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.

Mary Jane Lerner knew none of this.

Alan Jay Lerner ( August 31, 1918 – June 14, 1986 ) was an American lyricist and librettist.

Born in New York City, he was the son of Edith Adelson Lerner and Joseph Jay Lerner, whose brother, Samuel Alexander Lerner, was founder and owner of the Lerner Stores, a chain of dress shops.

Alan Jay Lerner was educated at Bedales School in England, The Choate School ( now Choate Rosemary Hall ) in Wallingford, Connecticut, ( where he wrote " The Choate Marching Song ") and Harvard.

At both Choate and Harvard, Lerner was a classmate of John F. Kennedy ; at Choate they had worked together on the yearbook staff.

During the summers of 1936 and 1937, Lerner studied at Juilliard.

In 1957, Lerner and Leonard Bernstein, another of Lerner's college classmates, collaborated on " Lonely Men of Harvard ," a tongue-in-cheek salute to their alma mater.

Due to his injury, Lerner could not serve in World War II.

While at the Lamb's, he came upon Lorenz Hart, and he helped transform Lerner into his protege.

Lerner and Loewe's first collaboration was a musical adaptation of Barry Conners's farce The Patsy called Life of the Party for a Detroit stock company.

In that same year Lerner also wrote the Oscar-winning original screenplay for An American in Paris, produced by Arthur Freed and directed by Vincente Minnelli.

This was the same team who would later join with Lerner and Loewe to create Gigi.

In 1956, Lerner and Loewe unveiled My Fair Lady.

Before finishing the musical, Lerner was eager to write while My Fair Lady was taking so long to complete.

Burton Lane and Lerner were working on a musical about Li ' l Abner.

Lerner and Loewe's run of success continued with their next project, a film adaptation of stories from Colette, the Academy Award winning film musical Gigi, starring Leslie Caron, Louis Jourdan and Maurice Chevalier.

Lerner was hospitalized with bleeding ulcers while Loewe continued to have heart troubles.

Camelot was a hit nonetheless, with a poignant coda ; immediately following the assassination of John F. Kennedy, his widow told Life magazine that JFK's administration reminded her of the " one brief shining moment " of Lerner and Loewe's Camelot.

In 1965 Lerner collaborated again with Burton Lane on the musical On a Clear Day You Can See Forever, which was adapted for film in 1970.

At this time, Lerner was hired by film producer Arthur P. Jacobs to write a treatment for an upcoming film project, Doctor Dolittle, but Lerner abrogated his contract after several non-productive months of non-communicative procrastination and was replaced with Leslie Bricusse.