: For an example of the simple algorithm " Add m + n " described in all three levels see Algorithm examples.

For example, if the water is 25 ft ( 8 m ) deep, and the anchor roller is 3 ft ( 1 m ) above the water, the scope is the ratio between the amount of cable let out and 28 ft ( 9 m ).

For small values of m like 1, 2, or 3, the Ackermann function grows relatively slowly with respect to n ( at most exponentially ).

For m ≥ 4, however, it grows much more quickly ; even A ( 4, 2 ) is about 2, and the decimal expansion of A ( 4, 3 ) is very large by any typical measure.

For example, with ( m, n )

For the case of a non-commutative base ring R and a right module M < sub > R </ sub > and a left module < sub > R </ sub > N, we can define a bilinear map, where T is an abelian group, such that for any n in N, is a group homomorphism, and for any m in M, is a group homomorphism too, and which also satisfies

For example, taking m to be 1 gives the triangular numbers 0, 1, 3, 6, ....

For instance, " Phnom Penh " will sometimes be shortened to " m ' Penh ".

For example, the cofinality of ω² is ω, because the sequence ω · m ( where m ranges over the natural numbers ) tends to ω² ; but, more generally, any countable limit ordinal has cofinality ω.

For example, the surface tension of distilled water is 72 dyn / cm at 25 ° C ( 77 ° F ); in SI units this is 72 x 10 < sup >− 3 </ sup > N / m or 72 mN / m.

For example, some references, and Wolfram's Mathematica software, define the complete elliptic integral of the first kind in terms of the parameter m, instead of the elliptic modulus k.

For if k, m, and n are integers, and k is a common factor of two integers A and B, then A = nk and B = mk implies A − B = ( n − m ) k, therefore k is also a common factor of the difference.

For example, it is 6-a-side rather than 11, the field is reduced to approximately 40 m x 20 m ; the shooting circles are 9m ; players may not raise the ball outside the circle nor hit it.

For biological molecules the diffusion coefficients normally range from 10 < sup >− 11 </ sup > to 10 < sup >− 10 </ sup > m < sup > 2 </ sup >/ s.

For positive integer m the derivative of gamma function can be calculated as follows ( here γ is the Euler – Mascheroni constant ):

For instance, German-speakers more often described, ( f .) " bridge " with words like ' beautiful ', ' elegant ', ' fragile ', ' peaceful ', ' pretty ', and ' slender ', whereas Spanish-speakers, which use puente ( m .) used terms like ' big ', ' dangerous ', ' long ', ' strong ', ' sturdy ', and ' towering '.

For that purpose, one needs a hash function that maps similar keys to hash values that differ by at most m, where m is a small integer ( say, 1 or 2 ).

For example, the orbital 1s < sup > 2 </ sup > ( pronounced " one ess two ") has two electrons and is the lowest energy level ( n

For this reason, orbitals with the same value of n are said to comprise a " shell ".

For instance, the n

For instance, binary search is said to run in a number of steps proportional to the logarithm of the length of the list being searched, or in O ( log ( n )), colloquially " in logarithmic time ".

For example, if the sorted list to which we apply binary search has n elements, and we can guarantee that each lookup of an element in the list can be done in unit time, then at most log < sub > 2 </ sub > n + 1 time units are needed to return an answer.

For example, n < sub > D </ sub > is measured at 589. 3 nm:

* For a finite field of prime order p, the algebraic closure is a countably infinite field which contains a copy of the field of order p < sup > n </ sup > for each positive integer n ( and is in fact the union of these copies ).

* The Ham sandwich theorem: For any compact sets A < sub > 1 </ sub >,..., A < sub > n </ sub > in ℝ < sup > n </ sup > we can always find a hyperplane dividing each of them into two subsets of equal measure.

* For all, takes n qubits as input and outputs 1 bit

For example, for n = 5:

For any set containing n elements, the number of distinct k-element subsets of it that can be formed ( the k-combinations of its elements ) is given by the binomial coefficient.

For natural numbers ( taken to include 0 ) n and k, the binomial coefficient can be defined as the coefficient of the monomial X < sup > k </ sup > in the expansion of.

For positive values of a and b, the binomial theorem with n = 2 is the geometrically evident fact that a square of side can be cut into a square of side a, a square of side b, and two rectangles with sides a and b. With n = 3, the theorem states that a cube of side can be cut into a cube of side a, a cube of side b, three a × a × b rectangular boxes, and three a × b × b rectangular boxes.

For example, there will only be one term x < sup > n </ sup >, corresponding to choosing x from each binomial.

For integer order α = n, J < sub > n </ sub > is often defined via a Laurent series for a generating function:

* For all S in M, a ( S ) ≥ 0.

For example, 7 ≥ 5 does not imply that 5 ≥ 7.

For arbitrary n ≥ 2 we may generalize this formula, as noted above, by interpreting the third equation for the harmonic mean differently.

For any information rate R < C and coding error ε > 0, for large enough N, there exists a code of length N and rate ≥ R and a decoding algorithm, such that the maximal probability of block error is ≤ ε ; that is, it is always possible to transmit with arbitrarily small block error.

Then N < sub > x </ sub > is a directed set, where the direction is given by reverse inclusion, so that S ≥ T if and only if S is contained in T. For S in N < sub > x </ sub >, let x < sub > S </ sub > be a point in S. Then ( x < sub > S </ sub >) is a net.

For any luminosity from a given distance L ( r ) N ( r ) proportional to r < sup > a </ sup >, is infinite for a ≥ − 1 but finite for a < − 1.

For n ≥ 2, the n-spheres are the simply connected n-dimensional manifolds of constant, positive curvature.

For arbitrary square matrices M, N we write M ≥ N if M − N ≥ 0 ; i. e., M − N is positive semi-definite.

For example, it follows that any closed oriented Riemannian surface can be C < sup > 1 </ sup > isometrically embedded into an arbitrarily small ε-ball in Euclidean 3-space ( there is no such C < sup > 2 </ sup >- embedding since from the formula for the Gauss curvature an extremal point of such an embedding would have curvature ≥ ε < sup >- 2 </ sup >).

For example, for any k ≥ 1, the following example meets the bounds exactly.

For example: no cube can be written as a sum of two coprime n-th powers, n ≥ 3.

For 1 < p, q < ∞ and f ∈ L < sup > p </ sup >( μ ) and g ∈ L < sup > q </ sup >( μ ), Hölder's inequality becomes an equality if and only if | f |< sup > p </ sup > and | g |< sup > q </ sup > are linearly dependent in L < sup > 1 </ sup >( μ ), meaning that there exist real numbers α, β ≥ 0, not both of them zero, such that α | f |< sup > p </ sup > = β | g |< sup > q </ sup > μ-almost everywhere.

For α ≥ 1 and β ≥ 1, the relative error ( the absolute error divided by the median ) in this approximation is less than 4 % and for both α ≥ 2 and β ≥ 2 it is less than 1 %.

For n ≥ 0, let C < sup > n </ sup >( G, M ) be the group of all functions from G < sup > n </ sup > to M. This is an abelian group ; its elements are called the ( inhomogeneous ) n-cochains.

For n ≥ 0, define the group of n-cocycles as:

Lemma: For all real numbers x ≥ 1, we have x # < 4 < sup > x </ sup >.

For n ≥ 1, the homotopy classes form a group.

For the case of k-in-a-row where the board is an n-dimensional hypercube with all edges with length k, Hales and Jewett proved that the game is a draw if k is odd and k ≥ 3 ^ n-1 or if k is even and k ≥ 2 ^( n + 1 )-2.