muscle or subcutaneously usually once every one to four weeks. | wikipedia |
3105bc11a5421346f0fb1acb71845ff9e72f4805 | tatyana olegovna shaposhnikova is a russian-born swedish mathematician. | tatyana olegovna shaposhnikova is a russian-born swedish mathematician. she is best known for her work in the theory of multipliers in function spaces, partial differential operators and history of mathematics, some of which was partly done jointly with vladimir maz'ya. she is also a translator of both scientific and l... | wikipedia |
569256452b14db048d74e34fb3bd5273a9fe2ef9 | in 2005 she began translating swedish children's books | in 2005 she began translating swedish children's books into russian. among them are " kerstin and i " by astrid lindgren, " mechanical santa claus " by sven nordqvist and two books of the " loranga " series by barbro lindgren. | wikipedia |
c93dacce95334360e8b6a5e13fb047b9dbed35ba | shaposhnikova began translating fiction while still living in | shaposhnikova began translating fiction while still living in russia. in the 1970s she translated into russian " the voyage of the dawn treader ", " the silver chair " and the " screwtape letters " by c. s. lewis. these translations were impossible to publish due to ideological reasons and were distributed as samizdat:... | wikipedia |
1f45b056cec257e1e82904566f02c213dbc4a4c8 | shaposhnikova has translated and edited several mathematical monographs: | shaposhnikova has translated and edited several mathematical monographs: it is worth to note the works by koshelev et al. (1975) and by mikhlin (1979), the book on sobolev spaces by maz'ya (1985), and the books by kresin & maz'ya (2007) and by maz'ya & soloviev (2010). however, her work is not restricted only to the tr... | wikipedia |
4f130e696c0f93bac017e03c3e685e9b588043b4 | her prize winning book on jacques hadamard, coauthored | her prize winning book on jacques hadamard, coauthored with v. maz'ya, was published in 1998 jointly by the american mathematical society and the london mathematical society. an earlier work on the same subject was written by her jointly with e. polishchuk (1990). her recent activity in this field includes the paper (s... | wikipedia |
bdd8fa3e89011a72a478e35f7c2ec21db9e2498e | based on her researches on the theory of | based on her researches on the theory of multipliers, t. shaposhnikova gave various applications of this theory to the study of solutions to second order linear and quasilinear elliptic partial differential equations and systems of such equations: this was a consequence of the fact that, in several cases, such solution... | wikipedia |
871d3b4a608429a51c0c352e9f8247a0d9faecb6 | from 1979 on, the theory of multipliers in | from 1979 on, the theory of multipliers in various spaces of differentiable functions has been the main research theme of her work. she found conditions for the boundedness of singular integrals and pseudodifferential operators acting between pairs of sobolev spaces in 1995. in 1989 she showed that multipliers in besse... | wikipedia |
5bc367e9b81b9cb87258229afef28a781af8949b | in march 2003 shaposhnikova and vladimir maz'ya were | in march 2003 shaposhnikova and vladimir maz'ya were awarded the verdaguer prize by the french academy of sciences for their work resulting in the first scientific biography of jacques hadamard. in may 2010 she was awarded the thureus prize by the royal society of sciences in uppsala " for her outstanding contribution ... | wikipedia |
d3b80c79156f960bcb0dcf290667756e19047f11 | t.o. shaposhnikova graduated from leningrad university in 1969. | t.o. shaposhnikova graduated from leningrad university in 1969. from 1969 to 1972 she was a graduate student at the same university. in 1973 she was awarded the kandidat nauk degree. from 1973 to 1990 she worked in the mathematics departments of a number of technical institutes in leningrad, first as an assistant and t... | wikipedia |
e6b48794b325b8c05501fc187e2e284e7b4baf32 | tatyana olegovna shaposhnikova (russian: Татьяна Олеговна Шапошникова, born | tatyana olegovna shaposhnikova (russian: Татьяна Олеговна Шапошникова, born 1946) is a russian -born swedish mathematician. she is best known for her work in the theory of multipliers in function spaces, partial differential operators and history of mathematics, some of which was partly done jointly with vladimir maz'y... | wikipedia |
7e7bd2190c4bb2b956211bb772949f41b8d01a0f | in mathematics, the integral of a non-negative function | in mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the x axis. the lebesgue integral, named after french mathematician henri lebesgue, is one way to make this concept rigorous and to extend it to more gen... | wikipedia |
2c0fad84207fbd9cb8fa05612d7309be1f6831f8 | the main purpose of the lebesgue integral is | the main purpose of the lebesgue integral is to provide an integral notion where limits of integrals hold under mild assumptions. there is no guarantee that every function is lebesgue integrable. but it may happen that improper integrals exist for functions that are not lebesgue integrable. one example would be the sin... | wikipedia |
b814965510c971faf9fcbf93030c3fd2222330f1 | more generally, when the measure space on which | more generally, when the measure space on which the functions are defined is also a locally compact topological space (as is the case with the real numbers r), measures compatible with the topology in a suitable sense (radon measures, of which the lebesgue measure is an example) an integral with respect to them can be ... | wikipedia |
24cb252d270717b388123639a02e63891350d39b | then c is a normed vector space (and | then c is a normed vector space (and in particular, it is a metric space.) all metric spaces have hausdorff completions, so let l be its completion. this space is isomorphic to the space of lebesgue integrable functions modulo the subspace of functions with integral zero. furthermore, the riemann integral ∫ is a unifor... | wikipedia |
3e356fb0e7ec55b6b357c3a407dadc7074954029 | let c be the space of all real-valued | let c be the space of all real-valued compactly supported continuous functions of r. define a norm on c by ‖ f ‖ = ∫ | f (x) | d x. {\displaystyle \left\|f\right\|=\int |f(x)|\,dx.} | wikipedia |
fd4b8bbb0a97c6c96ed10cab18a7634ed6bbed4c | there is also an alternative approach to developing | there is also an alternative approach to developing the theory of integration via methods of functional analysis. the riemann integral exists for any continuous function f of compact support defined on r (or a fixed open subset). integrals of more general functions can be built starting from these integrals. | wikipedia |
0aa12605ea82ceebc2a2a37f011945c529012a6c | two functions are said to be equal almost | two functions are said to be equal almost everywhere (f = a.e. g {\displaystyle f\ {\stackrel {\text{a.e.}}{=}}\ g} for short) if { x ∣ f (x) ≠ g (x) } {\displaystyle \{x\mid f(x)\neq g(x)\}} is a subset of a null set. measurability of the set { x ∣ f (x) ≠ g (x) } {\displaystyle \{x\mid f(x)\neq g(x)\}} is not require... | wikipedia |
83ace34316b50e052dd1d535db2666893279f424 | the function g is zero everywhere, except on | the function g is zero everywhere, except on a finite set of points. hence its riemann integral is zero. each g is non-negative, and this sequence of functions is monotonically increasing, but its limit as k → ∞ is 1, which is not riemann integrable. | wikipedia |
d1ebc0464e173ec38cd0158086c385a6ec3731ff | as shown above, the indicator function 1 on | as shown above, the indicator function 1 on the rationals is not riemann integrable. in particular, the monotone convergence theorem fails. to see why, let { a } be an enumeration of all the rational numbers in (they are countable so this can be done). then let g k (x) = { 1 if x = a j, j ≤ k 0 otherwise {\displaystyle... | wikipedia |
0b8b4cc1fec1571ee03795b208f72374fa03ccdf | a technical issue in lebesgue integration is that | a technical issue in lebesgue integration is that the domain of integration is defined as a set (a subset of a measure space), with no notion of orientation. in elementary calculus, one defines integration with respect to an orientation: ∫ b a f = − ∫ a b f. {\displaystyle \int _{b}^{a}f=-\int _{a}^{b}f.} generalizing ... | wikipedia |
67f4f0f16e6be21765dae71e45025113e75dcfe4 | if h = f + ig for real-valued | if h = f + ig for real-valued integrable functions f, g, then the integral of h is defined by ∫ h d μ = ∫ f d μ + i ∫ g d μ. {\displaystyle \int h\,d\mu =\int f\,d\mu +i\int g\,d\mu.} | wikipedia |
944294b24cafaa597fbde448e0cabfe4110a89dd | assuming that f is measurable and non-negative, the | assuming that f is measurable and non-negative, the function f ∗ (t) = def μ ({ x ∈ e ∣ f (x) > t }). {\displaystyle f^{*}(t)\ {\stackrel {\text{def}}{=}}\ \mu \left(\{x\in e\mid f(x)>t\}\right).} is monotonically non-increasing. the lebesgue integral may then be defined as the improper riemann integral of f: ∫ e f d μ... | wikipedia |
12679995c76d72703d7a81103d83b0656cceb789 | ∫ f d μ = ∫ f + | ∫ f d μ = ∫ f + d μ − ∫ f − d μ. {\displaystyle \int f\,d\mu =\int f^{+}\,d\mu -\int f^{-}\,d\mu.} | wikipedia |
a3244fcfce0cca99e80d7946e2c5515908bb48b4 | min (∫ f + d μ, ∫ f | min (∫ f + d μ, ∫ f − d μ) < ∞. {\displaystyle \min \left(\int f^{+}\,d\mu,\int f^{-}\,d\mu \right)<\infty.} | wikipedia |
1f01c5cff380f37b2a4bf1bc2d3d01d64f118536 | we say that the lebesgue integral of the | we say that the lebesgue integral of the measurable function f exists, or is defined if at least one of ∫ f + d μ {\textstyle \int f^{+}\,d\mu } and ∫ f − d μ {\textstyle \int f^{-}\,d\mu } is finite: | wikipedia |
9b74570b71b19aada993f2322a5c57a38b367b29 | where f + (x) = { f (x) | where f + (x) = { f (x) − if f (x) > 0, 0 otherwise, f − (x) = { − f (x) if f (x) < 0, 0 otherwise. {\displaystyle {\begin{aligned}f^{+}(x)&={\begin{cases}f(x){\hphantom {-}}&{\text{if }}f(x)>0,\\0&{\text{otherwise}},\end{cases}}\\f^{-}(x)&={\begin{cases}-f(x)&{\text{if }}f(x)<0,\\0&{\text{otherwise}}.\end{cases}}\end{... | wikipedia |
38c5d9dbfe97f688b808a6465a79b270812efb58 | to handle signed functions, we need a few | to handle signed functions, we need a few more definitions. if f is a measurable function of the set e to the reals (including ±∞), then we can write f = f + − f −, {\displaystyle f=f^{+}-f^{-},} | wikipedia |
3b75c6a6b7966af87a4b17ae8ef504eee1c503e2 | it is often useful to have a particular | it is often useful to have a particular sequence of simple functions that approximates the lebesgue integral well (analogously to a riemann sum). for a non-negative measurable function f, let s n (x) {\textstyle s_{n}(x)} be the simple function whose value is k / 2 n {\textstyle k/2^{n}} whenever k / 2 n ≤ f (x) < (k +... | wikipedia |
8e76b318fe53f94119f3ea0a6a0e57d776fe732c | we have defined the integral of f for | we have defined the integral of f for any non-negative extended real-valued measurable function on e. for some functions, this integral ∫ e f d μ {\textstyle \int _{e}f\,d\mu } is infinite. | wikipedia |
42a35612c8c52e8d40b2932e5e7f35884cdcec6c | we need to show this integral coincides with | we need to show this integral coincides with the preceding one, defined on the set of simple functions, when e is a segment. there is also the question of whether this corresponds in any way to a riemann notion of integration. it is possible to prove that the answer to both questions is yes. | wikipedia |
f1da2bc863a0e394e31797df6f0f7c80ddd17a83 | ∫ e f d μ = sup { | ∫ e f d μ = sup { ∫ e s d μ: 0 ≤ s ≤ f, s simple }. {\displaystyle \int _{e}f\,d\mu =\sup \left\{\,\int _{e}s\,d\mu:0\leq s\leq f,\ s\ {\text{simple}}\,\right\}.} | wikipedia |
c1b2c9dd5f27a0fc3243454ca5dc242e49468d62 | ∫ b s d μ = ∫ 1 | ∫ b s d μ = ∫ 1 b s d μ = ∑ k a k μ (s k ∩ b). {\displaystyle \int _{b}s\,\mathrm {d} \mu =\int 1_{b}\,s\,\mathrm {d} \mu =\sum _{k}a_{k}\,\mu (s_{k}\cap b).} | wikipedia |
982a7e582e1a7d29d45212fd27d8f2eeb93f3e60 | ∫ (∑ k a k 1 s k) | ∫ (∑ k a k 1 s k) d μ = ∑ k a k ∫ 1 s k d μ = ∑ k a k μ (s k) {\displaystyle \int \left(\sum _{k}a_{k}1_{s_{k}}\right)\,d\mu =\sum _{k}a_{k}\int 1_{s_{k}}\,d\mu =\sum _{k}a_{k}\,\mu (s_{k})} | wikipedia |
5063abfa1d8466b7e1644a49db37acfcfbb2b6d1 | one approach to constructing the lebesgue integral is | one approach to constructing the lebesgue integral is to make use of so-called simple functions: finite, real linear combinations of indicator functions. simple functions that lie directly underneath a given function f can be constructed by partitioning the range of f into a finite number of layers. the intersection of... | wikipedia |
885c6203fe9e76e9b1b6955705f2b6f8d435e91f | ∫ e f d μ = ∫ e | ∫ e f d μ = ∫ e f (x) d μ (x) = ∫ e f (x) μ (d x). {\displaystyle \int _{e}f\,d\mu =\int _{e}f(x)\,d\mu (x)=\int _{e}f(x)\,\mu (dx).} | wikipedia |
641b7349157437f8e2f3ba4b4aaa7d99f18a1b3d | sup k ∈ n f k, lim inf | sup k ∈ n f k, lim inf k ∈ n f k, lim sup k ∈ n f k {\displaystyle \sup _{k\in \mathbb {n} }f_{k},\quad \liminf _{k\in \mathbb {n} }f_{k},\quad \limsup _{k\in \mathbb {n} }f_{k}} | wikipedia |
4cb8186e3eaa5111d7e76c1f7547b1657fa5d7d5 | we can show that this is equivalent to | we can show that this is equivalent to requiring that the pre-image of any borel subset of r be in x. the set of measurable functions is closed under algebraic operations, but more importantly it is closed under various kinds of point-wise sequential limits: | wikipedia |
49353281f466b91db13d3be5795cf186151a9dcd | { x ∣ f (x) > t } | { x ∣ f (x) > t } ∈ x ∀ t ∈ r. {\displaystyle \{x\,\mid \,f(x)>t\}\in x\quad \forall t\in \mathbb {r}.} | wikipedia |
38f808478b749fe21ea2debf2f37331e2a56a87a | for example, e can be euclidean n -space | for example, e can be euclidean n -space r or some lebesgue measurable subset of it, x is the σ-algebra of all lebesgue measurable subsets of e, and μ is the lebesgue measure. in the mathematical theory of probability, we confine our study to a probability measure μ, which satisfies μ (e) = 1. | wikipedia |
892a1d92c148a14a24470ea875a65f66f5a7a9ae | in the development of the theory in most | in the development of the theory in most modern textbooks (after 1950), the approach to measure and integration is axiomatic. this means that a measure is any function μ defined on a certain class x of subsets of a set e, which satisfies a certain list of properties. these properties can be shown to hold in many differ... | wikipedia |
2e796c6c94d23e75f6283a2a3c7cd620477c3834 | the riemann integral uses the notion of length | the riemann integral uses the notion of length explicitly. indeed, the element of calculation for the riemann integral is the rectangle ×, whose area is calculated to be (b − a)(d − c). the quantity b − a is the length of the base of the rectangle and d − c is the height of the rectangle. riemann could only use planar ... | wikipedia |
8f6b71ae5787c1006682ea69d9838912033ab69d | measure theory was initially created to provide a | measure theory was initially created to provide a useful abstraction of the notion of length of subsets of the real line—and, more generally, area and volume of subsets of euclidean spaces. in particular, it provided a systematic answer to the question of which subsets of r have a length. as later set theory developmen... | wikipedia |
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