For instance, can hold an unboxed integer in a range supported by the hardware and implementation, permitting more efficient arithmetic than on big integers or arbitrary precision types.

For instance, there is no instruction to load an arbitrary “ immediate ” value into an accumulator ( although memory reference instructions do encode such a value to form an effective address ).

For who durst set himself in opposition to the crown and ministry, or aspire to the character of being a patron of freedom, while exposed to so arbitrary a jurisdiction?

For that reason it is beneficial to include keyframes at arbitrary intervals while encoding video.

Note: For any arbitrary number of propositional constants, we can form a finite number of cases which list their possible truth-values.

For a boost in an arbitrary direction with velocity v, it is convenient to decompose the spatial vector r into components perpendicular and parallel to v:

For an arbitrary n there exists a monotone formula for majority of size O ( n < sup > 5. 3 </ sup >).

For a number of years after the passing of the first Quarantine Act ( 1710 ) the protective practices in England were of the most haphazard and arbitrary kind.

For an arbitrary ring, let be the underlying additive group.

For intersexual selection to work, one sex must evolve a feature alluring to the opposite sex, sometimes resulting in a " fashion fad " of intense selection in an arbitrary direction.

For example, many theories of quantum gravity can create universes with arbitrary numbers of dimensions or with arbitrary cosmological constants.

It is important to realize that the word problem is in fact solvable for many groups G. For example, polycyclic groups have solvable word problems since the normal form of an arbitrary word in a polycyclic presentation is readily computable ; other algorithms for groups may, in suitable circumstances, also solve the word problem, see the Todd – Coxeter algorithm and the Knuth – Bendix completion algorithm.

For an initial value problem, the arbitrary functions and can be determined to satisfy initial conditions:

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

For example, in the IEEE 754-2008 standard it means the number of bits in the significand, so it is used as a measure for the relative accuracy with which an arbitrary number can be represented.

For any two arbitrary integers, m and n, form the equimultiples

For triclinic, orthorhombic, and cubic crystal systems the axis designation is arbitrary and there is no principle axis.

For example, in computer graphics rendering, the scene is divided until each node of the BSP tree contains only polygons that can render in arbitrary order.

For a flat spatial geometry, the scale of any properties of the topology is arbitrary and may or may not be directly detectable.

For an arbitrary heat engine, the efficiency is:

For example, with radiographic equipment such as a CT scanner, one would reasonably require the operator to have competence in radiological safety ; but for consumer electronics, the goal ( distant as it often may be ) is to shield the user from having to possess any arbitrary threshold of skill.

For trips into largely arbitrary Earth orbits, almost any time will do.

For example, it is not possible to obtain a filter which has both an arbitrary impulse response and arbitrary frequency function.

: 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