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 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 there was collusion between an outside murderer and a member of the household it would be an elementary precaution to check on the door later.

If an elementary particle truly has no substructure, then it is one of the basic building blocks of the universe from which all other particles are made.

If the arguments are both greater than zero then the algorithm can be written in more elementary terms as follows:

If one contemplates actual reaction mechanisms, stoichiometric coefficients will always be integers, since elementary reactions always involve whole molecules.

If the subsets of X in Σ correspond to numbers in elementary algebra, then the two set operations union ( symbol ∪) and intersection (∩) correspond to addition and multiplication.

If the outcome is the element of the elementary event of two pips on the first die and five on the second, then both of the events of " 7 pips " and " odd number of pips " have also happened.

If the Avogadro constant N < sub > A </ sub > and the Faraday constant F are independently known, the value of the elementary charge can be deduced, using the formula

If the former horn of the dilemma is chosen, arithmetical algebra becomes a mere shadow ; if the latter horn is chosen, the operations of algebra cannot be defined on the supposition that the elementary symbol is an integer number.

If both reactions are elementary reactions, then the rate of reaction is given by

If a Petri net is equivalent to an elementary net, then Z can be the countable set

If λ is any ordinal, κ is λ-strong means that κ is a cardinal number and there exists an elementary embedding j from the universe V into a transitive inner model M with critical point κ and

If λ is any ordinal, κ is λ-supercompact means that there exists an elementary embedding j from the universe V into a transitive inner model M with critical point κ, j ( κ )> λ and

If it is then assumed that the elementary opaque elements of all matter are identical ( i. e., having the same ratio of density to area ), it will follow that the shadow effect is, at least approximately, proportional to the mass ( P5 ).

If j is the elementary embedding mentioned in one of these axioms and κ is its critical point, then λ is the limit of as n goes to ω.

If this elementary product solution is substituted into the wave equation ( 2. 0 ), using the scalar Laplacian in rectangular coordinates:

If children were to begin training in Tang Te while in elementary school, then they will be well suited for military service.

If karate should be introduced beginning in the elementary schools, then we will produce many men each capable of defeating ten assailants.

If elementary events are assigned equal probabilities, then the probability of a disjunction of elementary events is just the number of events in the disjunction divided by the total number of elementary events.

If N is a substructure of M, then both N and M can be interpreted as structures in the signature σ < sub > N </ sub > consisting of σ together with a new constant symbol for every element of N. N is an elementary substructure of M if and only if N is a substructure of M and N and M are elementarily equivalent as σ < sub > N </ sub >- structures.