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P and .
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.
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.
Ten students act as managers for a high-school football team, and of these managers a proportion P are licensed drivers.
The word `` binomial '' means `` of two names '' or `` of two terms '', and both usages apply in our work: the first to the names of the two outcomes of a binomial trial, and the second to the terms P and Af that represent the probabilities of `` success '' and `` failure ''.
Obviously, very slight changes in P do not change the probabilities much, and a slight lack of independence may not make an appreciable difference.
Therefore each binomial trial can be thought of as producing a value of a random variable associated with that trial and taking the values 0 and 1, with probabilities Q and P respectively.
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.
As S increases we consider the two free corner points of the square, Af and Af, adjacent to P and Q respectively.
We erect a square with PQ as a side and with free corners Af and Af adjacent to P and Q respectively.
Now with P fixed at Af, Af-values occur when the corner Af crosses C, and are among the values of S such that Af.
Moreover, if Af and Af are two planes intersecting in a line l, tangent to Q at a point P, the two free intersections of the image curves Af and Af must coincide at P', the image of P, and at this point Af and Af must have a common tangent l'.
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.
P and J
The Tycoons: How Andrew Carnegie, John D. Rockefeller, Jay Gould, and J. P. Morgan Invented the American Supereconomy.
The current limited partners are Ronald E. Canakaris ; Warrick Dunn ; Douglas J. Hertz ; John P. Imlay, Jr .; Ed Mendel ; Derek V. Smith ; John A. Williams ; and Brian J. Barker.
J. P. de Florian's Gonsalve de Cordoue and Châteaubriand's Le dernier des Abencerrages are imitations of Perez de Hita's work.
* the Ligurian substrate hypothesis proposed in the 19th century by d ' Arbois de Joubainville, J. Pokorny, P. Kretschmer and several other linguists encompasses the Basco-Iberian hypothesis.
According to J. P. Mallory, in his 1989 book In Search of the Indo-Europeans, the hypothesis was also inspired by a Basque place-name ending in-adze.
The bundle theory of substance thus rejects the substance theories of Aristotle, Descartes, and more recently, J. P. Moreland, Jia Hou, Joseph Bridgman, Quentin Smith, and others.
* Sterling Quinlan, The Hundred Million Dollar Lunch ( Chicago, J. P. O ' Hara, 1974 ), ISBN 0-87955-310-3.
* Sweet, J. P. M., ( 1979, Updated 1990 ) Revelation, London: SCM Press, and Philadelphia: Trinity Press International.
Holly was offered a spot in the Winter Dance Party, a three-week tour across the Midwest opening on January 23, 1959, by the GAC agency, with other notable performers such as Dion and the Belmonts, Ritchie Valens, and J. P. " The Big Bopper " Richardson.
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