If the one- or two-digit address of an index word or electronic switch is used or is included in the operand of an XRESERVE or SRESERVE statement ( see page 99 ), the corresponding index word or electronic switch is reserved.

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

If the symbolic name or actual address of an index word or electronic switch appears or is included in the operand of an XRELEASE or SRELEASE statement ( see page 101 ), the specified index word or electronic switch will again be made available, regardless of the method by which it was reserved.

If, at any time during the assignment pass, the compiler finds that there are no more index words available for assignment, the warning message `` No More Index Words Available '' will be placed in the object program listing, the table will be altered to show that index words 1 through 96 are available, and the assignment will continue as before.

If the minimum legal value for every index is 0, then B is the address of the element whose indices are all zero.

If each node additionally records the size of its subtree ( including itself and its descendants ), then the nodes can be retrieved by index in O ( log n ) time as well.

If the keys match, then a matching element has been found so its index, or position, is returned.

If the index finger of the right hand is pointed forward, the middle finger bent inward at a right angle to it, and the thumb placed at a right angle to both, the three fingers indicate the relative directions of the x -, y -, and z-axes in a right-handed system.

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 list is stored as an array data structure, the location may be the index of the item found ( usually between 1 and n, or 0 and n − 1 ).

If the photons reside mainly in the higher index material, the confinement is due to total internal reflection.

If the reflected and direct image of the horizon are in line there is no index error.

If one is above the other adjust the index mirror until the two horizons merge.

If an index transforms like a vector with the inverse of the basis transformation, it is called contravariant and is traditionally denoted with an upper index, while an index that transforms with the basis transformation itself is called covariant and is denoted with a lower index.

If the refractive index is lower on the other side of the boundary and the incident angle is greater than the critical angle, no light can pass through and all of the light is reflected.

If for example, visible light were traveling through acrylic glass ( with an index of refraction of 1. 50 ) into air ( with an index of refraction of 1. 00 ), the calculation would give the critical angle for light from acrylic into air, which is

If an array is used to represent a cycle, it is convenient to obtain the index with a modulo operator, which can result in zero.

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.