: 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.

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 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: 1.

( For each State, make all computations set forth in items 1 to 8 above, and then add the results obtained for each State in item 8.

For an engine speed of 5000 rpm, a gear ratio of 4.00 to 1, and a tire radius of 13 inches, the equation would look like this: Af

For continuous control, feed 1 part phenothiazine to 9 parts minerals or salts.

For the albumin method, equal volumes of 30% bovine albumin, sample and 2% cells suspended in saline were allowed to stand at room temperature for 1 hr and then were centrifuged at 1000 rpm for 1 Aj.

For preparative ultracentrifugation, fractions from the column were concentrated by negative pressure dialysis to volumes of 1 ml or less, transferred to cellulose tubes and diluted to 12 ml with isotonic saline.

For example, the marked susceptibility of the monkey to respiratory infection might be related to its delicate, long alveolar ducts and short, large bronchioles situated within a parenchyma entirely lacking in protective supportive tissue barriers such as those found in types 1, and 3.

For an experiment to qualify as a binomial experiment, it must have four properties: ( 1 ) there must be a fixed number of trials, ( 2 ) each trial must result in a `` success '' or a `` failure '' ( a binomial trial ), ( 3 ) all trials must have identical probabilities of success, ( 4 ) the trials must be independent of each other.

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 Yokuts, I tabulated these 71 items in five columns, according as they were expressed by 1, 2, 3, 4, and more than 4 stems.

For Athabascan, with a greater range of stems, the first two of five corresponding columns were identical, 1 and 2 stems ; ;

For the initial development an oxidation pond was constructed as shown in Figure 1.

For example Test 3, E 1 refers to a specimen which has been washed by procedure `` 3 ( '' ( at 160-degrees-F ) for a total of 60 minutes in the machine, has been dried in a tumble dryer by procedure `` E '' and has been subjected to restorative forces on the Tension Presser by procedure `` 1 ''.

For hundreds of years, the evidence available consisted of ( 1 ) the captain's fragmentary journal, ( 2 ) a highly prejudiced account by one of the survivors, ( 3 ) a note found in a dead man's desk on board, and ( 4 ) several second-hand reports.

For example, the 2nd Academy Awards presented on April 3, 1930, recognized films that were released between August 1, 1928 and July 31, 1929.

For example Connecticut applies the following standard to review unpreserved claims: 1. the record is adequate to review the alleged claim of error ; 2. the claim is of constitutional magnitude alleging the violation of a fundamental right ; 3. the alleged constitutional violation clearly exists and clearly deprived the defendant of a fair trial ; 4. if subject to harmless error analysis, the state has failed to demonstrate harmlessness of the alleged constitutional violation beyond a reasonable doubt.

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 analysis of the debris at the testing site of the first U. S. hydrogen bomb, Ivy Mike, ( 1 November 1952, Enewetak Atoll ), revealed high concentrations of various actinides including americium ; due to military secrecy, this result was published only in 1956.

For each element a of a group G, conjugation by a is the operation φ < sub > a </ sub >: G → G given by ( or a < sup >− 1 </ sup > ga ; usage varies ).

For a 256-bit block, the first row is unchanged and the shifting for the second, third and fourth row is 1 byte, 3 bytes and 4 bytes respectively — this change only applies for the Rijndael cipher when used with a 256-bit block, as AES does not use 256-bit blocks.

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, 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 ).