Using the following observation, one can create a recursive algorithm that computes x < sup > n </ sup > for an integer n using squaring and multiplication:

Using K-theory, they constructed an example of a torsion Hodge class, that is, a Hodge class α such that for some positive integer n, n α = 0.

* Using variant data types as loop counters in Microsoft Visual Basic where an integer type is also available.

Using the floor function ( defined to be the largest integer less than or equal to the real number x ), one can construct several formulas that take only prime numbers as values for all positive integers n.

Using summation notation, it can be written as

Using base-10 ( the familiar decimal notation ) as an example, the number 152853. 5047, which has ten decimal digits of precision, is represented as the significand 1528535047 together with an exponent of 5 ( if the implied position of the radix point is after the first most significant digit, here 1 ).

Using asymptotic notation this result can be restated as

Using comparative analysis of all the material available to him, Vollaerts was able to show the internal logic and coherence in the neumatic notation with regard to a proper articulation of the verbal-melodic line.

Using the above given notation, suppose that some infinite cardinal.

Using the notation that is sometimes used to concisely describe diagonal matrices, we can write:

Using this notation, P ( t ) takes the form,

Using big-O notation, the performance of the interpolation algorithm on a data set of size N is O ( N ); however under the assumption of a uniform distribution of the data on the linear scale used for interpolation, the performance can be shown to be O ( log log N ).

Using the Dirac notation, let the system initially be in the state where

Using Leibniz notation: the substitution yields and thus, formally,, which is the required substitution for.

Using matrix notation, the first fundamental form becomes

Using Knuth's up-arrow notation, Graham's number G ( as defined in Gardner's Scientific American article ) is

Using this sigma notation the above summation is written as:

Using IUPAC notation in which the brackets contain the natural variables ( other than the main four ), we have:

Using the other notation:

Using index notation, the Hodge dual is obtained by contracting the indices of a k-form with the n-dimensional completely antisymmetric Levi-Civita tensor.

Using the limit notation of calculus, the volume may be expressed as:

Using the notation in the diagram on the right, the sides are ( AB ), ( BC ), ( CD ), ( DA ).

Using Lagrange's notation, we can write:

Using Newton's notation, we can write:

Using this notation, Z-X-Z would mean extrinsic composition.

Using hexadecimal notation, which includes the digits < tt > 0123456789ABCDEF </ tt >, it is possible to spell several words.

Using this notation and some knowledge about mathematical functions and probability theory we formalise different versions of the binding and hiding properties of commitments.

Using this notation, we can write Z = O < sub > Q </ sub > since Z is the ring of integers of the field Q of rational numbers.

Using index notation, an eigenvalue equation can be written as

Using Dirichlet's theorem on primes in arithmetic progressions, the law of quadratic reciprocity, and the Chinese remainder theorem ( CRT ) it is easy to see that for any M > 0 there are primes p such that the numbers 1, 2, …, M are all residues modulo p.

Using the known ionic radii for the trivalent rare earths and the linear correlation of log distribution coefficient with ionic radius, an average ionic radius of 0. 089 nm was estimated for Md < sup > 3 +</ sup > and a heat of hydration of –( 3654 ± 12 ) kJ / mol calculated using empirical models and the Born-Haber cycle.

* ( VEW04 ) " Using a Decompiler for Real-World Source Recovery ", M. Van Emmerik and T. Waddington, the Working Conference on Reverse Engineering, Delft, Netherlands, 9 – 12 November 2004.

Using Mason's knowledge of the prison, they quietly eliminate several small teams of marines and disable 12 of the 15 rockets, until Hummel threatens over the loudspeaker to execute a hostage if the remaining " Navy SEALs " do not surrender and return the guidance chips from the rockets.

Using the same S-IVB baseline, which does make sense in this case due to the lower thrust, we have an unfueled weight () of 10, 500 + 1, 650 = 12, 150 kg, and a fueled mass () of 22, 750 + 12, 150 = 34, 900 kg.

* Using the InterMezzo Distributed Filesystem, Bill von Hagen, LinuxPlanet, August 12, 2002.

" Using that measure, like the site at Purtan Bay, the site near Wicomico is also approximately 12 miles from Jamestown.

Using data from the 2001 Census, 74. 02 % of the population of the Hatch End ward identify as ' white ', 18. 94 % as ' Asian or Asian British ', 3. 12 % as ' Black or Black British ', 2. 16 % as ' mixed ', 1. 14 % as ' Other ethnic group ' and 0. 61 % as Chinese.

Using this shutter, in 1 second of video ( 60 fields equaling 30 frames ), 48 television fields ( totaling to 24 frames of video ) would be captured on 24 frames of film, and 12 additional fields would be omitted as the shutter closed and the film advanced.

Using existing multitrack master tapes — his small studio in fact had no capacity to record session musicians — Tubby would re-tape or ' dub ' the original after passing it through his 12 channel custom built MCI mixing desk, twisting the songs into unexpected configurations which highlighted the heavy rhythms of their bass and drum parts with minute snatches of vocals, horns and Piano / Organ.

Using government money, Bentham bought the land on behalf of the Crown for £ 12, 000 in November 1799.

Using the square format, there are 12 exposures per roll ; 4 × 3 and 4 × 6 give 16 and 8, respectively.

Using a circular track at Satory, Ader carried out taxiing trials on 12 October 1897 and two days later attempted a flight.

Using the commonalities among sequences and hydropathy plot analyses, it was predicted that there are 12 hydrophobic membrane spanning regions in the ‘ Classical ’ transporter family.

Using Royal Aircraft Factory BE. 12 and Martinsyde airplanes, they not only overcame enemy air attacks, but also bucked 65 mile per hour winds, antiaircraft fire, and malfunctioning equipment to complete their task circa 19 January 1918.

* August 7 – Using the probe-and-drogue aerial refueling system, a Royal Air Force Gloster Meteor Mk 3 remains aloft continuously for 12 hours 3 minutes, with pilot comfort appearing to be the only factor limiting an ability to stay aloft even longer.

* Using CH-53A Sea Stallion helicopters, the U. S. Navys Helicopter Mine Countermeasures Squadron 12 ( HM-12 ) and Mobile Mine Countermeaures Command begin the development of specifications for the U. S. Navys first air mine countermeasures aircraft.

On July 12, 1999, the U. S. House of Representatives censured ( in a 355-to-0 vote ) a scientific publication titled " A Meta-analytic Examination of Assumed Properties of Child Sexual Abuse Using College Samples ," by Bruce Rind, Philip Tromovich, and Robert Bauserman ; ( see Rind et al.

Using a consensus process, ASTM supports thousands of volunteer technical committees, which draw their members from around the world and collectively develop and maintain more than 12, 000 standards.

Using its hypnotic control, WOTAN enlists a workforce to construct 12 War Machines around London, which is the first capital to become controlled.