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 Blue Throat has his way he'll keep us all cooped up in here for days '', he said.

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 the man on the sidewalk is surprised at this question, it has served as an exclamation.

If any one of them has any power to veto the Secretary General's decisions the nature of the organization will have changed.

If this new Bible does not increase in significance by repeated readings throughout the years, it will not survive the ages as has the King James Version.

If the case is thus determined by us to be domestic, the court has no jurisdiction.

If this capacity had not failed them, they would see that their enemy has made a disastrous miscalculation.

If the new Soviet series has followed the general pattern of previous Russian tests, the shots were roughly half fission and half fusion, meaning a fission yield of 30 to 40 megatons thus far.

If the superintendents do not receive more cooperation from Handlers, it has been suggested that licensed Judges also be qualified to judge this Class.

If your state has no provisions for the numbering of pleasure boats, you must apply for a number from the U.S. Coast Guard for any kind of boat with mechanical propulsion rated at more than 10 horsepower before it can be used on Federal waterways.

If design head has a deep cavity, clay lid will be quite thick at this point ; ;

If you run into excess plug fouling on one truck, check to be sure that the rig has a thermostat.

If we manage to keep track of a Bombus queen after she has left her feeding place, we may discover the snug little hideout which she has fixed up for herself when she woke up from her winter sleep.

If cell Af has not previously been assigned as the information cell of a form in the text-form list, it is now assigned as the information cell of Af.

If of the founders of glottochronology Swadesh has escaped our steady plodding, and Lees has repudiated his own share in the founding, that is no reason why we should swerve.

If the transferor has substantial assets other than the claim, it seems reasonable to assume no corporation would be willing to acquire all of its properties in the dim hope of collecting a claim for refund of taxes.

If the patient can perceive figure kinesthetically when he cannot perceive it visually, then, it would seem, the sense of touch has immediate contact with the spatial aspects of things in independence of visual representations, at least in regard to two dimensions, and, as we shall see, even this much spatial awareness on the part of unaided touch is denied by the authors.

If the distant patron of the suburban branch has been frightened away from downtown by traffic problems, however, the city store can only pressure the politicians to do something about the highways or await the completion of the federal highway program.

If a statement has been assigned an address in the index word area

If she has not had such experiences, the female's normal adolescent degree of indecision will be compounded.

If the change, at first sight, seems minor, we may recall that it took the Italian painters about two hundred years to make an analogous change, and the Italian painters, by universal consent, were the most brilliant group of geniuses any art has seen.

If Barnett doesn't call a special session in 1961, it will be the first year in the last decade that the Legislature has not met in regular or special session.

If the church has followed the plan of cultivation of prospects and carried through a program of membership preparation as outlined earlier in this book, the process of assimilation and growth will be well under way.

:“ If an integer n is greater than 2, then has no solutions in non-zero integers a, b, and c. I have a truly marvelous proof of this proposition which this margin is too narrow to contain .”

If only the electric field () is non-zero, and is constant in time, the field is said to be an electrostatic field.

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 all the non-zero bits were counted, then the intermediate result register now holds the final result.

If the pump waves and the signal wave are superimposed in a medium with a non-zero χ < sup >( 3 )</ sup >, this produces a nonlinear polarization field:

If I is a right ideal of R, then I is simple as a right module if and only if I is a minimal non-zero right ideal: If M is a non-zero proper submodule of I, then it is also a right ideal, so I is not minimal.

If I is a right ideal of R, then R / I is simple if and only if I is a maximal right ideal: If M is a non-zero proper submodule of R / I, then the preimage of M under the quotient map is a right ideal which is not equal to R and which properly contains I.

If M is a module which has a non-zero proper submodule N, then there is a short exact sequence

If h has a fixed ( non-zero ) value instead of approaching zero, then the right-hand side of the above equation would be written

If it is assumed that each player is only concerned with lessening his own time in jail, the game becomes a non-zero sum game where the two players may either assist or betray the other.

is conformal if and only if it is holomorphic and its derivative is everywhere non-zero on U. If f is antiholomorphic ( that is, the conjugate to a holomorphic function ), it still preserves angles, but it reverses their orientation.

If the Jacobian g has a non-zero determinant, then is " conformal in the generalized sense " with respect to one of the three angle types, depending on the real matrix expressed by the Jacobian g.

If n is non-zero, and is neither 1 nor − 1, by the fundamental theorem of arithmetic, there exist primes p < sub > i </ sub >, and positive integers e < sub > i </ sub >, with.

If R = K is a field, then a series is invertible if and only if the constant term is non-zero, i. e., if and only if the series is not divisible by X.

If R is an ordered field, we can order R (( G )) by setting any element to have the same sign as its leading coefficient, defined as the least element of the index set I associated to a non-zero coefficient.

If D is non-zero ( so for planes not through the origin ) the values for a, b and c can be calculated as follows:

If R and B were to have been used, these would have non-zero values even in a B & W scene, requiring all three data-carrying signals.

# If h ( t ) is a non-zero scalar function of t, then Z ( t )

If the address field is non-zero, it is a disk address of the block, which has previously been rolled out, so the block is fetched from disk and the pbit is set to 1 and the physical memory address updated to point to the block in memory ( another pbit ).

If a is algebraic over K, then there are many non-zero polynomials g ( x ) with coefficients in K such that g ( a ) = 0.