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.

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.

This latter construal is sometimes expressed by saying " there is no fact of the matter as to whether or not P ." Thus, we may speak of anti-realism with respect to other minds, the past, the future, universals, mathematical entities ( such as natural numbers ), moral categories, the material world, or even thought.

Previous attempts have included Piece Of My Heart, which was to star Renée Zellweger or Brittany Murphy ; The Gospel According To Janis, with director Penelope Spheeris and starring either Zooey Deschanel or P! nk ; and an untitled film thought to be an adaptation of Laura Joplin's Off-Broadway play about her sister, with the show's star, Laura Theodore, attached.

In 1989, von Trier's mother revealed on her deathbed that the man who he thought was his father was not, and that she had had a tryst with her former employer, Fritz Michael Hartmann ( 1909 – 2000 ), who descended from a long line of Roman Catholic classical musicians ( his grandfather was Emil Hartmann, his great grandfather J. P. E.

Just as the class P can be thought of as the tractable problems ( Cobham's thesis ), so NC can be thought of as the problems that can be efficiently solved on a parallel computer.

* For essays exploring the relation of process thought to Wesleyan theology, see Bryan P. Stone and Thomas Jay Oord, Thy Nature and Thy Name is Love: Wesleyan and Process Theologies in Dialogue ( Nashville: Kingswood, 2001, ISBN 0-687-05220-3 ).

However, it was shown by Bennett that the type specimen later designated for Diopecephalus was a fossil belonging to P. kochi, and no longer thought to be separate from Pterodactylus.

In the end, Chief Justice Salmon P. Chase, who presided over the trial, was thought to have brought about Johnson's acquittal by the Senate with his statements from the bench.

There are some problems in both NP and co-NP, for example all problems in P or integer factorization, however it is not known if the sets are equal, although inequality is thought more likely.

Finding an efficient way to parallelize the solution to some P-complete problem would show that NC = P. It can also be thought of as the " problems requiring superlogarithmic space "; a log-space solution to a P-complete problem ( using the definition based on log-space reductions ) would imply L = P.

Maack's " Cape Kyrma " site ( thought by B. P.

As to the Cape Kyrma ruins, thought by Maack to be the remains of Achansk, B. P.

Hubble Space Telescope | Hubble's Wide Field Camera 3 clearly shows the slow evolution of the debris coming from asteroid P / 2010 A2, which is thought to be due to a collision with a smaller asteroid.

* Garagin, M., P. Woodruff, Early Greek Political thought from Homer to the Sophists, Cambridge 1995.

* Garagin, M., P. Woodruff, Early Greek Political thought from Homer to the Sophists, Cambridge 1995.

The other operators listed above can be expressed in similar ways ; for example, the universal quantifier can be thought of as an operator that evaluates to the logical conjunction of the boolean-valued function P applied over the ( possibly infinite ) set S.

The New Testament parables are thought by scholars such as John P. Meier to have been inspired by mashalim, a form of Hebrew comparison.

In an anisotropic medium, such as a crystal, the polarisation field P is not necessarily aligned with the electric field of the light E. In a physical picture, this can be thought of as the dipoles induced in the medium by the electric field having certain preferred directions, related to the physical structure of the crystal.

Daniel P. Todes, in his account of Russian naturalism in the 19th century, concludes that Kropotkin ’ s work " cannot be dismissed as the idiosyncratic product of an anarchist dabbling in biology " and that his views " were but one expression of a broad current in Russian evolutionary thought that pre-dated, indeed encouraged, his work on the subject and was by no means confined to leftist thinkers.

It is thought that ŭ was created by analogy with the Belarusian letter ў ( Cyrillic u with breve ), which was proposed by P. A.

The origin of P. × hybrida is thought to be by hybridisation between P. axillaris ( the large white or night-scented petunia ) and P. integrifolia ( the violet-flowered petunia ).

The first possibility requires that there be a line through P which meets **zg in Af points ; ;

the second requires that there be a line through P which meets **zg in Af points.

The photocathode sensitivities S, phosphor efficiencies P, and anode potentials V of the individual stages shall be distinguished by means of subscripts 1, and 2, in the text, where required.

In general, P is a function of V and the current density, but it shall here be assumed as a constant.

The policy may not be unique but the maximum value of P certainly is, and once the policy is specified this maximum can be calculated by ( 2 ) and ( 3 ) as a function of the feed state Af.

Quantum states of a particle and an antiparticle can be interchanged by applying the charge conjugation ( C ), parity ( P ), and time reversal ( T ) operators.

Let P be the root of the unbalanced subtree, with R and L denoting the right and left children of P respectively.

* If the balance factor of P is-2 then the right subtree outweighs the left subtree of the given node, and the balance factor of the right child ( R ) must be checked.

* If the balance factor of P is 2, then the left subtree outweighs the right subtree of the given node, and the balance factor of the left child ( L ) must be checked.

In addition to the balancing described above for insertions, if the balance factor for the tree is 2 and that of the left subtree is 0, a right rotation must be performed on P. The mirror of this case is also necessary.

In particular, we cannot in general claim that " P or not P " is true ( the law of Excluded Middle ), since in some cases we may not be able to prove the statement " P " nor prove the statement " not P ".

For example, P might be the circle with radius 1 and center ( 0, 0 ): P =

More technically, the law is concerned with the speedup achievable from an improvement to a computation that affects a proportion P of that computation where the improvement has a speedup of S. ( For example, if an improvement can speed up 30 % of the computation, P will be 0. 3 ; if the improvement makes the portion affected twice as fast, S will be 2.