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

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

* 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 ).

For example, with ( m, n )

* 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 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, 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 m, n ≥ 0 define

* For all S in M, a ( S ) ≥ 0.

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, 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.

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.

For example, the marksman gets 5 shots, but we take his score to be the number of shots before his first bull's-eye, that is, 0, 1, 2, 3, 4 ( or 5, if he gets no bull's-eye ).

For Euclid ’ s method to succeed, the starting lengths must satisfy two requirements: ( i ) the lengths must not be 0, AND ( ii ) the subtraction must be “ proper ”, a test must guarantee that the smaller of the two numbers is subtracted from the larger ( alternately, the two can be equal so their subtraction yields 0 ).

For example, the open interval ( 0, 1 ) does not have a least element: if x is in ( 0, 1 ), then so is x / 2, and x / 2 is always strictly smaller than x.

For example, the full moon has an angular diameter of approximately 0. 5 °, when viewed from Earth.

For example, the prime number theorem states that the number of prime numbers less than or equal to N is asymptotically equal to N / ln N. Therefore the proportion of prime integers is roughly 1 / ln N, which tends to 0.

For example, an array of 10 integer variables, with indices 0 through 9, may be stored as 10 words at memory addresses 2000, 2004, 2008, … 2036, so that the element with index i has the address 2000 + 4 × i.

For this reason, the C programming language specifies that array indices always begin at 0 ; and many programmers will call that element " zeroth " rather than " first ".

For example, P might be the circle with radius 1 and center ( 0, 0 ): P =

More technically, the law is concerned with the speedup achievable from an improvement to a computation that affects a proportion P of that computation where the improvement has a speedup of S. ( For example, if an improvement can speed up 30 % of the computation, P will be 0. 3 ; if the improvement makes the portion affected twice as fast, S will be 2.

For example, the graph contains the points ( 1, 1 ), ( 2, 0. 5 ), ( 5, 0. 2 ), ( 10, 0. 1 ), ... As the values of x become larger and larger, say 100, 1000, 10, 000 ..., putting them far to the right of the illustration, the corresponding values of y,. 01,. 001,. 0001, ..., become infinitesimal relative to the scale shown.

For example, the function has a horizontal asymptote at y = 0 when x tends both to −∞ and +∞ because, respectively,

For mammals, the relationship between brain volume and body mass essentially follows a power law with an exponent of about 0. 75.

For instance, division of real numbers is a partial function, because one can't divide by zero: a / 0 is not defined for any real a.

For example, when a woman lies on her back, the angle of the breast apex becomes a flat, obtuse angle ( less than 180 degrees ) while the base-to-length ratio of the breast ranges from 0. 5 to 1. 0.

For bipolar I, the ( probandwise ) concordance rates in modern studies have been consistently put at around 40 % in monozygotic twins ( same genes ), compared to 0 to 10 % in dizygotic twins.

For r = 0,

For any real numbers x, r > 0, one has