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

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.

Using whatever equipment the vassal could obtain by virtue of the revenues from the fief, the vassal was responsible to answer calls to military service on behalf of the lord.

Using this procedure, Groseclose and Milyo obtain the stark result that all sampled news providers-except Fox News ' Special Report and the Washington Times-are located to the left of the average Congress member, i. e. there are signs of a liberal bias in the US news media.

Using this legal device, a person can obtain immediate cessation of any action he considers to be an infringement of his rights.

Using this method with two bodies, navigators were finally able to cross two position lines and obtain their position – in effect determining both latitude and longitude.

Using a staggered board of directors can diminish the ability of minority factions to obtain representation by reducing the number of seats up for election at any given time.

Using polygamma functions we obtain:

Using primitive Burroughs, later IBM, minicomputers, Dunnigan put together a highly sophisticated system to obtain marketing information from his customers.

Using again just the syntactic transformations available in this formalism, one can obtain so called fixed point combinators ( the best-known of which is the Y combinator ); these, by definition, have the property that f ( Y ( f )) = Y ( f ) for all functions f.

Using OPAC to identify sources and catalog numbers, overseas users may obtain certain materials from the NDL through interlibrary loan.

Using it to improve the already improved estimator does not obtain a further improvement, but merely returns as its output the same improved estimator.

Using Arg z instead of arg z, we obtain the principal value of the logarithm, and we write

Using the traditional technique also makes the balls expensive and difficult to obtain.

Using X-rays, CT, ultrasound, MRI, and other imaging modalities, interventional radiologists obtain images which are then used to direct interventional instruments throughout the body.

Using funds from a taxpayer-funded government grant to obtain, recondition, and maintain 30 bicycles, project organizers announced plans to station the bicycles in downtown Tucson and areas adjacent to the University of Arizona.

Using differential calculus we can obtain conditions on x under which this holds.

Using DPA, an adversary can obtain secret keys by analyzing power consumption measurements from multiple cryptographic operations performed by a vulnerable smart card or other device.

Using his knowledge of the future, Strucker attempted to complete the atom bomb before the allies, sending HYDRA agents to obtain the materials.

( Using the callsign " Shanwick Oceanic ", the Prestwick Centre has 2 dedicated VHF frequencies specifically for the issue of Oceanic Clearances to westbound flights about to enter the Shanwick OCA, and also provides an ACARS based system called ORCA-Oceanic Route Clearance Authorisation-for suitably equipped aircraft to obtain such clearances without the need for voice communications ).

Using the HO model to obtain the impact force of real climbing ropes as a function of fall height h and climber's weight mg, one must know the experimental value for E of a given rope.

Using the binomial theorem also for the exponents m and n, and then the above formula for the product of polynomials, we obtain

Using this integrability condition, it is possible to define the exponential map of the Lie algebra and in this way obtain, locally, a group action on P.

Using this method, the aim of the game is to gather rings and Chaos Emeralds in order to obtain the highest score at the end of the game.

Using Newton's second law of motion, we obtain:

Using the commutation relation between and we obtain: