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After Serre we recognise l ( K − D ) as the dimension of H < sup > 1 </ sup >( D ), where now D means the line bundle determined by the divisor D. That is, Serre duality in this case relates groups H < sup > 1 </ sup >( D ) and H < sup > 0 </ sup >( KD *), and we are reading off dimensions ( notation: K is the canonical line bundle, D * is the dual line bundle, and juxtaposition is the tensor product of line bundles ).
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Some Related Sentences
After and Serre
After World War II Johnston spent most of his time at Jardin Serre de la Madone, his garden in the south of France ; and in 1947 he entrusted Hidcote to the National Trust.
After and we
After a while, Kitty murmured something to Cappy, and he held her close, answering, `` We'll just have to wait till we pull into Philly, honey ''.
After casting about for a way of describing this spirit, we decided that it would be better to use Mr. Lyford's introduction as an illustration.
After we got a script and the spots for the songs were blocked out, we'd get together for an hour or so every day.
After allowing for group exposures, it is apparent that other factors must be considered if we are to comprehend fanaticism.
After luncheon we took advantage of the siesta period to try to get in touch with a few people to whom our dear friend Deppy had written.
After figuring out how to regulate the barrels so that they shot to the same point of impact, we fired this little 20-inch-barrel job on my home range and in Marlin's underground test gallery.
After all, we did pretty well in some other areas of the Olympics competition.
After a few tortuous moments of wondering who `` he '' is, the camera pans across the room to the plaster statue, and we realize that Neitzbohr is trying to redeem himself in the eyes of a mute piece of sculpture.
After a while, we became aware that the money was disappearing as fast as we replenished it.
After that, we began to get `` visitors '' to class.
After a while we were perhaps less surprised, but still puzzled, when a friendly discussion would suddenly jump the track into the most irrelevant and illogical comparisons.
After the usual Honorable Sirs, it went on to say that there had been set off to the widow one full third part of the real estate of the deceased Salu Norberg, one lower room, on the Western side, privileges to the well and bake-oven and to one third of the cellar ( I can show you the cellar when we go up ), also one Cow Right, and lastly they set off to the widow her own land that she brought with her as dower, namely the Beech Pasture.
`` After inspiring this, I think we should certainly follow through on it '', he declared.
After all this destruction of old literature, it should be obvious why we have so little information about the early history and development of the Lo Shu, which was already semisecret anyhow.
After the spate of female vocalists we have been having, all of whom took Sarah as a point of departure and then tried to see what they could do that might make her seem old hat, it seemed that all that has happened is to make the real thing seem better than ever.
After a short time, both George and Donald joined the class with me so they wouldn't feel lonely, and we used to hang a sign on the door of the Brush-off reading out to work.
He later wrote: " After more than forty years, we have had no third world war, and the balance of nuclear terror ... may have helped to prevent one.
After we had done so, he shook his head in disbelief, still thinking that he had received the smaller portion.
After the first episode in 1960, the Daily Mirror printed: " The programme is doomed from the outset ... For there is little reality in this new serial, which apparently, we have to suffer twice a week.
They do not believe we can be sure of being saved, believing instead that salvation comes as a result of a life of obedience to the commands of Christ After death, believers are in a state of non-existence, knowing nothing until the Resurrection at the return of Christ.
In her journal for Christmas Eve 1832, the delighted 13-year-old princess wrote, " After dinner … we then went into the drawing-room near the dining-room … There were two large round tables on which were placed two trees hung with lights and sugar ornaments.
After we broke up, he would stand outside my home in Burbank and cry.
Director John Carpenter decided to add the original scenes into the special edition release as an extra only: " After we screened the rough cut, we realized that the movie didn't really start until Snake got to New York.
After and recognise
After pair formation has occurred calls also serve to help them reunite, the ability of individuals to recognise their own mate has also been demonstrated in several species.
After the independence of Bangladesh in 1971, East Germany was the third country in the world, and the first country in Europe, to officially recognise Bangladesh in 1972.
After four years Gwydion took the boy to see his mother, who refused to recognise him out of shame and anger that Gwydion should have nurtured the boy, and placed a tynged on him that he should have no name unless she gave it to him.
After the capture of Charles I, Hale was expected to defend him, and indeed offered to do so ; the King refused to submit to the court, claiming he did not recognise its jurisdiction.
After trial all the accused were acquitted but many refused to recognise the verdict of the courts.
After his death, even people in disagreement with some of those convictions now generally recognise his importance to the history of Quebec.
After his death in 1784 renowned architect James Gandon was appointed to finish the building, which we recognise today as the Four Courts.
After the Declaration of Independence on 4 July he was formally taken into custody by order of the Provincial Congress of New Jersey, an entity which William refused to recognise, regarding it as an " illegal assembly ".
After trial all the accused were acquitted but many refused to recognise the verdict of the courts.
After a strongly Catholic faction emerged in 1646, which opposed signing a peace treaty that did not recognise the position of the Catholic Church or return confiscated catholic land, the Confederates once again clashed with the Royalists, who abandoned most of their positions in Ireland to the Parliamentarians during 1646.
After this result the anti-treaty faction refused to recognise the new assembly and the Irish Civil War followed shortly afterwards.
After the introduction of the four-character codes, originally it was intended for the signallers to visually recognise the train by the number displayed on its front ; with the coming of automated systems the number has been retained internally within timetables and signalling systems but with some exceptions detailed above is no longer displayed on trains.
After his death his daughter sponsored a local award in Perton and Codsall, where her father lived until his death, to recognise local players achievements.
After that, the employer has 21 days in which to recognise the union. If the employer does not provide recognition within the stipulated period the matter will be taken to the Director General of Industrial Relations ( DGIR ) for arbitration.
After the short adjournment Barry announced “ As a soldier of the Irish Republic, I refuse to recognise the court .” Brigadier Onslow explained the prisoner ’ s “ perilous situation ” and that he was being tried on a capital charge.
After the ceremony, the Celtic Charity Fund presented a cheque of ₤ 5, 000 for St Mary's, to help the restoration fund for the church and to recognise the important link between club and community.
After 1945, it did recognise separate internal nations, with their own governments.
After this 45-minutes, everyone in Hong Kong started to recognise and notice him.
1.554 seconds.