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

Using these tables, often calculated out to 16 decimal places or more for some functions, one could look up values to plug into the formulas given and achieve very good numerical estimates of some functions.

Using his theory, he discovered homologies showing that slightly changed body parts served different functions to meet new conditions, and he found an intermediate stage in the evolution of distinct sexes.

Using newer versions of True BASIC, some of the older functions are blocked out.

Using floating-point variables and mathematical functions ( sin (), cos (), atan2 (), log (), exp (), sqrt () are the most popular ones ) of double precision as opposed to single precision comes at execution cost: the operations with double precision are usually slower.

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 a root finder on these sixty functions produces the exact collision times for the two given triangles and the two given trajectory.

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 power series expansions above, the Fresnel integrals can be extended to the domain of complex numbers, and they become analytic functions of a complex variable.

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

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

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

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

Using launch systems presently available, we are developing satellites to scout the world's weather ; ;

Hence S breaks up into uncountably many orbits under G. Using the axiom of choice, we could pick a single point from each orbit, obtaining an uncountable subset X of S with the property that all of its translates by G are disjoint from X.

Using the identity, we have

Using Zorn's lemma, it can be shown that every field has an algebraic closure, and that the algebraic closure of a field K is unique up to an isomorphism that fixes every member of K. Because of this essential uniqueness, we often speak of the algebraic closure of K, rather than an algebraic closure of K.

Using the above theorem it is easy to see that the original Borsuk Ulam statement is correct since if we take a map f: S < sup > n </ sup > → ℝ < sup > n </ sup > that does not equalize on any antipodes then we can construct a map g: S < sup > n </ sup > → S < sup > n-1 </ sup > by the formula

Using the right-hand rule, we expect the curl to be into the page.

Using the product rule in reverse, we get:

Using now the power series definition from above we see that for real values of x

Using these two examples, we can make some useful observations that motivate the major ideas in lambda calculus.

Using the definition of free variables, we may now define a capture-avoiding substitution.

Using our senses of sight, hearing, touch, smell, and taste, we perceive colors, sounds, textures, etc.

* Using the above methods for simple polygons, we discover that particular regions within the polygon may have their area multiplied by a factor which we call the density of the region.

Using this information we can perform a series of exchanges on the input array that puts it in order, moving items only once.

Using =, we find that the period of precession is given by:

Using relativistic relations for energy and momentum, we have

Using the chosen basis, we may write

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 previous example, we would get the following:

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

Using this Carnapian approach, Maher identifies a proposition which we intuitively ( and correctly ) know to be false, but which we easily confuse with the paradoxical conclusion.

Using Le Chatelier's principle, we can predict that the amount of methanol will increase, decreasing the total change in CO.