Add holes in top, forming `` S '' for salt and `` P '' for pepper.

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.

Since Af and P divides Af for Af, we have 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.

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 ''.

The constancy of P and the independence are the conditions most likely to give trouble in practice.

Obviously, very slight changes in P do not change the probabilities much, and a slight lack of independence may not make an appreciable difference.

We shall find a formula for the probability of exactly X successes for given values of P and N.

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 again consider a fixed point P at Af and a variable point Q at Af on C.

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'.

To do this we must first show that every line which meets **zg in a point P meets its image at P.

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.

* Every quadratic Bézier curve is also a cubic Bézier curve, and more generally, every degree n Bézier curve is also a degree m curve for any m > n. In detail, a degree n curve with control points P < sub > 0 </ sub >, …, P < sub > n </ sub > is equivalent ( including the parametrization ) to the degree n + 1 curve with control points P '< sub > 0 </ sub >, …, P '< sub > n + 1 </ sub >, where.

With the reduced state variables, i. e. V < sub > r </ sub >= V < sub > m </ sub >/ V < sub > c </ sub >, P < sub > r </ sub >= P / P < sub > c </ sub > and T < sub > r </ sub >= T / T < sub > c </ sub >, the reduced form of the Van der Waals equation can be formulated:

The formula provides the GoS ( grade of service ) which is the probability P < sub > b </ sub > that a new call arriving at the circuit group is rejected because all servers ( circuits ) are busy: B ( E, m ) when E Erlang of traffic are offered to m trunks ( communication channels ).

Like all subatomic particles, hadrons are assigned quantum numbers corresponding to the representations of the Poincaré group: J < sup > PC </ sup >( m ), where J is the spin quantum number, P the intrinsic parity ( or P-parity ), and C, the charge conjugation ( or C-parity ), and the particle's mass, m. Note that the mass of a hadron has very little to do with the mass of its valence quarks ; rather, due to mass – energy equivalence, most of the mass comes from the large amount of energy associated with the strong interaction.

Outside the SI system, a variety of different mass units are used, depending on context, such as the slug ( sl ), the pound ( lb ), the Planck mass ( m < sub > P </ sub >), and the solar mass.

A number a is a root of P if and only if the polynomial x − a ( of degree one in x ) divides P. It may happen that x − a divides P more than once: if ( x − a )< sup > 2 </ sup > divides P then a is called a multiple root of P, and otherwise a is called a simple root of P. If P is a nonzero polynomial, there is a highest power m such that ( x − a )< sup > m </ sup > divides P, which is called the multiplicity of the root a in P. When P is the zero polynomial, the corresponding polynomial equation is trivial, and this case is usually excluded when considering roots: with the above definitions every number would be a root of the zero polynomial, with undefined ( or infinite ) multiplicity.

* P. v. karelini, which breeds in Italy, south-east Europe south from southern Yugoslavia, Bulgaria, Asia Minor, northern Iran and south-west Turkmenistan.

* P. v. sharpei, which breeds in the Iberia north to the Spanish side of the western and central Pyrenees and south Roussillon, has upperparts like karelini but black on the head is limited to the moustachial stripe and lores ; the red stripe on males has only a narrow black border

* P. v. innominatus, which breeds in south-west and southern Iran, is like karelini but with nearly white cheeks, throat and chest and more defined barring on the tail.

P. m. blandfordi is like the nominate but with a greyer mantle and scapulars and pale yellow underparts, and P. m. karelini is intermediate between the nominate and P. m. blandfordi, and lacks white on the tail.

" Historian Donald described the speech as a " superb political move for an unannounced candidate, to appear in one rival's ( William H. Seward ) own state at an event sponsored by the second rival's ( Salmon P. Chase ) loyalists, while not mentioning either by name during its delivery.

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.

Robert M. Price described in his essay " H. P. Lovecraft and the Cthulhu Mythos " two stages in the development of the Cthulhu Mythos.

In a letter to P. Schuyler Miller and John D. Clark in 1936 only three months before Howard's death, Conan is described as standing and weighing when he takes part in an attack on Venarium at only 15 years old, though being far from fully grown.

Van Deinse points out that the " scrag whale ", described by P. Dudley in 1725 as one of the species hunted by the early New England whalers, was almost certainly the gray whale.

Van Deinse points out the " scrag whale ", described by P. Dudley in 1725, as one target of early New England whalers, was almost certainly the gray whale.

Geometric arrangement for Fresnel's calculation Consider the case of a point source located at a point P < sub > 0 </ sub >, vibrating at a frequency f. The disturbance may be described by a complex variable U < sub > 0 </ sub > known as the complex amplitude.

Regarding the CIA's refusal to inform the FBI about Mihdhar and Hazmi, author Lawrence Wright suggests the CIA wanted to protect its turf and was concerned about giving sensitive intelligence to FBI Agent John P. O ' Neill, who Alec Station chief Michael Scheuer described as duplicitous.

The improvements made by Mullis allowed PCR to become a central technique in biochemistry and molecular biology, described by The New York Times as " highly original and significant, virtually dividing biology into the two epochs of before P. C. R.

Auffenberg described the Komodo dragon as having septic pathogens in its saliva ( he described the saliva as " reddish and copious "), specifically the bacteria: E. coli, Staphylococcus sp., Providencia sp., Proteus morgani and P. mirabilis.

Parametric and lossy ' instantaneous ' ( i. e. electronic ) nonlinear optical phenomena, in which the optical fields are not too large, can be described by a Taylor series expansion of the dielectric Polarization density ( dipole moment per unit volume ) P ( t ) at time t in terms of the electrical field E ( t ):

In probability theory, the probability P of some event E, denoted, is usually defined in such a way that P satisfies the Kolmogorov axioms, named after the famous Russian mathematician Andrey Kolmogorov, which are described below.

Eduard Spranger's personality-model, consisting of six ( or, by some revisions, 6 + 1 ) basic types of value attitudes, described in his book Types of Men ( Lebensformen ; Halle ( Saale ): Niemeyer, 1914 ; English translation by P. J. W. Pigors-New York: G. E. Stechert Company, 1928 ).

These three species are very highly matrotrophic-so much so that in 1947, C. L. Turner described the follicular cells of P. turneri as " pseudo-placenta, pseudo-chorion, and pseudo-allantois ".

P. J. Plauger, an early adopter of structured programming, described his reaction to the structured program theorem:

Terms such as " stolen " were used in the context of taking children from their families – the Hon P. McGarry, a member of the Parliament of New South Wales, objected to the Aborigines Protection Amending Act 1915 which then enabled the Aborigines ' Protection Board to remove Aboriginal children from their parents without having to establish that they were in any way neglected or mistreated ; McGarry described the policy as " steal the child away from its parents ".

The Abduction of Figaro is a comic opera, described as " A Simply Grand Opera by P. D. Q. Bach ," which is actually the work of composer Peter Schickele.

For FM radio and TV broadcasting the path loss is most commonly predicted using the ITU model as described in P. 1546 ( former P. 370 ) recommendation.

* If X can be described as P and Y can be described as P, as well, then the sum of X and Y can also be described as P.