Using the example of a solution of two liquids, A and B, if no other gases are present, then the total vapor pressure p above the solution is equal to the weighted sum of the " pure " vapor pressures of the two components, p < sub > A </ sub > and p < sub > B </ sub >.

Using Vinogradov's method, Chudakov, van der Corput, and Estermann showed that almost all even numbers can be written as the sum of two primes ( in the sense that the fraction of even numbers which can be so written tends towards 1 ).

) Using more derivatives, and by tweaking the quadrature, we can do a similar error analysis using a Taylor series ( using a partial sum with remainder term ) for f. This error analysis gives a strict upper bound on the error, if the derivatives of f are available.

Using the periodicity of the trajectory, we can split the radiation integral into a sum over terms.

Using this property as definition, and extending this to the empty product, the right-hand side of this equation evaluates to b < sup > 0 </ sup > for the empty set, because the empty sum is defined to be zero, and therefore the empty product must equal one.

Using Fourier analysis, periodic waveforms are represented as the sum of an infinite series of sine waves.

Using this idea inductively we can write R ( x ) as a sum with denominators powers of irreducible polynomials.

Proof: Using the same sum for R ( p, n ) as in Lemma 2, we see that since p < sup > 2 </ sup > > 2n, in fact only the first term is nonzero ; this term is exactly the right-hand side of the above equation.

Using the Fourier series, just about any practical function of time ( the voltage across the terminals of an electronic device for example ) can be represented as a sum of sines and cosines, each suitably scaled ( multiplied by a constant factor ), shifted ( advanced or retarded in time ) and " squeezed " or " stretched " ( increasing or decreasing the frequency ).

Using Reynolds Decomposition to express the horizontal flow in the direction as the sum of a slowly varying component ,, and a turbulent component ,,:

Using such a frame, any point p in KP < sup > n </ sup > may be described by a projective version of barycentric coordinates: a collection of n + 2 scalars μ < sub > j </ sub > which sum to zero, such that p is represented by the vector

where e is the identity of G. Though the f ″ defined a couple of paragraphs back is only defined for G-irreducible representations, and though A ⊗ B is not a G-irreducible representation in general, we claim this argument could be made correct since A ⊗ B is simply the direct sum of copies of Bs, and we have shown that each copy all maps to the same G × G-irreducible subrepresentation of C, we have just showed that as an irreducible G × G-subrepresentation of C is contained in A ⊗ B as another irreducible G × G-subrepresentation of C. Using Schur's lemma again, this means both irreducible representations are the same.

Using the ring R as above, with residue field K, the identity element of G may be decomposed as a sum of mutually orthogonal primitive idempotents ( not necessarily

Using some of the huge sum of money Roma paid for Batistuta, Cecchi Gori brought in young striker Nuno Gomes from Benfica for € 17 million.

Using her ability, Molly wins a large sum of money from a local talent competition, by hypnotizing the crowd into believing that she is a talented singer and dancer.

Using series acceleration it is the sum of the n-th differences of ln ( k ) at k = 1 as given by:

Using the poisson-weighted mixture representation for, and the fact that the sum of chi-squared random variables is also chi-squared, completes the result.

Using a cheap brush, coat one side of the bottom with the resin and then apply the cloth.

Using kinship terms in political speeches increased audience agreement with the speaker in one study.

Using visual imagery of a soroban, one can arrive at the answer in the same time as, or even faster than, is possible with a physical instrument.

Using the Fibonacci sequence would keep the number of grains within any one field at minimum.

Using what became one of van Vogt's recurring themes, it told the story of a 9-year-old superman living in a world in which his kind are slain by Homo sapiens.

Using a core from the original, the Lemba priests constructed a new one.

* Using a suitable catalyst ( e. g. Wilkinson's catalyst ), it is possible to reduce just one of the double bonds of an allene.

Using these tools, one can write a more complex text editor, and a simple compiler for a higher-level language and so on, until one can have a graphical IDE and an extremely high-level programming language.

Using an existing language to bootstrap a new language is one way to solve the " chicken or the egg " causality dilemma.

Using the techniques of modern biotechnology, one or two genes ( Smartstax from Monsanto in collaboration with Dow AgroSciences will use 8, starting in 2010 ) may be transferred to a highly developed crop variety to impart a new character that would increase its yield.

Using the one-point compactification, one can also easily construct compact spaces which are not Hausdorff, by starting with a non-Hausdorff space.

Using more than one codec or encoding scheme successively can also degrade quality significantly.

Using a three-digit Julian day number saves one byte of computer storage over a two-digit month plus two-digit day, for example, " January 17 " is 017 in Julian versus 0117 in month-day format.

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 himself as the sole subject in his experiment, he memorized lists of three letter nonsense syllable words — two consonants and one vowel in the middle.

Using RPN, one could get the result of the mathematical expression this way:

Using uniformitarianism, which states that one cannot make an appeal to any force or phenomenon which cannot presently be observed ( see catastrophism ), Darwin theorized that the evolutionary process must occur gradually, not in saltations, since saltations are not presently observed, and extreme deviations from the usual phenotypic variation would be more likely to be selected against.

Using a modified Doom engine, Heretic was one of the first first-person games to feature inventory manipulation and the ability to look up and down.

Using the questions or processes of one domain on the subject matter of another domain will logically produce puzzles, paradoxes, and inconsistencies.

Using a defense in depth strategy, should one defensive measure fail there are other defensive measures in place that continue to provide protection.

Using the thoroughbred racing convention of one length equaling one-fifth of a second to calculate Sham ’ s time, he also finished in under two minutes.

Using a telescope, Rømer observed the motions of Jupiter and one of its moons, Io.

Using vernacular language that would be understandable to the majority of China's populace, he put forward his socialist political views, calling for revolution against the government ; in one notable article, he proclaimed the need for a " Great Union of the Popular Masses ", but failed to put forward an expressly Marxist analysis of how that revolution should proceed.

Using quantum theory Dirac showed that if magnetic monopoles exist, then one could explain the quantization of electric charge --- that is, why the observed elementary particles carry charges that are multiples of the charge of the electron.