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c2a071e24f5f9b77e4dc693006ddf91c803fd6cf | saballos, a., w. vermerris, l. rivera, and g. | saballos, a., w. vermerris, l. rivera, and g. ejeta. 2009. allelic association, chemical characterization and saccharification properties of brown midrib mutants of sorghum (s. bicolor (l.) moench). bioenerg. res. 1:193-204. | wikipedia |
bce038f4cc22ea6f0a5116118d3f951afe591707 | peters, p., m. jenks, p. rich, j, axtell, | peters, p., m. jenks, p. rich, j, axtell, and g. ejeta. 2009. mutagenesis, selection, and allelic analysis of epicuticular wax mutants in sorghum. crop sci. 49:1249-1258. | wikipedia |
f85143b55f8dc356dbed09d10d949ed5c2489387 | vogler, r., t. tesso, k. johnson, and g. | vogler, r., t. tesso, k. johnson, and g. ejeta. 2009. effect of allelic variation on forage quality of brown midrib sorghum mutants. african j. of biochem. 3(3):70-76. | wikipedia |
d7cc1abb5e7855ddbfb535d6765de5221ccbfa4b | saballos, a., g. ejeta, e. sanchez, c. kang, | saballos, a., g. ejeta, e. sanchez, c. kang, and w. vermerris. 2009. a genome-wide analysis of the cinnamyl alcohol dehydrogenase family in sorghum (sorghum bicolor (l.) moench) identifies sbcad2 as the brown midrib6 gene. genetics 181:783-795. | wikipedia |
e97bd3963e7e320699f384830c8ecea0ce975891 | gobena, d., m. shimels, p. rich, c. ruyter-spira, | gobena, d., m. shimels, p. rich, c. ruyter-spira, h. broumeester, s. kanuganti, t. mengiste, and g. ejeta. 2017. mutation in sorghum, low germination stimulant 1 alters strigolactones and causes striga resistance. proc. national acad. sci. 114(17): 4471-4476 | wikipedia |
10d3f5f84af38c75ce1ae5f1a3dd536d3f20dc21 | ongom, patrick o., j. volenec, g. ejeta. 2016. | ongom, patrick o., j. volenec, g. ejeta. 2016. selection for drought tolerance in sorghum using desiccants to simulate post-anthesis drought stress. field crops research. 198(2016): 312-321. | wikipedia |
fe6390e09387f00673fd348a72263c1bd8408515 | ongom, patrick o. and g. ejeta. 2018. mating | ongom, patrick o. and g. ejeta. 2018. mating design and genetic structure of a multi-parent advanced generation inter-cross (magic) population of sorghum (sorghum bicolor (l.) moench). g3 genes/genomes/genet. 8(1):331-341. | wikipedia |
de9bedf1a1759b51ae30127764fe2d63244e1e15 | working in sudan during the early 1980s, ejeta | working in sudan during the early 1980s, ejeta developed africa's first commercial hybrid variety of sorghum tolerant to drought. later, with a purdue university colleague in indiana, he discovered the chemical basis of the relationship between the deadly parasitic weed striga and sorghum, and was able to produce sorgh... | wikipedia |
db68e0d7bd3b04f3bffc0e4f59d09f2238defea2 | during primary school, ejeta planned to study engineering | during primary school, ejeta planned to study engineering when he reached college age. however, his mother convinced him he could do more working in agriculture. with assistance from the oklahoma state university, he attended an agricultural and technical secondary school in ethiopia, and also studied at what is now ha... | wikipedia |
4485230133dab32c837de9b41d325bfa3201cb96 | gebisa ejeta (born 1950) is an ethiopian american | gebisa ejeta (born 1950) is an ethiopian american plant breeder, geneticist and professor at purdue university. in 2009, he won the world food prize for his major contributions in the production of sorghum. | wikipedia |
b379a3bbe248ced0f803c719ec2ebde416c5395e | "line-up for yesterday: an abc of baseball immortals" | "line-up for yesterday: an abc of baseball immortals" is a poem written by ogden nash for the january 1949 issue of sport magazine. in the poem, nash dedicates each letter of the alphabet to a legendary major league baseball player. the poem pays tribute to 24 players altogether, plus one winking reference to himself a... | wikipedia |
8ea7652677165a53127ec9a9269901329d3a8d2d | "line-up for yesterday: an abc of baseball immortals" | "line-up for yesterday: an abc of baseball immortals" is a poem written by ogden nash for the january 1949 issue of sport magazine. in the poem, nash dedicates each letter of the alphabet to a legendary major league baseball player. the poem pays tribute to 24 players altogether, plus one winking reference to himself (... | wikipedia |
bbb6ceec14d9e92b90e605f2a44bc7fd306327b0 | hill was distinguished for his theoretical contributions to | hill was distinguished for his theoretical contributions to the study of the population and quantitative genetics of finite populations, in particular with respect to multilocus problems. he was the first to present formulae for the expected association of linked genes in finite populations due to random sampling of ga... | wikipedia |
7a6b851fe8babe33f5a01937a015164a2a9c1da7 | hill was educated at st albans school, hertfordshire | hill was educated at st albans school, hertfordshire and studied agriculture at wye college, university of london graduating with a bachelor of science degree in 1961. he studied genetics at the university of california, davis, graduating with a master of science degree in 1963, then moved to edinburgh to pursue a phd ... | wikipedia |
feaabddc190987b4cf818250690d9f3fcae42e80 | william george hill obe frs frse (7 august | william george hill obe frs frse (7 august 1940 – 17 december 2021) was an english geneticist and statistician. he was a professor at university of edinburgh. he is credited as co-discoverer of the hill–robertson effect with his doctoral advisor, alan robertson. | wikipedia |
5b38018f21ae2884dda8e1973b660e3b1ba956b1 | the harlem river lift bridge is a vertical | the harlem river lift bridge is a vertical lift bridge carrying the metro-north railroad's hudson line, harlem line, and new haven line across the harlem river between the boroughs of manhattan and the bronx in new york city. the average weekday ridership on the lines is 265,000. | wikipedia |
45490bf3feae48517fb937424c690c3422d4163e | between 1954 and 1956, the new york central | between 1954 and 1956, the new york central railroad built a fourth rail bridge on this site, this time a vertical-lift bridge, to replace the 1897 bridge. the new bridge opened in 1956. the four-track bridge remains in use today and consists of two parallel double-track spans, 340 feet (100 m) long. it has 25 feet (7.... | wikipedia |
3e0bd1cb24d3960874a2a786898d99c254c6b126 | on february 15, 1897, trains on the harlem | on february 15, 1897, trains on the harlem division started running over the new drawbridge over the harlem river and the elevated structure connecting to it. the department of war ordered that the bridge cannot be opened during peak hours, between 7 and 10 a.m. and 4 and 7 p.m. | wikipedia |
37fd67c43e72b3ab22a40b85aca7b86e424f4fa2 | during the course of construction, trains were to | during the course of construction, trains were to run over a temporary wooden structure along with a temporary two-track wooden drawbridge. the cost of the entire project was to be $2 million. at the time, construction was expected to begin on september 1, 1893. the work was divided into four sections. the bridge's des... | wikipedia |
1d065fde7add0324dc54d972a3bd06e1f0a67aa1 | the new bridge was to be 400 feet | the new bridge was to be 400 feet (120 m) -long and was built for about $500,000 by the king bridge company. the new bridge was to be 17 feet (5.2 m) higher than the old bridge, as mandated by the federal government, making it 24 feet (7.3 m) above the water. the park avenue line's grade had to be raised to allow it to... | wikipedia |
81d6ce26b298b3faf360b19a2677d0d320e2d36d | to remedy the situation, the central could have | to remedy the situation, the central could have raised the bridge to 24 feet (7.3 m) above the water to satisfy the department of war, allowing most vessels to cross under the bridge, for $300,000 or replaced it with a tunnel to satisfy the harlem community for $3 million. the railroad opted to raise the bridge, which ... | wikipedia |
ff46995455b82c0e9324051eac5a737a937a2b8b | the 1867 bridge was soon made obsolete by | the 1867 bridge was soon made obsolete by heavy traffic and dredging of the harlem river ship canal. in 1888, the united states department of war began work on the harlem river to allow for unrestricted shipping activity between the hudson river and the east river and through the new harlem river ship canal at 225th st... | wikipedia |
f4a444a7e6421b7dcde6cb26437897d7e7e6dd5a | the first bridge on this site was constructed | the first bridge on this site was constructed by the new york and harlem railroad in 1841. it was composed of four 90-foot (27 m) -long box truss spans, three of which were fixed iron spans, while the remaining span was a wooden swing span. in the closed position, the bridge had a clearance of only seven feet above mea... | wikipedia |
2ca2dccfc74b12f86631fdf83309ef357008579e | the harlem river lift bridge (also known as | the harlem river lift bridge (also known as the park avenue bridge) is a vertical lift bridge carrying the metro-north railroad 's hudson line, harlem line, and new haven line across the harlem river between the boroughs of manhattan and the bronx in new york city. the average weekday ridership on the lines is 265,000. | wikipedia |
cdc6962a9e2577b0808371e67407319c16229264 | mothobi mvala (born 14 june 1994) is a | mothobi mvala (born 14 june 1994) is a south african professional soccer player who plays as a defender for mamelodi sundowns and the south africa national team.he represented the south africa under-23 team at the 2016 summer olympics. | wikipedia |
44bc1a61e4617d85eb0935ae1538d8d452f6178d | on 1 september 2022, bachana joined macarthur fc | on 1 september 2022, bachana joined macarthur fc on a free transfer. he made his debut coming off the bench on 13 november 2022, and got the game winning assist against the central coast mariners in the 95th minute. bachana scored his first a-league goal on 18 december 2022, scoring the winning goal against perth glory... | wikipedia |
33bcdcec03bd92d448dac31662e0616f4d40972f | on 18 july 2019, arabuli signed a three-year | on 18 july 2019, arabuli signed a three-year contract with greek super league club panionios, who began the season on − 6 points because of financial irregularities. arabuli scored his first goal for the club in added time in an opening-day defeat at home to newly promoted volos. he scored a stoppage-time equaliser on ... | wikipedia |
bb76dcc8f87bbba16030fcd84140caf02a741929 | arabuli made his professional debut for dila gori | arabuli made his professional debut for dila gori on 30 march 2013 in a match against fc zugdidi. he has spent six months in spain where he played for ad alcorcón 's b team in the tercera division. he also played for georgian clubs tskhinvali, dinamo tbilisi and samtredia and for hungarian clubs balmazújváros and puská... | wikipedia |
80bd411d8b470fc44ed1c5eefd903c0e74f3c21e | bachana arabuli (georgian: ბაჩანა არაბული; born 5 january | bachana arabuli (georgian: ბაჩანა არაბული; born 5 january 1994) is a georgian professional footballer who most plays as a striker for k league 2 club gyeongnam. | wikipedia |
09129fbfd45b0d6fa8693a32d13f44f82b4c1c8c | sir david hull was a british paediatrician. hull | sir david hull was a british paediatrician. hull was most notable for research and for a paper he published in 1963 in the journal of physiology with michael dawkins, about research into brown fat, an adipose-like tissue found in hibernating animals and in the human infant and for later contributions considered outstan... | wikipedia |
a3f0e3cb47e40b773903ef6c743f9168dfa475ec | he served as president of the neonatal society | he served as president of the neonatal society from 1987 until 1991, as president of the british paediatric association from 1991 to 1994, and as an adviser on paediatrics to the government chief scientist. he received the royal college of paediatrics and child health 's james spence medal in 1996, "due to his contribu... | wikipedia |
3e6d904ba6f6a17a30b69beb68d551b5c8a31390 | on his return, he underwent further medical training | on his return, he underwent further medical training in london, and then obtained a post as nuffield research fellow at the institute for medical research in oxford., and then as lecturer in paediatrics, at the university of oxford from 1963, after which he was appointed in 1966 as consultant paediatrician, at the grea... | wikipedia |
cfb70ee6bde28e5b4eb1fba143099f08328a0b99 | hull was born in blackburn, lancashire, the second | hull was born in blackburn, lancashire, the second son of william hull and nellie hayes. he has a brother, derek hull (born 8 august 1931), almost exactly one year older. he attended poulton-le-fylde grammar school, before graduating from liverpool university. he then spent two years in the royal army medical corps, mo... | wikipedia |
b4d4d2c0b2cc39b7dd3d0a78b94e67a557116078 | sir david hull frcp frcpch (4 august 1932 | sir david hull frcp frcpch (4 august 1932 – 13 march 2021) was a british paediatrician. hull was most notable for research and for a paper he published in 1963 in the journal of physiology with michael dawkins, about research into brown fat, an adipose-like tissue found in hibernating animals and in the human infant an... | wikipedia |
c704f1284f9394039e185b892a4b7ec413f00449 | abner eliezer shimony was an american physicist and | abner eliezer shimony was an american physicist and philosopher. he specialized in quantum theory and philosophy of science. as a physicist, he concentrated on the interaction between relativity theory and quantum mechanics. he authored many works and research on complementarity in quantum entanglement as well as multi... | wikipedia |
372a03511d6887bc86f43e74e3ef2e6c257cbe43 | in 1996 he was awarded the lakatos award | in 1996 he was awarded the lakatos award in the philosophy of science forthe two-volume collection of papers, the search for a naturalistic world view, spanning his career up until 1992. he served as president of the philosophy of science association from 1995 to 1996. he died in new haven, connecticut, aged 87. | wikipedia |
85a1a0072b65d065e249faf8f85669bf041a1ab1 | he is also known for his inquiry into | he is also known for his inquiry into the question of the "peaceful coexistence" of quantum mechanics and special relativity. he wrote several books and numerous research articles on the foundations of quantum mechanics and related topics. shimony is credited with coining the phrase "passion at a distance" to character... | wikipedia |
659afaf51f86c588d84690db98d56ded295a5e7e | after receiving his second ph.d., shimony interacted with | after receiving his second ph.d., shimony interacted with both the philosophical academic world and the physics academic world. his most famous professional correspondence is with rudolf carnap. he taught philosophy of science at mit from 1959 until 1968 in the school’s department of humanities. in 1968 he transferred ... | wikipedia |
0d32c72b62650382745491d0f0480fe07e1503fa | shimony was born in columbus, ohio. he obtained | shimony was born in columbus, ohio. he obtained his ba in mathematics and philosophy from yale university in 1948, and an ma in philosophy from the university of chicago in 1950. he obtained his ph.d. in philosophy from yale university in 1953 under the supervision of rudolf carnap, and served in the u.s. army signal c... | wikipedia |
12678188382717eb73193a6a4a658af460374aab | abner eliezer shimony (/ ʃ ɪ ˈ m | abner eliezer shimony (/ ʃ ɪ ˈ m oʊ n i /; march 10, 1928 – august 8, 2015) was an american physicist and philosopher. he specialized in quantum theory and philosophy of science. as a physicist, he concentrated on the interaction between relativity theory and quantum mechanics. he authored many works and research on co... | wikipedia |
58752fe97f5577a00d874d878894adaceb5a354b | the 1976–77 season was the 75th season in | the 1976–77 season was the 75th season in which dundee competed at a scottish national level, playing in the second tier for the first time since the 1946–47 season. the club would fail to achieve promotion, finishing in 3rd place. dundee would also compete in both the scottish league cup and the scottish cup, where th... | wikipedia |
2dce64416af392b44c6130bc74602e39d4c6ffb5 | the 1976–77 season was the 75th season in | the 1976–77 season was the 75th season in which dundee competed at a scottish national level, playing in the second tier for the first time since the 1946–47 season. the club would fail to achieve promotion, finishing in 3rd place. dundee would also compete in both the scottish league cup and the scottish cup, where th... | wikipedia |
598b92f25d3ef55607e063aabe4a830bb050c71c | rajan sankaranarayanan is an indian structural biologist and | rajan sankaranarayanan is an indian structural biologist and a group leader at the centre for cellular and molecular biology (ccmb) in hyderabad. he is known for his research in the field of protein translation, especially for his contribution in chiral proofreading during protein biosynthesis. in 2020, sankaranarayana... | wikipedia |
5c8ec69775692294b5b760fca9d04c4c0514e65f | sankaranarayanan was awarded the shanti swarup bhatnagar prize | sankaranarayanan was awarded the shanti swarup bhatnagar prize for science and technology (2011), the highest science award in india, in the biological sciences category. he is also the recipient of the national bioscience award for career development. in 2020, sankaranarayanan has been awarded the infosys prize in lif... | wikipedia |
f2419907058dfd25ddb5f1cb3631b08c8fd6eff6 | recently, the group has also identified the role | recently, the group has also identified the role of a faal-like homolog in eukaryotes called disco-interacting protein 2 (dip2), in regulating a specific pool of diacylglycerols by converting it into triacylglycerols, thereby maintaining cellular homeostasis. | wikipedia |
670d77afd431e11c391e706649a018808c61333b | his laboratory is also interested in understanding the | his laboratory is also interested in understanding the roles of a class of lipid metabolising enzymes called fatty acyl-amp ligases (faal), which are involved in the production of lipidic secondary metabolites in bacteria. the group has identified the mechanistic underpinnings of faal’s incredible specificity towards i... | wikipedia |
cc526989d1042165238d2224bef98c2a0e3ae2f3 | the group has also identified the role of | the group has also identified the role of archaeal-derived chiral proofreader d-aminoacyl-trna deacylase 2 (dtd2) in removing n-ethyl adducts formed on d-aminoacyl-trna by acetaldehyde, an anaerobic fermentation intermediate. furthermore, the group has also went on to show how these two chiral proofreaders are involved... | wikipedia |
818081202efa8072bc522b3b612801720ba38a98 | his laboratory has elucidated the mechanism of d-aminoacyl-trna | his laboratory has elucidated the mechanism of d-aminoacyl-trna deacylase 1 (dtd1), where he has shown how an invariant ‘cross-subunit’ gly- cis pro dipeptide captures the chiral centre of incoming d-aminoacyl-trna. his group also identified a paralog of dtd1 in animals known as animalia-specific-trna deacylase (atd), ... | wikipedia |
7cdfbbedf21b31f6e9a24d67133702b382e2205d | sankaranarayanan's group is interested in understanding unique proofreading | sankaranarayanan's group is interested in understanding unique proofreading mechanisms that are operational in biological systems to maintain quality control during the translation of the genetic code. these processes are important in understanding how d-amino acids are kept away from getting incorporated during protei... | wikipedia |
eea9a33eb114fa5ee9cc2a8476cc5a719b954faf | he was born in papanasam project, tirunelveli, tamilnadu. | he was born in papanasam project, tirunelveli, tamilnadu. he pursued his master's degree in madurai kamaraj university (mku) followed by his ph.d. in indian institute of science (iisc) under the guidance of prof. m. vijayan. he did his postdoctoral research with prof. dino moras at laboratoire de biologie structurale, ... | wikipedia |
598b92f25d3ef55607e063aabe4a830bb050c71c | rajan sankaranarayanan is an indian structural biologist and | rajan sankaranarayanan is an indian structural biologist and a group leader at the centre for cellular and molecular biology (ccmb) in hyderabad. he is known for his research in the field of protein translation, especially for his contribution in chiral proofreading during protein biosynthesis. in 2020, sankaranarayana... | wikipedia |
a891a630dd3afe1c996f37544c3057661e60dd2d | the yanghwa bridge (korean: 양화대교), formerly known as | the yanghwa bridge (korean: 양화대교), formerly known as the second hangang bridge, is an eight lane bridge spanning the han river in seoul, south korea. the bridge connects mapo district on the north side of the river to yeongdeungpo district on the south side of the river. the bridge is buttressed by the eastern end of t... | wikipedia |
f19cfd8088c0f69724ed2d0880f8c0d2c0326037 | the bridge went through repairs and renovations in | the bridge went through repairs and renovations in 1996 and reopened in april 2002 with additional ramps. as of february 2010, the bridge is once again going through renovations by widening the space between bridge posts to allow 5000t ships to pass. yanghwa bridge also has a song named after it called "양화대교 yanghwa br... | wikipedia |
d7b1f74aa5f7807cc1e2320a65c7b97bab174bfb | the old bridge's upper structure has a width | the old bridge's upper structure has a width of 18 m (59 ft), length 1,053 m (3,455 ft), and is composed of steel plate girders and concrete box girders. the new bridge has a width of 16.2 m (53 ft), length 1,053 m (3,455 ft) and is a steel plate girder bridge. the lower structure has an open caisson well foundation. | wikipedia |
342fa65c6620cbf4eaa8bc12bb1061e52ec54aed | the bridge is a combination of two bridges: | the bridge is a combination of two bridges: the old bridge, originally called the "second han river bridge", completed in 1965; and the new bridge, completed in 1982. the old bridge was the first bridge built by korean technology after independence in 1945 and served as the gateway from seoul to the west coast. due to ... | wikipedia |
94df976d057968707b3e99fca207ee3f46c07fe7 | the yanghwa bridge (korean: 양화대교), formerly known as | the yanghwa bridge (korean: 양화대교), formerly known as the second hangang bridge (제2한강교; lit. second han river bridge), is an eight lane bridge spanning the han river in seoul, south korea. the bridge connects mapo district on the north side of the river to yeongdeungpo district on the south side of the river. the bridge... | wikipedia |
67876f454b4180827952ccf009e7190ad481f37e | naphthalenetetracarboxylic diimide (ntcdi) is a solid organic compound | naphthalenetetracarboxylic diimide (ntcdi) is a solid organic compound and one of the simplest naphthalenediimides (ndis). ntcdi is produced from the parent naphthalene via an intermediate compound naphthalenetetracarboxylic dianhydride. | wikipedia |
f60a456bdda132f341ebb52c3e487c5d069e370a | ntcdi is redox-active, forming stable radical anions near | ntcdi is redox-active, forming stable radical anions near -1.10 v vs. fc/fc. its ability to accept electrons reflects the presence of an extended conjugated ring system and the electron withdrawing groups (carbonyl centers). ndi is used in supramolecular chemistry owing to its tendency to form charge-transfer complexes... | wikipedia |
67876f454b4180827952ccf009e7190ad481f37e | naphthalenetetracarboxylic diimide (ntcdi) is a solid organic compound | naphthalenetetracarboxylic diimide (ntcdi) is a solid organic compound and one of the simplest naphthalenediimides (ndis). ntcdi is produced from the parent naphthalene via an intermediate compound naphthalenetetracarboxylic dianhydride. | wikipedia |
8601913a19983a4cd7cf7239187fba3b2c1b9a08 | in the mathematical field of complex analysis, contour | in the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane. contour integration is closely related to the calculus of residues, a method of complex analysis. one use for contour integrals is the evaluation of integrals along the real l... | wikipedia |
8db09db241cab9bfc5aed78fec6085e229b8dfa0 | is valid only for re(s) > 1. but | is valid only for re(s) > 1. but ζ (s) = − Γ (1 − s) 2 π i ∫ h (− t) s − 1 e t − 1 d t, {\displaystyle \zeta (s)=-{\frac {\gamma (1-s)}{2\pi i}}\int _{h}{\frac {(-t)^{s-1}}{e^{t}-1}}dt,} | wikipedia |
2dd4fbdf92ec0b505578121f59aedbc53ee8e531 | for example, the original definition of the riemann | for example, the original definition of the riemann zeta function ζ (s) via a dirichlet series, ζ (s) = ∑ n = 1 ∞ 1 n s, {\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}},} | wikipedia |
3cec5a8edebe44e0d827f84841f5395d4b8678d1 | an integral representation of a function is an | an integral representation of a function is an expression of the function involving a contour integral. various integral representations are known for many special functions. integral representations can be important for theoretical reasons, e.g. giving analytic continuation or functional equations, or sometimes for nu... | wikipedia |
8ea48960fb5df1a8d8d283c7d64af86603027d02 | thus, we can evaluate a contour integral with | thus, we can evaluate a contour integral with n = 4 {\displaystyle n=4}. we can use the same method to evaluate contour integrals for any vector field with n > 4 {\displaystyle n>4} as well. | wikipedia |
70abd9c5889e41b553772b83a0d822c3335a0cc0 | = ⨌ v (∂ f u ∂ u | = ⨌ v (∂ f u ∂ u + ∂ f x ∂ x + ∂ f y ∂ y + ∂ f z ∂ z) d v = ⨌ v (∂ u 4 ∂ u + ∂ x 5 ∂ x + ∂ y 6 ∂ y + ∂ z − 3 ∂ z) d v = ⨌ v 4 u 3 z 4 + 5 x 4 z 4 + 5 y 4 z 4 − 3 z 4 d v = ⨌ v 4 u 3 z 4 + 5 x 4 z 4 + 5 y 4 z 4 − 3 z 4 d v = ∫ 0 1 ∫ − 10 2 π ∫ 4 5 ∫ − 1 3 4 u 3 z 4 + 5 x 4 z 4 + 5 y 4 z 4 − 3 z 4 d v = ∫ 0 1 ∫ − 10 2 π ... | wikipedia |
06bc0c07908020622b2ee7728e70376f00835276 | = ⨌ v (∂ f u ∂ u | 105 210 π 2 ≈ 576468.77 {\displaystyle {\begin{aligned}&=\iiiint _{v}\left({\frac {\partial f_{u}}{\partial u}}+{\frac {\partial f_{x}}{\partial x}}+{\frac {\partial f_{y}}{\partial y}}+{\frac {\partial f_{z}}{\partial z}}\right)\,dv\\&=\iiiint _{v}\left({\frac {\partial u^{4}}{\partial u}}+{\frac {\partial x^{5}}{\pa... | wikipedia |
a4fb8fefe7d6fc9a10377de361b554bbfbaaf41d | = ⨌ v (∂ f u ∂ u | 91z^{3}+3)}{3z^{3}}}\right)\,dy\,dz\,du\\&=\int _{0}^{1}\int _{-10}^{2\pi }\left(4u^{4}+{\frac {743440}{21}}+{\frac {4}{z^{3}}}\right)\,dz\,du\\&=\int _{0}^{1}\left(-{\frac {1}{2\pi ^{2}}}+{\frac {1486880\pi }{21}}+8\pi u^{4}+40u^{4}+{\frac {371720021}{1050}}\right)\,du\\&={\frac {371728421}{1050}}+{\frac {14869136\pi ... | wikipedia |
f3e6f1dc79db0ecc8c9508407b5df8e97bb168f6 | to evaluate this, we must utilize the divergence | to evaluate this, we must utilize the divergence theorem as stated before, and we must evaluate ∇ ⋅ f {\displaystyle \nabla \cdot \mathbf {f} }. let d v = d x d y d z d u {\displaystyle dv=dx\,dy\,dz\,du} | wikipedia |
48b48d29a36541d07fd7b3c388948c610947a427 | let the vector field f = u 4 | let the vector field f = u 4 e u + x 5 e x + y 6 e y + z − 3 e z {\displaystyle \mathbf {f} =u^{4}\mathbf {e} _{u}+x^{5}\mathbf {e} _{x}+y^{6}\mathbf {e} _{y}+z^{-3}\mathbf {e} _{z}}, and remark that there are 4 parameters in this case. let this vector field be bounded by the following: 0 ≤ x ≤ 1 − 10 ≤ y ≤ 2 π 4 ≤ z ≤... | wikipedia |
3bb159dd667362ec747193cebc6a5a8a2b6d4b2e | we now evaluate ∇ ⋅ f {\displaystyle \nabla | we now evaluate ∇ ⋅ f {\displaystyle \nabla \cdot \mathbf {f} }. meanwhile, set up the corresponding triple integral: = ∭ v (∂ f x ∂ x + ∂ f y ∂ y + ∂ f z ∂ z) d v = ∭ v (∂ sin (2 x) ∂ x + ∂ sin (2 y) ∂ y + ∂ sin (2 z) ∂ z) d v = ∭ v 2 (cos (2 x) + cos (2 y) + cos (2 z)) d v = ∫ 0 1 ∫ 0 3 ∫ − 1 4 2 (cos (2 x) + cos (2 ... | wikipedia |
50667b8174d9c85d40ab588ebc5cfe650cdab3b4 | we now evaluate ∇ ⋅ f {\displaystyle \nabla | {\partial z}}\right)dv\\&=\iiint _{v}2\left(\cos(2x)+\cos(2y)+\cos(2z)\right)dv\\&=\int _{0}^{1}\int _{0}^{3}\int _{-1}^{4}2(\cos(2x)+\cos(2y)+\cos(2z))\,dx\,dy\,dz\\&=\int _{0}^{1}\int _{0}^{3}(10\cos(2y)+\sin(8)+\sin(2)+10\cos(z))\,dy\,dz\\&=\int _{0}^{1}(30\cos(2z)+3\sin(2)+3\sin(8)+5\sin(6))\,dz\\&=18\sin(2)+3\sin(... | wikipedia |
8737884bd65146dec4409f3d63279cbf60c9a77f | let the vector field f = sin (2 | let the vector field f = sin (2 x) e x + sin (2 y) e y + sin (2 z) e z {\displaystyle \mathbf {f} =\sin(2x)\mathbf {e} _{x}+\sin(2y)\mathbf {e} _{y}+\sin(2z)\mathbf {e} _{z}} and be bounded by the following 0 ≤ x ≤ 1 0 ≤ y ≤ 3 − 1 ≤ z ≤ 4 {\displaystyle {0\leq x\leq 1}\quad {0\leq y\leq 3}\quad {-1\leq z\leq 4}} | wikipedia |
3d999370c309ef907e5b3b41e28999b529e9db85 | in addition, we also need to evaluate ∇ | in addition, we also need to evaluate ∇ ⋅ f {\displaystyle \nabla \cdot \mathbf {f} } where ∇ ⋅ f {\displaystyle \nabla \cdot \mathbf {f} } is an alternate notation of div (f) {\displaystyle \operatorname {div} (\mathbf {f})}. the divergence of any dimension can be described as div (f) = ∇ ⋅ f = (∂ ∂ u, ∂ ∂ x, ∂ ∂ y, ∂... | wikipedia |
5f807eb3082304ab93e2eb40694bfc388cf3c16c | to solve multivariable contour integrals (i.e. surface integrals, | to solve multivariable contour integrals (i.e. surface integrals, complex volume integrals, and higher order integrals), we must use the divergence theorem. for right now, let ∇ {\displaystyle \nabla } be interchangeable with div {\displaystyle \operatorname {div} }. these will both serve as the divergence of the vecto... | wikipedia |
1cd81ecc6a597429e3386f26564f4e985e5435bf | thus, using the residue theorem, we can determine: | thus, using the residue theorem, we can determine: ∮ c e z z 3 d z = π i. {\displaystyle \oint _{c}{\frac {e^{z}}{z^{3}}}dz=\pi i.} | wikipedia |
3d812d02418ace5b64941da800636bf02f9b1d5a | f (z) {\displaystyle f(z)} has only one pole, | f (z) {\displaystyle f(z)} has only one pole, 0 {\displaystyle 0}. from that, we determine that the residue of f (z) {\displaystyle f(z)} to be 1 2 {\displaystyle {\tfrac {1}{2}}} ∮ c f (z) = ∮ c e z z 3 = 2 π i ⋅ res z = 0 f (z) = 2 π i res z = 0 e z z 3 = 2 π i ⋅ 1 2 = π i {\displaystyle {\begin{aligned}\oint _{c}f(z... | wikipedia |
b7bf5ac82f3b785dfce2481690fa924e71923b56 | where res {\displaystyle \operatorname {res} } is the | where res {\displaystyle \operatorname {res} } is the residue of f (z) {\displaystyle f(z)}, and the a k {\displaystyle a_{k}} are the singularities of f (z) {\displaystyle f(z)} lying inside the contour c {\displaystyle c} (with none of them lying directly on c {\displaystyle c}). | wikipedia |
cf4f89b517d4b63d5c83ffda0c8c4f36d319a1bd | recall that the residue theorem states ∮ c | recall that the residue theorem states ∮ c f (z) = 2 π i ⋅ ∑ res (f, a k), {\displaystyle \oint _{c}f(z)=2\pi i\cdot \sum \operatorname {res} (f,a_{k}),} | wikipedia |
a1a99aaa85b3e27b3bc16c1fbbb7601e49d46f38 | finally, it follows that the value of i | finally, it follows that the value of i is i = 2 π i e 1 4 π i − 1 + i (17 4 − 5 3 4 2 1 4) = 2 π 2 − 1 2 (17 4 − 5 3 4 2 1 4) {\displaystyle i=2\pi i{\frac {e^{{\frac {1}{4}}\pi i}}{-1+i}}\left({\frac {17}{4}}-5^{\frac {3}{4}}2^{\frac {1}{4}}\right)=2\pi 2^{-{\frac {1}{2}}}\left({\frac {17}{4}}-5^{\frac {3}{4}}2^{\fra... | wikipedia |
e4f51bc77eb5fa4b4f3ae1cfcd388dac8b8018f7 | the conclusion is that res z = ∞ | the conclusion is that res z = ∞ f (z) 5 − z = e 1 4 π i (5 − 3 4) = e 1 4 π i 17 4. {\displaystyle \operatorname {res} _{z=\infty }{\frac {f(z)}{5-z}}=e^{{\frac {1}{4}}\pi i}\left(5-{\frac {3}{4}}\right)=e^{{\frac {1}{4}}\pi i}{\frac {17}{4}}.} | wikipedia |
6ff4a58fd321c24a0a25aa849251735a04cfdb6c | substituting, we find 1 5 − 1 z | substituting, we find 1 5 − 1 z = − z (1 + 5 z + 5 2 z 2 + 5 3 z 3 + ⋯) {\displaystyle {\frac {1}{5-{\frac {1}{z}}}}=-z\left(1+5z+5^{2}z^{2}+5^{3}z^{3}+\cdots \right)} and (1 z 3 (3 − 1 z)) 1 4 = 1 z (3 z − 1) 1 4 = 1 z e 1 4 π i (1 − 3 z) 1 4, {\displaystyle \left({\frac {1}{z^{3}}}\left(3-{\frac {1}{z}}\right)\right)... | wikipedia |
8bbb534a0cd10f038f327e4b82c261fc08a245b8 | we use the following formula for the residue | we use the following formula for the residue at infinity: res z = ∞ h (z) = res z = 0 (− 1 z 2 h (1 z)). {\displaystyle \operatorname {res} _{z=\infty }h(z)=\operatorname {res} _{z=0}\left(-{\frac {1}{z^{2}}}h\left({\frac {1}{z}}\right)\right).} | wikipedia |
16e45ba7951b5a4ab2e87afc3892b3ec54ceebed | the pole is shown in blue in the | the pole is shown in blue in the diagram. the value simplifies to − 5 3 4 e 1 4 (log 2 + π i) = − e 1 4 π i 5 3 4 2 1 4. {\displaystyle -5^{\frac {3}{4}}e^{{\frac {1}{4}}(\log 2+\pi i)}=-e^{{\frac {1}{4}}\pi i}5^{\frac {3}{4}}2^{\frac {1}{4}}.} | wikipedia |
5cd8bd1d01fb0376512100972d602c5fa094f213 | now using the cauchy residue theorem, we have | now using the cauchy residue theorem, we have (− i + 1) i = − 2 π i (res z = 5 f (z) 5 − z + res z = ∞ f (z) 5 − z). {\displaystyle (-i+1)i=-2\pi i\left(\operatorname {res} _{z=5}{\frac {f(z)}{5-z}}+\operatorname {res} _{z=\infty }{\frac {f(z)}{5-z}}\right).} where the minus sign is due to the clockwise direction aroun... | wikipedia |
fb0b7b21db8730e8ce6553a477b9c54376872937 | similarly, for the circle c on the right, | similarly, for the circle c on the right, we have | ∫ c r f (z) 5 − z d z | ≤ 2 π ρ 3.001 3 4 ρ 1 4 1.999 ∈ o (ρ 5 4) → 0. {\displaystyle \left|\int _{c_{\mathrm {r} }}{\frac {f(z)}{5-z}}dz\right|\leq 2\pi \rho {\frac {3.001^{\frac {3}{4}}\rho ^{\frac {1}{4}}}{1.999}}\in {\mathcal {o}}\left(\rho ^{\frac {5}{4}}\right)\... | wikipedia |
5d092ce92685f973ceca3fee9c5a6761bf2c378d | if we can show that the integrals along | if we can show that the integrals along the two green circles vanish in the limit, then we also have the value of i, by the cauchy residue theorem. let the radius of the green circles be ρ, where ρ < 0.001 and ρ → 0, and apply the ml inequality. for the circle c on the left, we find | ∫ c l f (z) 5 − z d z | ≤ 2 π ρ ρ ... | wikipedia |
3a69f72cfa06eb5fbc1e830e82261826115bbc62 | let z = r (in the limit, i.e. | let z = r (in the limit, i.e. as the two green circles shrink to radius zero), where 0 ≤ r ≤ 3. along the upper segment, we find that f (z) has the value r 3 4 e 0 4 π i (3 − r) 1 4 e 2 4 π i = i r 3 4 (3 − r) 1 4 {\displaystyle r^{\frac {3}{4}}e^{{\frac {0}{4}}\pi i}(3-r)^{\frac {1}{4}}e^{{\frac {2}{4}}\pi i}=ir^{\fra... | wikipedia |
5fd18be418ab65149507f7908ace1dcfcc25a506 | when we approach from below, f (z) has | when we approach from below, f (z) has the value r 3 4 e − 3 4 π i (3 + r) 1 4 e 0 4 π i = r 3 4 (3 + r) 1 4 e − 3 4 π i. {\displaystyle r^{\frac {3}{4}}e^{-{\frac {3}{4}}\pi i}(3+r)^{\frac {1}{4}}e^{{\frac {0}{4}}\pi i}=r^{\frac {3}{4}}(3+r)^{\frac {1}{4}}e^{-{\frac {3}{4}}\pi i}.} | wikipedia |
179ec47dda4e24ea1a156cbc5cbeb832d764474b | the cut of z is therefore (−∞, 0 | the cut of z is therefore (−∞, 0 ] and the cut of (3 − z) is (−∞, 3 ]. it is easy to see that the cut of the product of the two, i.e. f (z), is, because f (z) is actually continuous across (−∞, 0). this is because when z = − r < 0 and we approach the cut from above, f (z) has the value r 3 4 e 3 4 π i (3 + r) 1 4 e 2 4... | wikipedia |
0309a5c3c8895986046c6529307bd11806716868 | we will construct f (z) so that it | we will construct f (z) so that it has a branch cut on, shown in red in the diagram. to do this, we choose two branches of the logarithm, setting z 3 4 = exp (3 4 log z) where − π ≤ arg z < π {\displaystyle z^{\frac {3}{4}}=\exp \left({\frac {3}{4}}\log z\right)\quad {\mbox{where }}-\pi \leq \arg z<\pi } and (3 − z) 1 ... | wikipedia |
965814d141bd0700500e9b2fdfc97c4d863f50f0 | this requires a close study of f (z) | this requires a close study of f (z) = z 3 4 (3 − z) 1 4. {\displaystyle f(z)=z^{\frac {3}{4}}(3-z)^{\frac {1}{4}}.} | wikipedia |
5dc963e293c1005adf9dbbcdcbd142e55913152a | we seek to evaluate i = ∫ 0 | we seek to evaluate i = ∫ 0 3 x 3 4 (3 − x) 1 4 5 − x d x. {\displaystyle i=\int _{0}^{3}{\frac {x^{\frac {3}{4}}(3-x)^{\frac {1}{4}}}{5-x}}\,dx.} | wikipedia |
18b8ff039ffc1f02b832972888afd41c1375c0eb | which gives ∫ 0 ∞ log x (1 | which gives ∫ 0 ∞ log x (1 + x 2) 2 d x = − π 4. {\displaystyle \int _{0}^{\infty }{\frac {\log x}{\left(1+x^{2}\right)^{2}}}\,dx=-{\frac {\pi }{4}}.} | wikipedia |
3bed886fa630307a369b36f218b9c16603a9da8b | − i π 2 = (∫ r + | − i π 2 = (∫ r + ∫ m + ∫ n + ∫ r) f (z) d z = (∫ m + ∫ n) f (z) d z ∫ r, ∫ r vanish = − ∫ ∞ 0 (log (− x + i ε) 1 + (− x + i ε) 2) 2 d x − ∫ 0 ∞ (log (− x − i ε) 1 + (− x − i ε) 2) 2 d x = ∫ 0 ∞ (log (− x + i ε) 1 + (− x + i ε) 2) 2 d x − ∫ 0 ∞ (log (− x − i ε) 1 + (− x − i ε) 2) 2 d x = ∫ 0 ∞ (log x + i π 1 + x 2) 2 d ... | wikipedia |
d4f6436d2423b6b00237840889688ef949b63331 | − i π 2 = (∫ r + | right)f(z)\,dz&&\int _{r},\int _{r}{\mbox{ vanish}}\\&=-\int _{\infty }^{0}\left({\frac {\log(-x+i\varepsilon)}{1+(-x+i\varepsilon)^{2}}}\right)^{2}\,dx-\int _{0}^{\infty }\left({\frac {\log(-x-i\varepsilon)}{1+(-x-i\varepsilon)^{2}}}\right)^{2}\,dx\\&=\int _{0}^{\infty }\left({\frac {\log(-x+i\varepsilon)}{1+(-x+i\var... | wikipedia |
eeb92fffb6ce1bb7c2901252b0083ed96e65febe | let r be the radius of the large | let r be the radius of the large circle, and r the radius of the small one. we will denote the upper line by m, and the lower line by n. as before we take the limit when r → ∞ and r → 0. the contributions from the two circles vanish. for example, one has the following upper bound with the ml lemma: | ∫ r f (z) d z | ≤ ... | wikipedia |
da278c0f9139dcd2e826defdc213eda82eb4a3d2 | (∫ r + ∫ m + ∫ n | (∫ r + ∫ m + ∫ n + ∫ r) f (z) d z = 2 π i (res z = i f (z) + res z = − i f (z)) = 2 π i (− π 4 + 1 16 i π 2 − π 4 − 1 16 i π 2) = − i π 2. {\displaystyle {\begin{aligned}\left(\int _{r}+\int _{m}+\int _{n}+\int _{r}\right)f(z)\,dz=&\ 2\pi i{\big (}\operatorname {res} _{z=i}f(z)+\operatorname {res} _{z=-i}f(z){\big)}\\=... | wikipedia |
346806debe92e17a7fa3647dcea86f7f28a39682 | to calculate this integral, one uses the function | to calculate this integral, one uses the function f (z) = (log z 1 + z 2) 2 {\displaystyle f(z)=\left({\frac {\log z}{1+z^{2}}}\right)^{2}} and the branch of the logarithm corresponding to −π < arg z ≤ π. | wikipedia |
04380710bfd5e10c6cd1deb96bbbe0731c129f4e | this section treats a type of integral of | this section treats a type of integral of which ∫ 0 ∞ log x (1 + x 2) 2 d x {\displaystyle \int _{0}^{\infty }{\frac {\log x}{\left(1+x^{2}\right)^{2}}}\,dx} is an example. | wikipedia |
a14b0b2583420145fd475180ab133fa046cabc3b | by using the residue theorem or the cauchy | by using the residue theorem or the cauchy integral formula (first employing the partial fractions method to derive a sum of two simple contour integrals) one obtains π i (i 2 − i) = ∫ 0 ∞ x x 2 + 6 x + 8 d x = π (1 − 1 2). ◻ {\displaystyle \pi i\left({\frac {i}{\sqrt {2}}}-i\right)=\int _{0}^{\infty }{\frac {\sqrt {x}... | wikipedia |
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