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 p is a non-zero real number, we can define the generalized mean with exponent p ( or power mean with exponent p ) of the positive real numbers as:

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 the target can change its position significantly during the 30 minutes the missile is in the air on its way, the probability of the missile destroying the target is drastically reduced.

`` If asked to choose between a terrible probability and a more terrible possibility, most men will choose the latter ''.

If the response variable is expected to follow a parametric family of probability distributions, then the statistician may specify ( in the protocol for the experiment or observational study ) that the responses be transformed to stabilize the variance.

If a defendant has been convicted and can prove that his lawyer did not adequately handle his case and that there is a reasonable probability that the result of the trial would have been different had the lawyer given competent representation, he is entitled to a new trial.

If there are g ( E ) dE states with energy E to E + dE, then the Boltzmann distribution predicts a probability distribution for the energy:

If this is not possible, probability proportionate to size sampling is used.

If the program hasn't halted yet, then it never will, since its contribution to the halting probability would affect the first N bits.

If a probability distribution has a density function f ( x ), then the mean is

If we think of the atoms as impenetrable steel discs and the impinging particle as a bullet of negligible diameter, this ratio is the probability that the bullet will strike a steel disc, i. e., that the atomic projectile will be stopped by the foil.

If the affected bits are independently chosen at random, the probability of a two-bit error being undetected is 1 / n.

If two molecules react, not only molecules of the final reaction products are formed, but also some unstable molecules, having the property of being able to further react with the parent molecules with a far larger probability than the initial reactants.

If a database cannot tolerate even such smaller probability of data loss, then the RAID group itself is replicated ( mirrored ).

If a database cannot tolerate even such smaller probability of data loss, then the RAID group itself is replicated ( mirrored ).

If it is " observed " ( measured with a photon ) not at a particular slit but rather at the screen, then there is no " which path " information as part of the interaction, so the electron's " observed " position on the screen is determined strictly by its probability function.

If multiple fermions have the same spatial probability distribution, then at least one property of each fermion, such as its spin, must be different.

If the coin is fair, then the probability of the next flip being heads is 1 / 2.

If one flips a coin 21 times in a row and obtains 21 heads, one might rationally conclude a high probability of bias towards heads, and hence conclude that future flips of this coin are also highly likely to be heads.

If these bits are known ahead of transmission ( to be a certain value with absolute probability ), logic dictates that no information has been transmitted.

If is the set of all messages that could be, and is the probability of given some, then the entropy of is defined:

If μ is an unknown constant and W is random noise with probability density, then has probability density and is therefore a location family ..

If a random experiment can result in N mutually exclusive and equally likely outcomes and if N < sub > A </ sub > of these outcomes result in the occurrence of the event A, the probability of A is defined by

If these relative frequencies are to define the probability, the probability will be slightly different every time it is measured.

If we acknowledge the fact that we only can measure a probability with some error of measurement attached, we still get into problems as the error of measurement can only be expressed as a probability, the very concept we are trying to define.