Using mathematical rules and procedures based on properties of reducible configurations, Appel and Haken found an unavoidable set of reducible configurations, thus proving that a minimal counterexample to the four-color conjecture could not exist.

Using his new rules vigorously, he won passage of the Lodge Fair Elections Bill in the House in 1890.

Using it for such rules makes the relationship between the result of the rule and the environment obvious.

Using that, it follows that addition and composition of matrices obey the usual rules for matrix addition and matrix multiplication.

Using software to analyze one's own histories is generally accepted, however acquiring histories of hands in which a player did not participate violates the rules of most cardrooms.

Using the rules for complex numbers, the magnitude of this function is

Using political connections, Mayer called on the Justice Department's anti-trust unit to block the merger despite that MGM itself coupled with Loews Theatres was considered being in violation of anti-trust rules.

Using cellular automata and treating each bird in the flock as an autonomous agent, with simple local rules controlling agent behaviour relative to closest neighbouring birds, a system can be constructed where the overall behaviour of the collective agents reflects the behaviour of the real flock.

Using these systems in international trials is also allowed, as long as the qualifying criteria are within the general FCI framework of rules.

Using formation rules, a player could move three to five adjacent figures while using only one action.

* Using outdated testing rules and loose standards to award Energy Star ratings.

* Most commonly, it refers to a programming language feature making use of the method lookup rules for dispatching so-called self-calls as defined by Lieberman in his 1986 paper " Using Prototypical Objects to Implement Shared Behavior in Object-Oriented Systems ".

Using pro-wrestling rules – no closed fisted punching to the head ( closed fisted punches were allowed to the body ), breaks on the ropes, but fighting for real – Shamrock beat his friend and mentor, MMA legend Masakatsu Funaki by arm-triangle choke in the main event of the very first Pancrase show on September 21, 1993.

Using trails and other pathways only accessible to those on foot is against the rules.

These encoding rules are available from NARA, upon request, in the form of General Information Leaflet 55, " Using the Census Soundex ".

Using the hand or arm by anyone other than a goalkeeper is illegal according to the rules of association football.

Using various rules, the program can then detect various errors, such as agreement in tense, number, word order, and so on.

Using the advanced rules a character may well collapse from particular types of mortal wounds, or in combat from sheer exhaustion, especially if wearing heavy armour.

Using standard boxing rules, 16 oz.

Using bots is incidentally against the rules of all of the current main Massively multiplayer online role-playing games ( MMORPGs ).

Using out-of-character communications to influence game play is against the rules, and the creators reserve the right to delete and even ban problematic abusers.

Using a small number of simple rules, each Feynman diagram can be readily expressed in its corresponding algebraic form.

Using these rules a player who rolls a 20 on a 20 sided die does double the normal damage, and a 20 followed by a 19 or 20 counts as a killing blow: " this simulates the ' lucky hit ' on a vital organ.

Using the convenience that Laplace transforms follow the rules of linearity, one can solve the above example for y by performing a Laplace transform on both sides of the differential equation, substituting in the initial values, solving for the transformed function, and then performing an inverse transform.

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

Using significant with the layperson meaning " of practical significance " in such contexts would qualify as punning, such as the webcomic xkcd's double pun " statistically significant other ".

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.

Using contacts from among the Montoneros ' many double agents ( allegedly including the leader, Mario Firmenich ), the agency kept the Born brothers in a known SIDE safehouse for nine months until their June 1975 release without public suspicion of SIDE involvement, a successful false flag operation that led to others ( albeit less ambitious ones ) in the following months.

( Using the flow rates of Niagara Falls as a standard of comparison, this is more than double its average of 1, 833 cubic metres and about a third over its maximum.

Using a radiate crown to indicate double value was also used on the antoninianus ( double denarius ) and double sestertius.

Using a BRST antifield formalism with a duality gauge fixing, a consistent quantization in spaces of double dual curvature is obtained.

Using the express lanes as a lone driver without a transponder during charged periods, or crossing the double white lines will result in a large fine placed upon the driver.

Using the basic flintlock action in common usage at the time, the pistols were muzzle loading single shots, or in some cases, double barreled in an over-under manner.

Using double salvo cannon fire, the Koreans threw a barrage which quickly began damaging Japanese vessels.

Using middle C ( C4 ) as a guide, the next higher C is C5 or tenor C. The next C is C6 or soprano high C. The next C, C7 or double high C, is again one octave higher.

Using his clone brother as a double, Miles is able to persuade the Cetagandans that Miles Vorkosigan and Admiral Naismith are not the same person, preserving the Vorkosigan identity as a place for him to hide between missions.

Using a dual-fisted double Turbo Punch delivers force equal to a full-powered punch from a superhuman with Class 100 strength.

Using this method, the customer does not need to authenticate his or her identity by PIN entry or signature, as on a regular machine.

Using Green Tipped Bourbon Coffee plants imported in 1733, crops were grown on several sites, including the Bamboo Hedge Estate Sandy Bay estate used for the 1851 Great Exhibition entry.

This rivalry went far enough that the OS-9 version 2. 4 manual had this entry describing UNIX in the Glossary of Appendix C of " Using Professional OS-9 ":

Using this device, Max Fleischer was able to secure a contract with the John R. Bray Studios, and in 1919 they began Out of the Inkwell as an entry in each monthly in the Bray Pictograph Screen Magazine released through Paramount ( 1919 – 1920 ), and later Goldwyn ( 1921 ).

High platform design ( Using outside steps to avoid having a level entry from the platform ) is troublesome.

# Using asymmetric key cryptography, the originator uses the public key ( obtained from the directory ) of the first node in the circuit, known as the entry node, to send it an encrypted message, called a create cell, containing:

Using the already established encrypted link, the originator sends the entry node a relay extend cell, which is like any relay cell, only that it contains a create cell intended for the next node ( known as the relay node ) in the chain, encrypted using the relay node's public key and relayed to it by the entry node, containing the following:

Using the same hole, the operators gained entry to the plane but in the confusion opened fire indiscriminately and killed and injured more passengers.

Using the bell bottom hatch for this purpose has the advantage of only needing one hatch, and the disadvantage of having to lift the bell up and place it over a vertical entry to the chamber.

Using admissions software developed by its own students, the institute organizes the counseling process ( invitation of applications, scrutiny of forms, preparation of merit list, dispatch of call letters and allotment of seats ) for direct entry into second year B. E / B. Tech courses all over the state.

Using the details from an index, a copy ( certificate ) of the corresponding birth, marriage or death entry could be applied for at the cashiers ' section on the same floor.

Using WikiScanner they found out that the IP address matched the address of the party responsible for a recent alteration to the Buckcherry Wikipedia entry which added band manager Josh Klemme's name .. After emailing Klemme with their findings, they found that his IP-address was the same as the torrent uploader.

Using the character sheet to cross-reference their action with that of their opponent, players then turn to a specific entry in the book they are holding in order to determine the results.

Using a protein called Nucleoplasmin, the archetypal ‘ molecular chaperone ’, they identified a domain in the protein which acted as a signal for nuclear entry.

Using the actual membership of the index, applying entry and exit dates to gain the appropriate return during inclusion in the index, would allow for a bias-free output.

Using far heavier materials, Dallara created an extremely strong chassis from square-section tubing which provided easy entry and exit through the doors, aided in the quietness of the car, and provided a solid platform on which to mount the body, much like the Aston Martin DB4.