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 the Council accepts these amendments then the legislation is approved.

If the Supreme Court grants certiorari and accepts the case, it will receive written briefs from each side ( and any amici curiae or friends of the court — usually interested third parties with some expertise to bear on the subject ) and schedule oral arguments.

If one accepts that Luke was in fact the author of the Gospel bearing his name and also the Acts of the Apostles, certain details of his personal life can be reasonably assumed.

If any branch of the tree halts with an " accept " condition, we say that the NTM accepts the input.

If the defendant accepts these suggestions and changes his penalty proposition, the court approves it and passes the verdict according to the plea agreement.

If the state accepts them ( or fails to act at all ), both the reserving state and the accepting state are relieved of the reserved legal obligation as concerns their legal obligations to each other ( accepting the reservation does not change the accepting state's legal obligations as concerns other parties to the treaty ).

If the accused accepts NJP, he or she, plus a representative if desired, will attend the hearing conducted by the commander.

If the first person accepts the hypothesis that " All ravens are black " then, according to the argument, about 50 objects whose colors were previously in doubt ( the ravens ) are now thought to be black, while nothing different is thought about the remaining objects ( the non-ravens ).

: If x accepts P as true, then Q.

" If one accepts this theory, then the Induction becomes vital to interpretation, as it serves to undermine any questions of the seriousness of Katherina's closing sentiments.

If one accepts the theory that the speech is sincere, then the final scene must be interpreted literally.

If she accepts, she is tonsured in a formal service during which she is given the outer robe ( Exorassa ) and veil ( Epanokamelavkion ) to wear, and ( because she is now dead to the world ) receives a new name.

If the publisher accepts unsolicited manuscripts, then the manuscript is placed in the slush pile, which publisher's readers sift through to identify manuscripts of sufficient quality or revenue potential to be referred to acquisitions editors for review.

If the chair accepts the point of order, it is said to be sustained or ruled well-taken.

If the dealer accepts then the elder hand must propose a discard and the dealer should deal the same number of fresh cards from the pack ; following which the dealer must then also make an exchange of at least one card.

If the proposal was accepted, then the elder hand can make another proposal, if desired, and can go on making proposals as long as the dealer accepts them.

* If, after the trump card is turned up, and before the dealer accepts or refuses, or, if there is no proposal, before he plays, it be discovered that he has dealt himself more than five cards, the non-dealer may claim a fresh deal.

* If, after the trump card is turned up, and before the dealer accepts or refuses, or, if there is no proposal, before he plays, it be discovered that the dealer has fewer than five cards, the non-dealer may permit the dealer to complete his hand from the stock, or may claim a fresh deal.

If the final state is an accepting state, then the automaton accepts the word.

When Achilles demands that " If you accept A and B and C, you must accept Z ," the Tortoise remarks that that's another hypothetical proposition, and suggests even if it accepts C, it could still fail to conclude Z if it did not see the truth of:

If the next unelected candidate has died or declines the offer, it is offered to each subsequent candidate on the list until one accepts the seat, such as was the case upon the resignation of Darren Hughes in March 2011, where Louisa Wall was elected after the five list candidates above her on the New Zealand Labour Party's list declined the seat.

If it is valid, it accepts ; otherwise, it rejects.

If one accepts the validity of the idea of social reality, scientifically, it must be amenable to measurement, something which has been explored particularly in relation to trust.

If one accepts this connection, then the Tukharas (= Rishikas