Using the orthogonality relations derived above, we find that the q characters for the q inequivalent irreducible representations forms a basis set.

Using a lattice basis, neither orthogonality nor unit length is guaranteed, only independence.

Using this relation, it is possible to construct a partial order on the quotient set of the equivalence, S / ~, the set of all equivalence classes of ~.

Using the corresponding strict relation "<", one can also define the interval ( a, b ) as the set of points x satisfying a < x and x < b, also written a < x < b. An open interval may be empty even if a < b.

Using this, we can get the relation, again choosing the right sign:

where the reference variables may be functions of the number of particles N. Using the definition of the heat capacity at constant volume for the first differential and the appropriate Maxwell relation for the second we have:

Using this etymology, scholar John Lindow gives the meanings " in-law-relationship ", scholar Andy Orchard provides " relation ", and scholar Rudolf Simek gives " relation by marriage ".

The position of Bradman's left foot in relation to the stumps is an example of how he Crease ( cricket )# Using the crease | used the crease when batting.

Using this period-luminosity relation, in 1913 the distance to the SMC was first estimated by Ejnar Hertzsprung.

Using the Gibbs – Duhem relation it is possible to translate the change in solvent vapor pressures with concentration into activities for the solute.

Using freezing point depression techniques, it is possible to calculate the activity of a weak acid from the relation,

Using the ideal properties, it is not difficult to check that ~ is a congruence relation.

** Protective Coating: Bearing the Weight of Pynchon Using the Spectrum of Freud's Insight An analysis of Pynchon's use of parody in relation to Freudian theory

Using the accessibility relation we can translate this scenario as follows: At all of the worlds accessible to our own world, it is not the case that humans can travel faster than the speed of light, but at one of these accessible worlds there is another world accessible from those worlds but not accessible from our own at which humans can travel faster than the speed of light.

Using a reciprocity relation for on this equation and reordering gives a cyclic relation ( the triple product rule ),

Using this relation, putting ( 7 ) into ( 6 ) and transforming to one gets

Using the notation for the relation of being infinitely close as above,

Using generating functionals, the relation is expressed as

Using his mastery of the Quran and his innovative subject-based approach to Quranic exegesis, Al-Sadr extracted two concepts from the Holy text in relation to governance: khilafat al-insan ( Man as heir or trustee of God ) and shahadat al-anbiya ( Prophets as witnesses ).

Using the assumption that the cross-shore velocity v is zero at the coast, v = 0, one may solve a frequency relation for the phase speed of coastal Kelvin waves, which are among the class of waves called boundary waves, edge waves, trapped waves, or surface waves ( similar to the Lamb waves ).

Using the above recurrence relation, all higher can be expressed as polynomials in and.

Using the commutation relation between and we obtain:

Using the relativistic relation between momentum and energy, when Δp exceeds mc then the uncertainty in energy is greater than mc < sup > 2 </ sup >, which is enough energy to create another particle of the same type.

Using this relation we see that for itself.

" Freedom of contract is a qualified and not an absolute right ...," Hughes declared in upholding an Iowa law that voided contracts limiting the legal rights of railroad workers: The state may " interfere where the parties do not stand upon an equality ...." Using similar reasoning, the associate justice upheld a California law that mandated a forty-eight-hour work-week for women in various industries and allowed a federal statute to override a contract between an interstate railroad and its employees.

Using non-cooperation as a technique, Gandhi initiated a signature campaign where peasants pledged non-payment of revenue even under the threat of confiscation of land.

Using their flycycles, they try to reach the rim of the ring, where they hope to find some technology that will help them.

Using this process, investigators can create a profile of places where the serial killer and victim both frequent.

can be calculated from the specific impulse, I < sub > sp </ sub >, where ( when I < sub > sp </ sub > is in seconds and g < sub > n </ sub > is the standard, not local, acceleration of gravity ), Using the J-2 on the S-IVB as a baseline design, we have P = ( 1014 kN )( 414 s )( 9. 81 m / s < sup > 2 </ sup >)/ 2

Using Global Positioning System ( GPS ) technology as a reference, these antennas automatically re-aim to the satellite no matter where or how the vehicle ( on which the antenna is mounted ) is situated.

Using the eggbeater, swimmers can also perform " boosts ", where they use their legs to momentarily propel themselves out of the water to their hips or higher.

Using the above definition, it is useful to think of the function as a matrix ( called the input matrix ) where each row of the matrix corresponds to X and each column corresponds to Y.

Using the disintegration theorem we can define a restriction of Lebesgue measure to the-dimensional affine subspace of where the Gaussian distribution is supported, i. e..

Using Fibonacci numbers, he proved that when finding the greatest common divisor of integers a and b, the algorithm runs in no more than 5k steps, where k is the number of ( decimal ) digits of b. He also proved a special case of Fermat's last theorem.

Using Landau symbols, the worst case running time of CYK is, where n is the length of the parsed string and | G | is the size of the CNF grammar G. This makes it one of the most efficient parsing algorithms in terms of worst-case asymptotic complexity, although other algorithms exist with better average running time in many practical scenarios.

Using a specific literary device or technique in a type of work where, historically, it has been uncommon.

where r is the full displacement vector from the wire element to the point at which the field is being computed and r ̂ is the unit vector of r. Using this the equation can be equivalently written

Using the authorization granted to him by Congress, on 13 November 2001, President Bush issued a Presidential Military Order: " Detention, Treatment, and Trial of Certain Non-Citizens in the War Against Terrorism " which allowed " individuals ... to be detained, and, when tried, to be tried for violations of the laws of war and other applicable laws by military tribunals ", where such individuals are a member of the organization known as al Qa ' ida ; or has conspired or committed acts of international terrorism, or have as their aim to cause, injury to or adverse effects on the United States, its citizens, national security, foreign policy, or economy.

Using careful observation, the officers were able to locate where the opposition was hidden in the brush and entrenchments and they were able to target their men properly to overcome them.

Using atomic authorization is an alternative to per-system authorization management, where a trusted third party securely distributes authorization information.

Using the copy-text method, the critic examines the base text and makes corrections ( called emendations ) in places where the base text appears wrong to the critic.

Using v = lω, where ω is the angular velocity, one obtains:

; Mobile Code: Using languages such as Java where code can be streamed / slurped from one location to another via URLClassloaders and Codebases the commands can enable new behavior to be delivered to remote locations ( EJB Command, Master Worker )

Using person-time rather than just time handles situations where the amount of observation time differs between people, or when the population at risk varies with time.

Using, where possible, the functional organs of the diseased surplus body may partially alleviate this concern.

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

Using modular exponentiation by repeated squaring, the running time of this algorithm is O ( k log < sup > 3 </ sup > n ), where k is the number of different values of a we test ; thus this is an efficient, polynomial-time algorithm.

Using the definition, the solution contained a term of the form ; where the value of making this term singular has come to be known as the Schwarzschild radius.