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If either p < sub > n </ sub ># + 1 or p < sub > n </ sub ># − 1 is a primorial prime, it means that there are larger primes than the nth prime ( if neither is a prime, that also proves the infinitude of primes, but less directly ; note that each of these two numbers has a remainder of either p − 1 or 1 when divided by any of the first n primes, and hence cannot be a multiple of any of them ).
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If and either
If Mrs. Wright doesn't accept the terms in the morning, I'll go either to Tokyo or to Holland, to do what I can.
If it proclaims that the best is yet to be, it always arouses, at least in the young, either a suspicious question or perhaps the exclamation of the Negro youth who saw on a tombstone the inscription, `` I am not dead but sleeping ''.
If either one ever started making promises, there is no telling where the promises would end.
If the UN troops are not mercenaries then the Hessians were not mercenaries either.
If the intruder persisted, a biting lunge was usually launched at either the tail region or the naso-labial grooves.
** Trichotomy: If two sets are given, then either they have the same cardinality, or one has a smaller cardinality than the other.
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* Continuity: If a certain mass of fluid enters a volume, it must either exit the volume or change the mass inside the volume.
If not, the system may either be rejected or accepted on conditions previously agreed between the sponsor and the manufacturer.
If X is a Banach space and K is the underlying field ( either the real or the complex numbers ), then K is itself a Banach space ( using the absolute value as norm ) and we can define the continuous dual space as X ′ = B ( X, K ), the space of continuous linear maps into K.
If the Allies were not to be outnumbered on the Danube, Eugene realised he must either try to cut Tallard off before he could get there, or, he must hasten to reinforce Marlborough.
If given subcutaneously, it may induce local infection and spread to the regional lymph nodes, causing either suppurative and nonsuppurative lymphadenitis.
If players cannot move either die in a roll, given the position of their checkers then that turn is over and the turn passes to the opponent.
If either one die or the other but not both can be moved, the higher must be used.
" If both opponents roll the same opening number, the doubling cube is incremented on each occasion yet remains in the middle of the board, available to either player.
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If the player to move has no legal moves, the game is over ; it is either a checkmate — if the king is under attack — or a stalemate — if the king is not.
In the words of Henry Chadwick, " If the Consolation contains nothing distinctively Christian, it is also relevant that it contains nothing specifically pagan either ... is a work written by a Platonist who is also a Christian, but is not a Christian work.
If two sets are played, the pieces can either go into the opponent's starting corners, or one of the players ' two sets can go into an opposite empty corner.
If the player either hits the ball through the correct hoop (" runs " the hoop ), or hits another ball ( a " roquet "), the turn continues.
* If the offence fumbles ( a ball carrier drops the football, or has it dislodged by an opponent, or if the intended player fails to catch a lateral pass or a snap from centre, or a kick attempt is blocked by an opponent ), the ball may be recovered ( and advanced ) by either team.
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If and p
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
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