If the opponents win the rally and their new score is even, the player in the right service court serves ; if odd, the player in the left service court serves.

If a group has an involution with a 2-component that is a group of Lie type of odd characteristic, the goal is to show that it has a centralizer of involution in " standard form " meaning that a centralizer of involution has a component that is of Lie type in odd characteristic and also has a centralizer of 2-rank 1.

# 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 the remainder is 2, swap 1 and 3 in odd list and move 5 to the end ( i. e. 3, 1, 7, 5 )

# If the remainder is 3, move 2 to the end of even list and 1, 3 to the end of odd list ( i. e. 4, 6, 8, 2-5, 7, 9, 1, 3 )

# If p is an odd prime, then any prime q that divides 2 < sup > p </ sup > − 1 must be 1 plus a multiple of 2p.

# If p is an odd prime, then any prime q that divides must be congruent to ± 1 ( mod 8 ).

# If move leaves no heap of size 2 or larger, leave an odd ( misère ) or even ( normal ) number of heaps of size 1

If there was an odd number of competitors, there would be a bye ( ἔφεδρος — ephedros " reserve ") in every round until the last one.

If it were rational, it could be expressed as a fraction a / b in lowest terms, where a and b are integers, at least one of which is odd.

If there is an odd number of larger chips, smaller chips from the pot can be used to equalize stacks or make change as necessary.

* If playing a high-low split game and dividing a pot between the high and low hands, always award the odd chip to the high hand.

* If splitting a pot because of tied hands, award the odd chip to the hand that contains the highest-ranking single card, using suits to break ties if necessary ( clubs ranking the lowest, followed by diamonds, hearts, and spades as in bridge ).

If n is odd, this representation is irreducible.

If n = 2k is even, then let W ′ be an isotropic subspace complementary to W. If n = 2k + 1 is odd let W ′ be a maximal isotropic subspace with W ∩ W ′

If a strip with an odd number of half-twists is cut in half along its length, it will result in a longer strip, with the same number of loops as there are half-twists in the original.

If n is odd, multiply it by 3 and add 1 to obtain 3n + 1.

* If the number is odd, triple it and add one.

< li > If r is odd, go back to step 1 .</ li >

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 one pursues this heuristic, one might expect the total number of ways to write a large even integer n as the sum of two odd primes to be roughly

( a ) If k is odd,.

If in any one calculation Ptolemy had had to invoke 83 epicycles all at once, while Copernicus never required more than one third this number, then ( in the sense obvious to Margenau ) Ptolemaic astronomy would be simpler than Copernican.

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 a nation wished to get a head start in physical fitness over all other nations, it would start its kindergarten students on a program of gymnastics the day they entered and thus eliminate a large number of the problems that plague American schools.

If no specific organization plan exists limiting the number of scientists at each salary level, the result is a department top-heavy with high-level, high-salaried personnel ''.

If we thus spent our very first day in the midst of a large number of your people honoring a new hero and a great national achievement, our last day, to us at least, was equally impressive and very moving, even though the crowds were absent and there was almost complete silence.

If a wedge-shaped coating of increasing thickness is removed from a substrate by an instrument like the Hesiometer with a knife of constant rake angle, a number of removal mechanisms are often observed which depend upon the thickness of the coating.

If all the operating variables were varied simultaneously, Af operations would be required to do the same job, and as R increases this increases very much more rapidly than the number of operations required by the dynamic program.

If Rock shows up during the number -- or you hear anything -- give me the signal ''.

If one takes the middle number, 5, and multiplies it by 3 ( the base number of the magic square of three ), the result is 15, which is also the constant sum of all the rows, columns, and two main diagonals.

If so, it might be worth while to assign a future jazz show to a different department -- one with enough confidence in the musical material to cut down on the number of performers and give them a little room to display their talents.

* 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 its minimal polynomial has degree, then the algebraic number is said to be of degree.

If K is a number field, its ring of integers is the subring of algebraic integers in K, and is frequently denoted as O < sub > K </ sub >.

* Let Q be a set enclosed between two step regions S and T. A step region is formed from a finite union of adjacent rectangles resting on a common base, i. e. S ⊆ Q ⊆ T. If there is a unique number c such that a ( S ) ≤ c ≤ a ( T ) for all such step regions S and T, then a ( Q )

If a description of s, d ( s ), is of minimal length ( i. e. it uses the fewest number of characters ), it is called a minimal description of s. Thus, the length of d ( s ) ( i. e. the number of characters in the description ) is the Kolmogorov complexity of s, written K ( s ).

If a ) testimonies conflict one another, b ) there are a small number of witnesses, c ) the speaker has no integrity, d ) the speaker is overly hesitant or bold, or e ) the speaker is known to have motives for lying, then the epistemologist has reason to be skeptical of the speaker's claims.

If we take in our hand any volume ; of divinity or school metaphysics, for instance ; let us ask, Does it contain any abstract reasoning concerning quantity or number?

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.

If the two carbon atoms at the double bond are linked to a different number of hydrogen atoms, the halogen is found preferentially at the carbon with fewer hydrogen substituents ( Markovnikov's rule ).

If F is an antiderivative of f, and the function f is defined on some interval, then every other antiderivative G of f differs from F by a constant: there exists a number C such that G ( x ) = F ( x ) + C for all x.

If there is no number to its left, simply look at the column headed " 1 " in the previous row.

If the game has been scored correctly, the total number of plate appearances for a team should equal the total of that team's runs, men left on base, and men put out.

If the other player also knows to offer sacrifices, the expert also has to manipulate the number of available sacrifices through earlier play.