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fact and polynomial-time
This problem is solved by the polynomial-time Horn-satisfiability algorithm, and is in fact P-complete.
This makes the calculation using the Legendre symbol significantly slower than the one using the Jacobi symbol, as there is no known polynomial-time algorithm for factoring integers .< ref > The number field sieve, the fastest known algorithm, requires operations to factor N. See Cohen, p. 495 </ ref > In fact, this is why Jacobi introduced the symbol.
This yields a nonconstructive proof that there is a polynomial-time algorithm for determining if a given graph can be embedded on a torus, despite the fact that no concrete algorithm is known for this problem.

fact and machine
that any procedure which could " naturally " be called effective, can in fact be realized by a ( simple ) machine.
In fact, BQP is low for PP, meaning that a PP machine achieves no benefit from being able to solve BQP problems instantly, an indication of the possible difference in power between these similar classes.
Feeling sympathy for the Japanese, Emmy reveals the truth behind the Futurians ' mission: In the future, Japan became very wealthy and its economy surpassed that of the United States, Russia and China-the Futurians in fact stole the time machine and plan to use King Ghidorah to alter the future by devastating or subjugating present-day Japan, thus preventing its future economic reign.
Whereas Infocom's games had benefited significantly from the portability offered by running on top of a virtual machine, this strategy did not prove to be a significant advantage for Cornerstone ; in fact, the virtual machine significantly slowed the database's execution speed.
The whimsical Svetz series consists of a collection of short stories, The Flight of the Horse, and a novel, Rainbow Mars, which involve a nominal time machine sent back to retrieve long-extinct animals, but which travels, in fact, into alternate realities and brings back mythical creatures such as a Roc and a Unicorn.
The Steyr TMP ( Tactical Machine Pistol ) is a 9 mm blowback-operated, rotating-barrel weapon that is 282 mm long and that can fire 800 – 900 rounds per minute ; despite its small size, lack of a stock, and the fact that it is called a " tactical machine pistol ", it is often classed as a compact submachine gun.
In fact, a common setup of the machine was to keep parts of the projector in a separate, adjoining room with only the aperture visible, to make it seem more magical and scare people.
The equivalence of the two definitions follows from the fact that an algorithm on such a non-deterministic machine consists of two phases, the first of which consists of a guess about the solution which is generated in a non-deterministic way, while the second consists of a deterministic algorithm which verifies or rejects the guess as a valid solution to the problem.
" The machine thus spoke for itself, and made known the fact that it was the phonograph ..."
In fact, many light machine guns ( such as the Russian RPK ) are adaptations of existing automatic rifle designs.
Despite the fact that the result of the next use would be displayed on the machine, both courts ruled that " The inducement for each play was the chance that by that play the machine would be set to indicate that it would pay checks on the following play.
There were potential patent infringements in FreeType 1 because parts of the TrueType hinting virtual machine were patented by Apple, a fact not mentioned in the TrueType standards.
They do not hide the fact that communication takes place, and as such do not attempt to present the cluster as a single parallel machine.
In fact, the time machine transported Havok to a parallel world, populated by twisted versions of Marvel characters.
In fact, the only real change made in most markets involved the video output frequency ( the ZX80 used an external power transformer, so differences in AC line frequency and outlet were not an issue to the machine itself ).
* it is recommended to use manufacturer's alignment target values to set up the machine train to a defined non-zero alignment, due to the fact that later when the machine is at operation temperature the alignment condition is perfect
The name combine is derived from the fact that the two steps are combined in a single machine.
His machine did, in fact, have a rotating brush like a traditional carpet sweeper, and also possessed an elaborate bellows mechanism on top of the body to generate suction of dust and dirt.
In fact, operational errors, chiefly in key choice, made the system less secure than it could have been ; in that way the Purple code shared the fate of the German Enigma machine.
The first production X-ray CT machine ( in fact called the " EMI-Scanner ") was limited to making tomographic sections of the brain, but acquired the image data in about 4 minutes ( scanning two adjacent slices ), and the computation time ( using a Data General Nova minicomputer ) was about 7 minutes per picture.
This, together with the fact that the motors were made by Polymotor in Italy, started the urban myth that the C5 was powered by a washing machine motor.

fact and only
And, as a matter of fact, Nicolas had slept in the park only part of one night, when he discovered that Munich's early mornings even in summer are laden with dew.
In fact it has caused us to give serious thought to moving our residence south, because it is not easy for the most objective Southerner to sit calmly by when his host is telling a roomful of people that the only way to deal with Southerners who oppose integration is to send in troops and shoot the bastards down.
Today's evidence, such as the fact that only three Southern states ( South Carolina, Alabama and Mississippi ) still openly defy integration, would have astounded many of yesterday's Southerners into speechlessness.
To my knowledge, Lincoln remains the only Head of State and Commander-in-Chief who, while fighting a fearful war whose issue was in doubt, proved man enough to say this publicly -- to give his foe the benefit of the fact that in all human truth there is some error, and in all our error, some truth.
If our sincerity is granted, and it is granted, the discrepancy can only be explained by the fact that we have come to believe hearsay and legend about ourselves in preference to an understanding gained by earnest self-examination.
In fact, I can only say this seems to me to follow from a wide, continuous, and properly guided exposure to literary art.
The only fact that holds any weight in the article is the result of the tea party.
I was drunk with excitement and the smell of gunpowder that came floating down from the road, and the fact that I was not afraid now, but only waiting to know what to do next.
One indication of the merits of the new management is found in the fact that during the period 1951-1956, while total annual mileage put on the vehicles increased 35%, the total maintenance cost increased only 11%.
As this year marks the centennial of the beginning of the Civil War, this fact is being commemorated with several exhibits throughout the State, but most of all paying tribute to the first Rhode Island Volunteers who rushed to the defense of the City of Washington, putting at the disposal of President Lincoln the only fully equipped and best trained regiment at this time.
The fact that the Deerstalker and the Jet were the only completely new designs this year doesn't mean that 1961 didn't see changes in models, actions and calibers.
The fact that even the larger letters weighed only 5 lb. each made it possible to secure the letters to the building through clear acrylic angle brackets cemented to the letters.
The photochemical reaction cells consisted of 10 mm. i.d. Pyrex tubing, 5.5 cm. long, diffraction effects being minimized by the fact that the light passed through only liquid-glass interfaces and not gas-glass interfaces.
In fact agglutination of Af cells in saline could be produced by the insoluble material from sera containing `` only '' incomplete antibody activity.
In the indirect method, this was evident from the fact that tumor sections were stained light green even when stained with NS and Af or with Af only.
Since the writer had not noticed this characteristic in married students scattered throughout the various sections previous to this experiment, nor, as a matter of fact, in those who were continuing in `` single sections '', he can only conclude that there must have been something `` contagious '' within the specific group which caused this to occur.
A state, the highest form of human organization in fact and theory, could be subjected to Law only by a manifestation of self-will, or consent.
The change was not quite so dramatic as it sounds because in fact common norms continued to be invoked by municipal courts and were only gradually changed by legislation, and then largely in marginal situations.
The fact seems to be that very many large branch stores are uneconomical, that the choice of location in the suburbs is as important as it was downtown, and that even highly suburbanized cities will support only so many big branches.
In fact, his only disciple -- the only person to imitate his style -- was W. S. Graham, who seems to have imitated him without much understanding, and who has since moved on to other methods.
But even more important than this is the fact that the direct search by simultaneously varying all operating conditions has produced only one optimal policy, namely, that for the given feed state and R stages.
This recommendation was based on the fact that the hymen was not the only barrier to smooth consummation of the sex act.
it is, in fact, the only college in the Ivy group.
This is a significant fact in Japan, for only a few years ago he would have had no choice.

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