The first major application was the relative version of Serre's theorem showing that the cohomology of a coherent sheaf on a complete variety is finite dimensional ; Grothendieck's theorem shows that the higher direct images of coherent sheaves under a proper map are coherent ; this reduces to Serre's theorem over a one-point space.

When storing records in a large unsorted file, one may use a hash function to map each record to an index into a table T, and collect in each bucket T a list of the numbers of all records with the same hash value i. Once the table is complete, any two duplicate records will end up in the same bucket.

To date, a complete map of the area has been made, including not only the Ash Altar and temenos, but also two fountains, including the Hagno fountain mentioned by Pausanias, the hippodrome, the stadium, a building that was probably a bathhouse, the xenon ( hotel ), a stoa, several rows of seats, and a group of statue bases.

That same year, Fiery Dragon Productions of Canada produced a 30th Anniversary Edition under license in a tin box complete with CD, map, and monster counters, two new versions of the rules.

A map editor and overall complete modification program called UnrealEd also came with the package.

As the initiator of scientific meteorology, he devised the first weather map, proposed a theory of anticyclones, and was the first to establish a complete record of short-term climatic phenomena on a European scale .< ref >

The largest statue ever erected for Lenin was replaced with a globe, complete with a geographic map of Uzbekistan over it.

The effort to automate map plotting began in the United States in 1969, with the process complete in the 1970s.

A DLSM implies that elevation is available continuously at each location in the study area, i. e. that the map represents a complete surface.

See Chapter 9, " Wonder Cave ", pages 161-178 for a complete history of the cave, including numerous photographs and a location map.

The film shows an actual map of the Middleburg area, complete with actual surrounding towns and Route 15 pictured on the map.

Once both discs were complete they were packaged together with extensive liner notes and a " visual time map " of the sources used in the compositions.

* John Rudd ( c. 1498-c. 1579 ) and his apprentice Christopher Saxton ( c. 1540-c. 1610 ), both clerics and cartographers who would be the first to complete the map of England, the latter finishing the job after the former's death.

* 1777 map of the complete length of the Canal.

*" Visual History Map " A complete visual map of remote viewing History and its participants

Finally, with the complete set of link-state advertisements ( one from each node in the network ) in hand, it is obviously easy to produce the graph for the map of the network.

Jedediah, in order to make his map complete, needed first hand information on the Southwest, an area he had not extensively explored.

The first complete geologic map showing the confines of this basin was published in 1978 by the United States Geological Survey.

# The Cartan – Hadamard theorem states that a complete simply connected Riemannian manifold M with nonpositive sectional curvature is diffeomorphic to the Euclidean space R ^ n with n = dim M via the exponential map at any point.

The challenge of a complete theory of population genetics is to provide a set of laws that predictably map a population of genotypes ( G < sub > 1 </ sub >) to a phenotype space ( P < sub > 1 </ sub >), where selection takes place, and another set of laws that map the resulting population ( P < sub > 2 </ sub >) back to genotype space ( G < sub > 2 </ sub >) where Mendelian genetics can predict the next generation of genotypes, thus completing the cycle.

On entering the brainstem, sensory fibers from V, VII, IX, and X are sorted out and sent to the trigeminal nucleus, which thus contains a complete sensory map of the face and mouth.

The spinal counterparts of the trigeminal nucleus ( cells in the dorsal horn and dorsal column nuclei of the spinal cord ) contain a complete sensory map of the rest of the body.

Matsuo puzzled and grew anxious over the complete passiveness, concluding that he was the butt of a devilish joke.

But a few days after Fred's return he began hemorrhaging and that was the beginning of early and complete disintegration.

He was a learned and brilliant man, one of the best jurists in Europe and with flashes of penetrating insight, and yet in his dealings with other people, particularly when he tried to be ingratiating, he was capable of an abysmal stupidity that can have come only from a complete incomprehension of human nature and human motives.

They, however much they were in disagreement with the late Victorians over the method by which Britain was Germanized, agreed with them that the end result was the complete extinction of the previous Celtic population and civilization.

A recent editorial discussing a labor-management agreement reached between the Southern Pacific Co. and the Order of Railroad Telegraphers has been criticized on the grounds that it was not based on complete information.

The prize was an old-fashioned, woven cloth hammock, complete with cross-top pillow, fringed side pieces, and hooks for hanging.

Despite several years of front-page stories, the average citizen was unable to get a complete picture of McCarthy until he saw on the television screen what the reporters had been seeing all along but had no effective way of communicating.

The work program, as it was originally proposed, was to take five years to complete.

Another recent achievement was the successful development of a method for the complete combustion in a bomb calorimeter of a metal in fluorine when the product is relatively non-volatile.

Cuban S.S.R.: Whatever may have been the setbacks resulting from the unsuccessful attempt of the Cuban rebels to establish a beachhead on the Castro-held mainland last week, there was at least one positive benefit, and that was the clear-cut revelation to the whole world of the complete conversion of Cuba into a Russian-dominated military base.

This was later known to be the result of concentrating the minute amount of complete antibody found in these sera ; ;

when the insoluble fraction was suspended in a volume of saline equal to that of the original serum sample, no complete antibody activity could be detected.

It was nevertheless almost incredible that four years after Yalta there should be a complete split over Germany, with hot heads on both sides planning to use the Germans against their former allies, and with Nazi-minded Germans expecting to recover their power by fighting on one side or the other.

For the use of students and future restorers, a full, day-by-day record was kept of all three undertakings, complete technical reports on what we found and what we did.

If we thus spent our very first day in the midst of a large number of your people honoring a new hero and a great national achievement, our last day, to us at least, was equally impressive and very moving, even though the crowds were absent and there was almost complete silence.

Milman Parry rigorously defended the observation that the extant Homeric poems are largely formulaic, and was led to postulate that they could be shown entirely formulaic if the complete corpus of Greek epic survived ; ;

Some time later the missing part of the relic was found and the complete inscription, together with other new evidence, fully corroborated the ancient priest's information.

As a natural outgrowth of this approach it was often suggested that the doctor should complete the preparation for painless intercourse by dilating the vagina.

So filled was Mel Chandler with the spirit of Garryowen that after Korea was over, he took on the job of writing the complete history of the regiment.