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

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 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 instant John longed for the sound of the bells of Noyon-la-Sainte, the touch of his mother's hand, the lilt of Charles's voice in the square raftered rooms, his father's bass tones rumbling to the canons, and the sight of the beloved bishop.

For any such square the middle corner of these will be called the vertex of the square and the corner not on the curve will be called the diagonal point of the square.

For the Lo Shu square was a remarkably complete compendium of most of the chief religious and philosophical ideas of its time.

For example, a certain landowner might have been said to own 32, 000 acres of land, not 50 square miles of land.

For example, a vertex configuration of ( 4, 6, 8 ) means that a square, hexagon, and octagon meet at a vertex ( with the order taken to be clockwise around the vertex ).

For example, adjacent to square 5F are squares 4F, 6F, 5E, and 5G, but not 6E or 4G.

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 square matrices with entries in a non-commutative ring, for instance the quaternions, there is no unique definition for the determinant, and no definition that has all the usual properties of determinants over commutative rings.

** For example, consider the map ( which is the " realification " of the complex square function ) where U

For example, suppose the word square is defined by " A figure is square if and only if it has four perpendicular sides of equal length.

For example, JPEG compression uses a variant of the Fourier transformation ( discrete cosine transform ) of small square pieces of a digital image.

For example, both the Egyptians and the Babylonians were aware of versions of the Pythagorean theorem about 1500 years before Pythagoras ; the Egyptians had a correct formula for the volume of a frustum of a square pyramid ;

For example, a 4-cycle ( square ) has girth 4.

For houses in and around Geneva, the average price was 11, 595 Swiss francs ( CHF ) per square metre () ( June 2011 ), with a lowest price per square metre () of 4, 874 Swiss francs ( CHF ), and a maximum price of 21, 966 Swiss francs ( CHF ).

For instance, the geometric mean of two numbers, say 2 and 8, is just the square root of their product ; that is.

# For a given orbit, the ratio of the cube of its semi-major axis to the square of its period is constant.

For a number written in scientific notation, this logarithmic rounding scale requires rounding up to the next power of ten when the multiplier is greater than the square root of ten ( about 3. 162 ).

For problems with all four properties, the time for a quantum computer to solve this will be proportional to the square root of the number of inputs.

For other examples, see proof that the square root of 2 is not rational and Cantor's diagonal argument.