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 >.

If the numbers describing the position of the pyramid were changed and this process repeated, the pyramid would appear to move.

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 the condition is thought to be cellulitis rather than abscess, consideration should be given to possibility of strep species as cause that are still sensitive to traditional anti-staphylococcus agents such as dicloxacillin or cephalexin in patients able to tolerate penicillin ).

# Did the government agents obtain testimonial evidence as a result of the interrogation? If “ Yes ” go to 4 ; If “ No ”, Miranda does not apply.

# Before any custodial interrogation began did the government agents properly advise the defendant of her Miranda rights? If “ Yes ” go to 6 ; If “ No ”, go to 18 ..

If the infection is of bacterial origin, the most common three causative agents are Streptococcus pneumoniae, Haemophilus influenzae, and Moraxella catarrhalis.

If there should happen to be an irreconcilable variance between two, that which has the superior obligation and validity ought, of course, to be preferred ; or, in other words, the constitution ought to be preferred to the statute, the intention of the people to the intention of their agents .< ref name =" fed78 ">

If the claim succeeds the recovery agents share a portion of the damages.

If the Federal Reserve attempts to lower unemployment through expansionary monetary policy economic agents will anticipate the effects of the change of policy and raise their expectations of future inflation accordingly.

If a private agent promises to pay something later, it has a debt, and this debt is enforceable by public agents.

If the man was not interested in joining the regiment, the king resorted to forced recruitment and kidnapping — his agents kidnapped tall priests, monks, innkeepers, etc., from all over Europe.

If undercover agents have been used without proper justification, punishment for the committed offense may be reduced ( 1st Senate's decision in 1 StR 148 / 84-23 May, 1984 ).

If the hostile action is in one's own country, or in a friendly one with cooperating police, the hostile agents may be arrested, or, if diplomats, declared persona non grata.

If one agent defects and the opponent cooperates, then both agents will end up alternating cooperate and defect, yielding a lower payoff than if both agents were to continually cooperate.

If private agents ( consumers and firms ) believe that policymakers are committed to lowering inflation, they will anticipate future prices to be lower than otherwise ( how those expectations are formed is an entirely different matter ; compare for instance rational expectations with adaptive expectations ).

If an announcement about low-level inflation targets is made but not believed by private agents, wage-setting will anticipate high-level inflation and so wages will be higher and inflation will rise.

If policymakers believe that private agents anticipate low inflation, they have an incentive to adopt an expansionist monetary policy ( where the marginal benefit of increasing economic output outweighs the marginal cost of inflation ); however, assuming private agents have rational expectations, they know that policymakers have this incentive.

If the contact list is poor, the performance of the predictive dialing campaign is at risk as agents are not connected to live contacts and are not able to do business.

" If, in so doing, the institutions continue to satisfy certain structural conditions, both in the sense of conditions which delimit the scope for institutional variation and the conditions which underlie the operation of structural differentiation, then the agents may be said to reproduce social structure.

# If government agents suspect anyone applying for benefits of being illegal immigrants, the agents must report their suspicions in writing to the appropriate enforcement authorities.

If any buyer's broker ( or any of his / her agents ) brings the buyer for the property, the buyer's broker would typically be compensated with a co-op commission coming from the total offered to the listing broker, often about half of the full commission from the seller.