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* P. N. Mathur.
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P and .
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.
P and N
The choice of P and N determines the binomial distribution uniquely, and different choices always produce different distributions ( except when Af ; ;
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.
* 2012 – N. K. P. Salve, Indian politician ( b. 1921 )
In number theory, if P ( n ) is a property of positive integers, and if p ( N ) denotes the number of positive integers n less than N for which P ( n ) holds, and if
* Slightly more generally, given four sets M, N, P and Q, with h: M to N, g: N to P, and f: P to Q, then
* Coffin, Tristram P .; Cohen, Hennig, ( editors ), Folklore in America ; tales, songs, superstitions, proverbs, riddles, games, folk drama and folk festivals, Garden City, N. Y.: Doubleday, 1966.
In the case of parallelization, Amdahl's law states that if P is the proportion of a program that can be made parallel ( i. e., benefit from parallelization ), and ( 1 − P ) is the proportion that cannot be parallelized ( remains serial ), then the maximum speedup that can be achieved by using N processors is
In the limit, as N tends to infinity, the maximum speedup tends to 1 / ( 1 − P ).
In practice, performance to price ratio falls rapidly as N is increased once there is even a small component of ( 1 − P ).
P and Mathur
* Mathur, P. C., 1995.
* Mathur, D. P., B. Rangarajan, V. Gupta.
. and N
The completeness of the connections provide that, for N people, there are Af lines of communication between the pairs, which can become a large number ( 1,225 ) for a party of fifty guests.
In spite of the dreams of the host for oneness in the group, the Af incoming messages for each guest overload his receiving system beyond comprehension if N exceeds about six.
Such items recall the California journalist who reported an accident involving a movie star: `` The area in which Miss N -- was injured is spectacularly scenic ''.
For example, to move ( as the score requires ) from the lowest F-major register up to a barely audible N minor in four seconds, not skipping, at the same time, even one of the 407 fingerings, seems a feat too absurd to consider, and it is to the flautist's credit that he remained silent throughout the passage.
The approximate equation is Af, where N is the number of Af with electron line-density greater than or equal to Af, and Q is proportional to the mass of the meteorite.
Therefore, N is inversely proportional to the radius cubed and in fair agreement with the inverse 7/2 power derived from 1958 Alpha and 1959 Eta data.
Let N be a linear operator on the vector space V.
We say that N is nilpotent if there is some positive integer R such that Af.
Then there is a diagonalizable operator D on V and a nilpotent operator N in V such that ( A ) Af, ( b ) Af.
The diagonalizable operator D and the nilpotent operator N are uniquely determined by ( A ) and ( B ) and each of them is a polynomial in T.
We have just observed that we can write Af where D is diagonalizable and N is nilpotent, and where D and N not only commute but are polynomials in T.
hence they commute with D and with N.
for, using the fact that N and N' commute Af and so when R is sufficiently large every term in this expression for Af will be 0.
Then every linear operator T in V can be written as the sum of a diagonalizable operator D and a nilpotent operator N which commute.
These operators D and N are unique and each is a polynomial in T.
Let N be a positive integer and let V be the space of all N times continuously differentiable functions F on the real line which satisfy the differential equation Af where Af are some fixed constants.
If Af denotes the space of N times continuously differentiable functions, then the space V of solutions of this differential equation is a subspace of Af.
Each of the N trials is either a success or a failure.
Each performance of an n-trial binomial experiment results in some whole number from 0 through N as the value of the random variable X, where Af.
Some predictions had been made concerning factors R, N, I and Co on these inventories which appeared to be directly related to control and security aspects of personality functioning which were hypothesized as being of importance in differential Kohnstamm reactivity.
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