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This is in P, since an XOR-SAT formula is a system of linear equations mod 2, and can be solved by Gaussian elimination.
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is and P
If Af is the change per unit volume in Gibbs function caused by the shear field at constant P and T, and **yr is the density of the fluid, then the total potential energy of the system above the reference height is Af.
It is clear that each vector in the range of Af is in Af for if **ya is in the range of Af, then Af and so Af because Af is divisible by the minimal polynomial P.
Thus Af is divisible by the minimal polynomial P of T, i.e., Af divides Af.
In the primary decomposition theorem, it is not necessary that the vector space V be finite dimensional, nor is it necessary for parts ( A ) and ( B ) that P be the minimal polynomial for T.
If T is a linear operator on an arbitrary vector space and if there is a monic polynomial P such that Af, then parts ( A ) and ( B ) of Theorem 12 are valid for T with the proof which we gave.
If D denotes the differentiation operator and P is the polynomial Af then V is the null space of the operator p (, ), because Af simply says Af.
On C, from the point P at Af to the point Q at Af, we construct the chord, and upon the chord as a side erect a square in such a way that as S approaches zero the square is inside C.
But if this is the case, then an arbitrary pencil of lines having a point, P, of **zg as vertex is transformed into a ruled surface of order Af having Af generators concurrent at P.
Since a ruled surface of order N with N concurrent generators is necessarily a cone, it follows finally that every line through a point, P, of **zg meets its image at P, as asserted.
Now consider the transformation of the lines of a bundle with vertex, P, on **zg which is effected by the involution as a whole.
Moreover, in this involution there is a cone of invariant lines of order Af, namely the cone of secants of **zg which pass through P.
Af is the friction force between chip and knife surfaces, and P is the normal force acting on the face of the knife.
is and since
Even the knowledge that she was losing another boy, as a mother always does when a marriage is made, did not prevent her from having the first carefree, dreamless sleep that she had known since they dropped down the canyon and into Bear Valley, way, way back there when they were crossing those other mountains.
since Bourbon whiskey, though of Kentucky origin, is at least as much favored by liberals in the North as by conservatives in the South.
But what a super-Herculean task it is to winnow anything of value from the mud-beplastered arguments used so freely, particularly since such common use is made of cliches and stereotypes, in themselves declarations of intellectual bankruptcy.
Its massive contours are rooted in the simple need of man, since he is always incomplete, to complete himself.
The young William Faulkner in New Orleans in the 1920's impressed the novelist Hamilton Basso as obviously conscious of being a Southerner, and there is no evidence that since then he has ever considered himself any less so.
Without saying or seeming to say that in portraying the Sartoris and the Compson families Faulkner's chief concern is social criticism, we can say nevertheless that through those families he dramatizes his comment on the planter dynasties as they have existed since the decades before the Civil War.
Circular motion, however, since it is eternal and perfectly continuous, lacks termini.
At the same time, I am aware that my recoil could be interpreted by readers of the tea leaves at the bottom of my psyche as an incestuous sign, since theirs is a science of paradox: if one hates, they say it is because one loves ; ;
But when these expectations are once too often ground into the dust, innocence can falter, since its strength is according to the strength of him who possesses it.
It is true that this distinction between style and idea often approaches the arbitrary since in the end we must admit that style and content frequently influence or interpenetrate one another and sometimes appear as expressions of the same insight.
It has been a long time since he has seen any campaign money, and when the proposition is laid down to him as the friends of Mr. Hearst are laying it down these days he is quite likely to get aboard the Hearst bandwagon ''.
One's daily work becomes sacred, since it is performed in the field of influence of the moral law, dealing as it does with people as well as with matter and energy.
My argument is that there was no Saxon Shore prior to that time even though the forts had been in existence since the time of Carausius.
This, naturally, will be difficult to do since both the archaeological and place-name evidence in this period, with some fortunate exceptions, is insufficient for precise chronological purposes.
I would, however, like to suggest that, wrong though I may be, the tendency to see dilemmas rather than solutions is one of which I have been a victim ever since I can remember, and therefore not merely a senile phenomenon.
But since last fall the United States has been moving toward a pro-neutralist position and now is ready to back the British plan for a cease-fire patrolled by outside observers and followed by a conference of interested powers.
`` The cannery '', said Mrs. Lewellyn Lundeen, an active booster of the cannery since its opening during the war and rationing years of 1941, to handle the `` victory garden '' produce, `` is a service to the taxpayer.
Morrison points out that since our country is more urbanized than the Soviet Union or Red China, it is the most vulnerable of the great powers -- Europe of course must be written off out of hand.
Perhaps Khrushchev is in a more difficult position than any since 1957, when the `` anti-party group '' nearly liquidated him.
It is vitally important that the new U.S. aid program should encourage all of them, since the main thrust for development must come from the less developed countries themselves.
Mr. Sulzberger's successor as publisher is Mr. Orvil E. Dryfoos, who is president of the New York Times Co., and who has been with the Times since 1942.
is and formula
Actually, only two men know what the formula is, Blake and '' -- He stopped and looked at Thor's body.
First, the present situation is too varied to be systematized by any single formula.
These factors practically insure that no single financing formula is feasible ; ;
As is the case with the allotment provisions for support of vocational rehabilitation services, the matching requirements are also based on a statutory formula.
A mathematical formula is nothing more than a pattern for solving a specific problem.
The first step in using a formula is to insert the numerical values of the factors involved in their correct positions in the formula.
A constant is a number that remains the same regardless of the other numbers used in the formula and the resultant equation.
When it is needed, one formula is as important as another.
Actually, the engine displacement formula is the standard formula for computing the volume of a cylinder of any type with an added factor that represents the number of cylinders in the engine.
The formula is: Af.
Cylinder volume is determined in exactly the same manner as for the displacement formula: Af.
The only precaution is that all volumes used in the formula be quoted in the same terms.
The formula for this step is: Af.
When tire size is measured in this manner a constant of 168 is used in the formula.
To determine car speed for a given combination of engine speed, gear ratio, and tire size, the formula is: Af.
To determine engine speed for a given combination of the other three factors the formula is: Af.
To determine the rear axle gear ratio for a combination of the other three factors, the formula is: Af.
This formula is given by Rumscheidt and Mason.
If it is assumed that the formula given by Lodge of cosec Af applies, the pressure difference along the major axes can be calculated from the angle of inclination of the major axis, and from this the interfacial tension can be calculated.
Since the complex of singular lines is of order K and since there is no complex of invariant lines, it follows from the formula Af that the order of the involution is Af.
0.086 seconds.