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.

If F is algebraically closed and p ( x ) is an irreducible polynomial of F, then it has some root a and therefore p ( x ) is a multiple of x − a.

If denotes the quantum state of a particle ( n ) with momentum p, spin J whose component in the z-direction is σ, then one has

If the first allele is dominant to the second, then the fraction of the population that will show the dominant phenotype is p < sup > 2 </ sup > + 2pq, and the fraction with the recessive phenotype is q < sup > 2 </ sup >.

If T is a ( p, q )- tensor ( p for the contravariant vector and q for the covariant one ), then we define the divergence of T to be the ( p, q − 1 )- tensor

If f is a polynomial and p is big enough, then the remainder term vanishes.

If all outcomes x < sub > i </ sub > are equally likely ( that is, p < sub > 1 </ sub >

If we believe in this mental component and call it God, then we can say that we are small pieces of God's mental apparatus " ( p. 297 ).

This can be done for all m of the p < sub > i </ sub >, showing that m ≤ n. If there were any q < sub > j </ sub > left over we would have

Given a trigonometric series f ( x ) with S as its set of zeros, Cantor had discovered a procedure that produced another trigonometric series that had S ' as its set of zeros, where S ' is the set of limit points of S. If p ( 1 ) is the set of limit points of S, then he could construct a trigonometric series whose zeros are p ( 1 ).

* If f is an irreducible polynomial of prime degree p with rational coefficients and exactly two non-real roots, then the Galois group of f is the full symmetric group S < sub > p </ sub >.

If the field of scalars of the vector space has characteristic p, and if p divides the order of the group, then this is called modular representation theory ; this special case has very different properties.

If we compress data in a manner that assumes q ( X ) is the distribution underlying some data, when, in reality, p ( X ) is the correct distribution, the Kullback – Leibler divergence is the number of average additional bits per datum necessary for compression.

If Alice knows the true distribution p ( x ), while Bob believes ( has a prior ) that the distribution is q ( x ), then Bob will be more surprised than Alice, on average, upon seeing the value of X.

# If a and p are natural numbers such that a < sup > p </ sup > − 1 is prime, then a

If the circumstances are faced frankly it is not reasonable to expect this to be true.

If his dancers are sometimes made to look as if they might be creatures from Mars, this is consistent with his intention of placing them in the orbit of another world, a world in which they are freed of their pedestrian identities.

If a work is divided into several large segments, a last-minute drawing of random numbers may determine the order of the segments for any particular performance.

If they avoid the use of the pungent, outlawed four-letter word it is because it is taboo ; ;

If Wilhelm Reich is the Moses who has led them out of the Egypt of sexual slavery, Dylan Thomas is the poet who offers them the Dionysian dialectic of justification for their indulgence in liquor, marijuana, sex, and jazz.

If he is the child of nothingness, if he is the predestined victim of an age of atomic wars, then he will consult only his own organic needs and go beyond good and evil.

If it is an honest feeling, then why should she not yield to it??

If he thus achieves a lyrical, dreamlike, drugged intensity, he pays the price for his indulgence by producing work -- Allen Ginsberg's `` Howl '' is a striking example of this tendency -- that is disoriented, Dionysian but without depth and without Apollonian control.

If love reflects the nature of man, as Ortega Y Gasset believes, if the person in love betrays decisively what he is by his behavior in love, then the writers of the beat generation are creating a new literary genre.

If he is good, he may not be legal ; ;

If the man on the sidewalk is surprised at this question, it has served as an exclamation.

If the existent form is to be retained new factors that reinforce it must be introduced into the situation.

If we remove ourselves for a moment from our time and our infatuation with mental disease, isn't there something absurd about a hero in a novel who is defeated by his infantile neurosis??

If many of the characters in contemporary novels appear to be the bloodless relations of characters in a case history it is because the novelist is often forgetful today that those things that we call character manifest themselves in surface behavior, that the ego is still the executive agency of personality, and that all we know of personality must be discerned through the ego.

If he is a traditionalist, he is an eclectic traditionalist.

If our sincerity is granted, and it is granted, the discrepancy can only be explained by the fact that we have come to believe hearsay and legend about ourselves in preference to an understanding gained by earnest self-examination.

If to be innocent is to be helpless, then I had been -- as are we all -- helpless at the start.

:“ If an integer n is greater than 2, then has no solutions in non-zero integers a, b, and c. I have a truly marvelous proof of this proposition which this margin is too narrow to contain .”

If only the electric field () is non-zero, and is constant in time, the field is said to be an electrostatic field.

If all the non-zero bits were counted, then the intermediate result register now holds the final result.

If the pump waves and the signal wave are superimposed in a medium with a non-zero χ < sup >( 3 )</ sup >, this produces a nonlinear polarization field:

If I is a right ideal of R, then I is simple as a right module if and only if I is a minimal non-zero right ideal: If M is a non-zero proper submodule of I, then it is also a right ideal, so I is not minimal.

If I is a right ideal of R, then R / I is simple if and only if I is a maximal right ideal: If M is a non-zero proper submodule of R / I, then the preimage of M under the quotient map is a right ideal which is not equal to R and which properly contains I.

If M is a module which has a non-zero proper submodule N, then there is a short exact sequence

If N has a non-zero proper submodule, then this process can be repeated.

If h has a fixed ( non-zero ) value instead of approaching zero, then the right-hand side of the above equation would be written

If it is assumed that each player is only concerned with lessening his own time in jail, the game becomes a non-zero sum game where the two players may either assist or betray the other.

is conformal if and only if it is holomorphic and its derivative is everywhere non-zero on U. If f is antiholomorphic ( that is, the conjugate to a holomorphic function ), it still preserves angles, but it reverses their orientation.

If the Jacobian g has a non-zero determinant, then is " conformal in the generalized sense " with respect to one of the three angle types, depending on the real matrix expressed by the Jacobian g.

If n is non-zero, and is neither 1 nor − 1, by the fundamental theorem of arithmetic, there exist primes p < sub > i </ sub >, and positive integers e < sub > i </ sub >, with.

If R = K is a field, then a series is invertible if and only if the constant term is non-zero, i. e., if and only if the series is not divisible by X.

If R is an ordered field, we can order R (( G )) by setting any element to have the same sign as its leading coefficient, defined as the least element of the index set I associated to a non-zero coefficient.

If D is non-zero ( so for planes not through the origin ) the values for a, b and c can be calculated as follows:

If R and B were to have been used, these would have non-zero values even in a B & W scene, requiring all three data-carrying signals.

# If h ( t ) is a non-zero scalar function of t, then Z ( t )

If the address field is non-zero, it is a disk address of the block, which has previously been rolled out, so the block is fetched from disk and the pbit is set to 1 and the physical memory address updated to point to the block in memory ( another pbit ).

If a is algebraic over K, then there are many non-zero polynomials g ( x ) with coefficients in K such that g ( a ) = 0.