Using the F-distribution is a natural candidate because the test statistic is the ratio of two scaled sums of squares each of which follows a scaled chi-squared distribution.

Using terms from formal language theory, the precise mathematical definition of this concept is as follows: Let S and T be two finite sets, called the source and target alphabets, respectively.

Using the composition of functions we can rewrite this statement as follows:

Using the rotation angle as a parameter, this group can be parametrized as follows:

Using the current official IAU constellation boundaries — and taking into account the variable precession speed and the rotation of the ecliptic — the solstices shift through the constellations as follows ( expressed in astronomical year numbering in which the year 0

Using differential calculus, we can determine the limit, or the value that Δy / Δx approaches as Δy and Δx get closer to zero ; it follows that this limit is the exact slope of the tangent.

Using the Greek numerical prefixes, these would be as follows for series with between two and ten entries:

Using the above rule, we can complete the proof of Fermat's little theorem quite easily, as follows.

Using the simple formula, the percent change of the average amount of each parameter, considered independently, to cause a one degree Celsius change in steady-state average earth temperature ( i. e., the climate sensitivity ) is as follows:

There are thus ( 2n + 1 ) coefficients for every value of n. Using these facts and the well known formula,, it follows that the total number of coefficients is given by

Using the more modern interpretation, the cognitive functions of the ISTP are as follows:

Using the more modern interpretation, the cognitive functions of the INTP are as follows:

) Using the < gml: pos > element instead of the < gml: coordinates > element, the same point can be represented as follows:

Using the story on a white-owned factory that controls the economy of the local society and the effects of racism and intimidation on the lives of the African-American community, the film follows the psychological transition of a family and community members from belief in a strict non-violent stance to belief in self-defense.

Using the reification vocabulary, a reification of the statement about the person's membership would be given by assigning the statement a URIref such as so that describing statements can be written as follows:

Using the more modern interpretation, the cognitive functions of the INTJ are as follows:

Using the more modern interpretation, the cognitive functions of the ENTJ are as follows:

Using the more modern interpretation, the cognitive functions of the ENFJ are as follows:

Using the more modern interpretation, the cognitive functions of the ESTP are as follows:

Using the more modern interpretation, the cognitive functions of the ESTJ are as follows:

Using the more modern interpretation, the cognitive functions of the ISTJ are as follows:

Using the more modern interpretation, the cognitive functions of the ENTP are as follows:

Using the more modern interpretation, the cognitive functions of the ENFP are as follows:

Using the more modern interpretation, the cognitive functions of the ESFP are as follows:

Using an orchestra does not make a piece symphonic rock ; it must meet the critera for being progressive rock in addition to the qualities listed for being symphonic.

Using the Curry – Howard isomorphism-types also serve to model implication and universal quantification: e. g. a term inhabiting is a function which assigns to any pair of natural numbers a proof that addition is commutative for that pair and hence can be considered as a proof that addition is commutative for all natural numbers.

* Using sign-magnitude representation requires only complementing the sign bit of the subtrahend and adding, but the addition / subtraction logic needs to compare the sign bits, complement one of the inputs if they are different, implement an end-around carry, and complement the result if there was no carry from the most significant bit.

Using the LCD ( or any multiple of it, such as the product of the denominators ) as a denominator enables addition, subtraction or comparison of fractions:

Using a limited number of color inks, or specific color inks in addition to the primary colors, is referred to as " spot color " printing.

Using a geometry in addition to special surfaces is often fruitful.

Using FUSE, MacFUSE and Callback File System respectively for file system integration, in addition to a Java-based app accessible from any Java-enabled web browser.

Using a computer with common and well-documented hardware, the manual configuration required will be considerably simpler than the alternative option of manually ' building ' ( installing ) MythTV, in addition to ' building ' numerous other programs that MythTV depends on.

Using competition reaction experiments, relative rates of addition for different cycloaddition reactions have been found to offer general findings on factors in reactivity.

Using a wide spectrum of components, his work often includes organic materials, such as flowers, food and sand, in addition to deconstructed computer parts, auto-parts and a multitude of found and collected objects.

Using this approach any plain Java class can be converted into a web service just by the addition of annotations into the Java source files.

* Using a ( rarer type of ) credit card that gives interest free periods on balance transfers, in addition to normal purchases, to repay debt on an expiring card-then repaying this debt in full as a credit card account settled in full each month attracts no interest.

Using his North America base, Krupp has brought several young German players to North America for a variety of tournaments and camps, in addition to opening his home to two Hurricane Katrina refugees who played on his son's youth team.

Using this guideline, to achieve the 80-bit strength recommended for high security ( non-military ) by NIST, a passphrase would need to be 58 characters long, assuming a composition that includes uppercase and alphanumeric.

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

Using the equivalence between Euler angles and rotation composition, it is possible to change to and from matrix convention.

Using electron microprobe analysis, a new chemical composition for murdochite was determined that included chlorine and bromine.

Using the multiplication by a function f model of a self-adjoint operator given by the spectral theorem, this is multiplication by the composition of h with f.

Using aqueous electrodes introduces undefined, variable, and unmeasurable junction potentials to the cell in the form of a liquid-liquid junction as well as different ionic composition between the reference compartment and the rest of the cell.

* Using a deck of cards with non-standard composition.

Using the oxygen isotopic composition, pallasites are divided into 3 subgroups.