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 he can't play with Mommy's magazines, he should have some old numbers of his own.

If we are discussing differentiable complex-valued functions, then Af and V are complex vector spaces, and Af may be any complex numbers.

If he does, it's still better than an even chance he won't notice the transposition of the numbers, and if he should notice it, the thing can be passed off as an honest mistake.

* If numbers have mean X, then.

* If it is required to use a single number X as an estimate for the value of numbers, then the arithmetic mean does this best, in the sense of minimizing the sum of squares ( x < sub > i </ sub > − X )< sup > 2 </ sup > of the residuals.

** If the set A is infinite, then there exists an injection from the natural numbers N to A ( see Dedekind infinite ).

If, however, the numbers are referring to allies it could possibly include forces from Tyre, Judah, Edom, and Moab.

If a is algebraic over K, then K, the set of all polynomials in a with coefficients in K, is not only a ring but a field: an algebraic extension of K which has finite degree over K. In the special case where K = Q is the field of rational numbers, Q is an example of an algebraic number field.

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 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 exponent r is even, then the inequality is valid for all real numbers x.

If the convention B < sub > 1 </ sub >=− is used, this sequence is also known as the first Bernoulli numbers ( / in OEIS ); with the convention B < sub > 1 </ sub >=+ is known as the second Bernoulli numbers ( / in OEIS ).

If the shooter rolls one of these numbers on the come-out roll, this establishes the " point "-to " pass " or " win ", the point number must be rolled again before a seven.

If is a convex set, for any in, and any nonnegative numbers such that, then the vector

If n ≥ 1 and is an integer, the numbers coprime to n, taken modulo n, form a group with multiplication as operation ; it is written as ( Z / nZ )< sup >×</ sup > or Z < sub > n </ sub >< sup >*</ sup >.

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.

This is a Cauchy sequence of rational numbers, but it does not converge towards any rational limit: If the sequence did have a limit x, then necessarily x < sup > 2 </ sup > = 2, yet no rational number has this property.

If x and y are real numbers, and if the graph of y is plotted against x, the derivative measures the slope of this graph at each point.

If we look for solutions that have the form, where C is a constant, we discover the relationship, and thus must be one of the complex numbers or.

If the matrix entries are real numbers, the matrix can be used to represent two linear mappings: one that maps the standard basis vectors to the rows of, and one that maps them to the columns of.

* If we think of as the set of real numbers, then the direct product is precisely just the cartesian product,.

* If we think of as the group of real numbers under addition, then the direct product still consists of.

If we take the simple molecular orbital description of the ground state and combine that function with the functions describing all possible excited states using unoccupied orbitals arising from the same set of atomic orbitals, we also reach the full configuration interaction wavefunction.

If the position was found to be r < sub > 0 </ sub > then in an interpretation satisfying CFD, the statistical population describing position and momentum would contain all pairs ( r < sub > 0 </ sub >, p ) for every possible momentum value p, whereas an interpretation that rejects counterfactual values completely would only have the pair ( r < sub > 0 </ sub >,⊥) where ⊥ denotes an undefined value.

If they are describing experiments or calculations, they must supply enough details that an independent researcher could repeat the experiment or calculation to verify the results.

" Sports Illustrated considered the activity a sport in 1987, describing the Drum Corps International World Championships “ one of the biggest sporting events of the summer .” In the same article, Sports Illustrated quoted basketball coach Bobby Knight, " If a basketball team trained as hard as these kids do, it would be unbelievable.

If the dynamics of a system is known, the equations are the solutions to the differential equations describing the motion of the dynamics.

The song involves Marvin describing his views on life (" I'd feel a little better if they broke me up for spares ", " If I had my time again, I'd rather be a lemming "), to a synthesiser backing.

If the software is completely automated ( meaning no user or user interface ), a software design may be as simple as a flow chart or text describing a planned sequence of events.

If the ascription of the Shatapatha Brahmana and the Brhadaranyaka Upanishad is genuine, Yājñavalkya has made important contributions to both philosophy, including the apophatic teaching of ' neti neti ', and to astronomy, describing the 95-year cycle to synchronize the motions of the sun and the moon.

If — for example — a wall describing an east-west line is under construction, then bricks oriented to point north-south may be built into the width of the wall, their length spanning two widths of brick and tying the brickwork on the transverse plane.

If we assume that indeed Svarog was believed to be Dažbog ’ s father, the question arises of his relation with Svarožič, another deity who is mentioned as god of fire and war in several other medieval documents describing the beliefs of pagan Slavs.

If one assumes that Svarog was believed to be Dažbog ’ s father, the question arises of his relation with Svarožič, another deity who is mentioned as a god of fire and war in several other medieval documents describing the beliefs of pagan Slavs.

If the official observers ( OO ) hear something that, in their judgment, is in violation of Federal Communications Commission ( FCC ) regulations, they will send a postcard to the amateur radio station describing the infraction.

If one studies, analyzes, and describes a language, the language used for studying, analyzing, and describing the object language is a metalanguage.

Most What If stories begin with Uatu describing an event in the mainstream Marvel Universe, then introducing a point of divergence in that event and then describing the consequences of the divergence.

If more than three ranks are listed in describing a taxon, the " classification " is being specified, but only three parts make up the " name " of the taxon: a genus name, a specific epithet, and an infraspecific epithet.

If the particle displacement rotates through an angle dθ in time dt, so does, describing an arc on the unit circle of magnitude dθ.

If M is a model of describing a set theory and N is a class of M such that

If the wavepacket describing a quantum billiard ball is made up of only the closed-form solutions, its motion will not be chaotic, but if any amount of non-closed-form solutions are included, the quantum billiard motion becomes chaotic.

" " If you watch me, it doesn't look like I'm doing much ," he said in an interview, describing his subtle but highly musical style.

If Richard was unpopular in Aquitaine, Philip II was equally unliked by his contemporaries with comments describing him as: astute, manipulative, calculating, penurious and ungallant ruler.

* So-and-so a neutral placeholder name, for example, If an idea came not from you but from a previous writer, when describing it you should say, " So-and-so said that ..." ; ( separately ) sometimes used as a euphemism for a possibly vulgar epithet, for example, " that stupid so-and-so!

Proof: If we have a description in, we can convert it into a description in our optimal language by first describing as a computer program ( part 1 ), and then using the original description as input to that program ( part 2 ).

If v is a vector in and k is a unit vector describing an axis of rotation about which we want to rotate v by an angle θ ( in a right-handed sense ), the Rodrigues formula is: