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If N is a prime number, then the set of non-zero indices n = 1 ,..., N – 1 forms a group under multiplication modulo N. One consequence of the number theory of such groups is that there exists a generator of the group ( sometimes called a primitive root ), an integer g such that n = g < sup > q </ sup > ( mod N ) for any non-zero index n and for a unique q in 0 ,..., N – 2 ( forming a bijection from q to non-zero n ).
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If and N
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.
** If the set A is infinite, then there exists an injection from the natural numbers N to A ( see Dedekind infinite ).
Let ( m, n ) be a pair of amicable numbers with m < n, and write m = gM and n = gN where g is the greatest common divisor of m and n. If M and N are both coprime to g and square free then the pair ( m, n ) is said to be regular, otherwise it is called irregular or exotic.
If ( m, n ) is regular and M and N have i and j prime factors respectively, then ( m, n ) is said to be of type ( i, j ).
If A is expressed as an N × N matrix, then A < sup >†</ sup > is its conjugate transpose.
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 S is an arbitrary set, then the set S < sup > N </ sup > of all sequences in S becomes a complete metric space if we define the distance between the sequences ( x < sub > n </ sub >) and ( y < sub > n </ sub >) to be, where N is the smallest index for which x < sub > N </ sub > is distinct from y < sub > N </ sub >, or 0 if there is no such index.
If the expression that defines the DFT is evaluated for all integers k instead of just for, then the resulting infinite sequence is a periodic extension of the DFT, periodic with period N.
If a filter is implemented using, for instance, biquad stages using op-amps, N / 2 stages will be needed.
# If the remainder from dividing N by 6 is not 2 or 3 then the list is simply all even numbers followed by all odd numbers ≤ N
If f: M → N is any function, then we have f id < sub > M </ sub >
If we neglect the possibility of overlapping states, which is valid if the temperature is high, then the number of times we count each state is approximately N < nowiki >!</ nowiki >.
If the molecule is symmetrical, e. g. N < sub > 2 </ sub >, the band is not observed in the IR spectrum, but only in the Raman spectrum.
# If M, N ∈ Λ, then ( M N ) ∈ Λ
If the alphabet consists of alternative symbols, each symbol represents a message consisting of N bits.
If μ is a positive measure, then N is null ( or zero measure ) if its measure μ ( N ) is zero.
If μ is not a positive measure, then N is μ-null if N is | μ |- null, where | μ | is the total variation of μ ; equivalently, if every measurable subset A of N satisfies μ ( A )
If and is
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 and prime
If Mahzeer was alone with the prime minister he could be arranging his execution while Hoag stood out here shivering in the darkening street.
If Mahzeer was planning to set up the prime minister for Muller he would have to do it in the next few minutes.
** If S is a set of sentences of first-order logic and B is a consistent subset of S, then B is included in a set that is maximal among consistent subsets of S. The special case where S is the set of all first-order sentences in a given signature is weaker, equivalent to the Boolean prime ideal theorem ; see the section " Weaker forms " below.
From the criterion it also follows that Carmichael numbers are cyclic .< ref > Proof sketch: If is square-free but not cyclic, for two prime factors and of.
If the rational number's denominator has any prime factors other than 2 or 5, it cannot be expressed as a finite decimal fraction, and has a unique eventually repeating infinite decimal expansion.
If GCD ( a, b ) = 1, then a and b are said to be coprime ( or relatively prime ).
If two numbers have no prime factors in common, their greatest common divisor is 1 ( obtained here as an instance of the empty product ), in other words they are coprime.
Assume it is true for all numbers less than n. If n is prime, there is nothing more to prove.
If Matthew's prime concern was to preserve the Jewish character of the church, he failed: Christianity became a Gentile religion, and Christianity and Judaism came to view each other as opposites.
* If q is a prime power, and if F
* 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 a and p are natural numbers such that a < sup > p </ sup > − 1 is prime, then a
#* Proof: Then so Thus However, is prime, so or In the former case, hence ( which is a contradiction, as neither 1 nor 0 is prime ) or In the latter case, or If however, which is not prime.
# If 2 < sup > p </ sup >-1 is prime, then p is prime.
# If p is an odd prime, then any prime q that divides 2 < sup > p </ sup > − 1 must be 1 plus a multiple of 2p.
# If p is an odd prime, then any prime q that divides must be congruent to ± 1 ( mod 8 ).
If a government cannot get its appropriation ( budget ) legislation passed by the House of Representatives, or the house passes a vote of " no confidence " in the government, the prime minister is bound by convention to resign immediately.
If, however, an opposition party wins a majority of seats in the House of Commons, the prime minister may resign or be dismissed by the governor general.
If a person elected prime minister is neither a member of the Lok Sabha nor Rajya Sabha, then he must become a member of the Lok Sabha or Rajya Sabha within six months.
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