Given the properties of a concept, these are necessary and sufficient conditions to classify the concept.

Given a complete set of axioms ( see below for one such set ), modus ponens is sufficient to prove all other argument forms in propositional logic, and so we may think of them as derivative.

Given a base for the topology, in order to prove convergence of a net it is necessary and sufficient to prove that there exists some point x, such that ( x < sub > α </ sub >) is eventually in all members of the base containing this putative limit.

Given sufficient time, oxygen, and water, any iron mass will eventually convert entirely to rust and disintegrate.

Given sufficient resources for growth, a biofilm will quickly grow to be macroscopic ( visible to the naked eye ).

Given a base for a topology, in order to prove convergence of a net or sequence it is sufficient to prove that it is eventually in every set in the base which contains the putative limit.

Given its strategic value, Winston Churchill would later quote a telegram he sent to the Chief of the Imperial General Staff on 4 June 1940: " To lose Crete because we had not sufficient bulk of forces there would be a crime.

Given typesetters of sufficient skill, experienced proofreaders familiar with their typesetters ' work can accurately scan their pages without reading the text for errors that neither they nor their typesetters are responsible for.

Given sufficient time, these variations may lead to reproductive isolation.

Given a fire of sufficient duration and intensity, the pressure being generated by the boiling and expanding gas can exceed the ability of the valve to vent the excess.

Given sufficient magic energy in her surroundings she has even effortlessly turned into a female copy of the Silver Surfer.

* Given a sufficient power source, laser weapons could essentially have limitless ammunition.

Given sufficient time under fire or heat conditions, these products char away, crumble, and disappear.

Given a statistic T that is not sufficient, an ancillary complement is a statistic U that is ancillary to T and such that ( T, U ) is sufficient.

Given sufficient time, the solvent will permeate through the polymer and out into the environment, so that the chains lose their mobility.

Given the fact that HIT predisposes strongly to new episodes of thrombosis, it is not sufficient to simply discontinue the heparin administration.

Given that the increase in encryption strength afforded by four-square over Playfair is marginal and that both schemes are easily defeated if sufficient ciphertext is available, Playfair has become much more common.

Given that High Sensitivity GPS receivers may be up to 30 dB more sensitive, this is sufficient to track through 3 layers of dry bricks, or up to 20 cm of steel reinforced concrete for example.

Given sufficient voltage, the electron can be accelerated sufficiently fast to exhibit measurable relativistic effects, and the velocity must be accounted for relativistically.

Given the large number of casualties, since the ambulances and emergency vehicles were not sufficient for the transport of the injured to the city's hospitals, firefighters employed also buses, in particular the line 37, private cars and taxis.

Given these assumptions, the constraint that time travel must not lead to inconsistent outcomes could be seen merely as a tautology, a self-evident truth that cannot possibly be false, because if you make the assumption that it is false this would lead to a logical paradox.

Given initial speed u, one can calculate how high the ball will travel before it begins to fall.

Given the geography of the abbey, his landing site, and the account of his flight, to travel for " more than a furlong " ( 220 yards, 201 metres ) he would have had to have been airborne for about 15 seconds.

Given that the number of women needing to travel has doubled since 1995, there is a very strong demand for these kinds of services.

Given Amtrak's own projected 41 hour travel time, the average speed is still in excess of 55 mph ( including stops ).

Given how critical transportation is to every human culture, it is unsurprising that in an interstellar culture, which must deal with distances orders of magnitude greater than terrestrial cultures, the unique ways in which interstellar travel is described in various fictional universes tends to create major plot elements in that universe.

Given the needs of the large volume of Saints who would travel west, Church leaders decided to avoid potential conflicts over grazing rights, water access and campsites.

Given the lack of air and rail transit, most goods have to travel by truck or ship, which makes exporting and importing more difficult.

Given the amount of damage, and the travel time to Jupiter Station, Archer decides that it is time for someone to help them out for once, and orders Ensign Sato to send a general distress-call.

Given the problem of the aerodynamic design of the nose cone section of any vehicle or body meant to travel through a compressible fluid medium ( such as a rocket or aircraft, missile or bullet ), an important problem is the determination of the nose cone geometrical shape for optimum performance.

Given their relative isolation, several mistaken rumours regarding the Torres Islands and their people have become widespread by way of travel guides, inter-island gossip and the mass media.

Given the problem of the aerodynamic design of the nose cone section of any vehicle or body meant to travel through a compressible fluid medium ( such as a rocket or aircraft, missile or bullet ), an important problem is the determination of the nose cone geometrical shape for optimum performance.

Given a fertile mind and boundless energy, he pursued other fields of interest: growing vegetables in Puerto Rico ; owning a beautician school, restaurants, lounges, a marina, marine dredging, data processing services, multi-level marketing of vitamins and cosmetics, cement block manufacturing, country-western music publishing and recording studios, telecommunications companies, a high-end boutique shop, general insurance agency, internet connection and data center, a walk-in medical clinic, wholesale gasoline and diesel fuel distribution operation, a travel agency, home security monitoring, mail order medical and diabetic supply company, a Big & Tall men's clothing store, an AM radio station, a mobile medical screening company, a book publishing company, a warehouse fulfillment and distribution company, an advertising media consulting firm, television stations, mobile home parks, a 225 room hotel in Nassau, Bahamas, a 500, 000 square foot commercial warehouse complex in Las Vegas, and a five-star 23 story resort hotel and residences on Ft. Lauderdale Beach.

# Given his interest in travel and foreign cultures, many of Bjorn ’ s videos feature men of clearly identifiable ethnic backgrounds and nationalities in typical surroundings.

Given that travel to London was an onerous burden during the medieval period, however, the Statute of Westminster II provided in 1285 for trial of fact in civil cases at the local assizes.

Given the poor condition of municipal roads, travel times may exceed 8 hours from one side to the other.

Given the typically remote location and the absence of any travel infrastructure at the mines, coal camps became a part of being a coal miner.

Given more time to prepare for trial, Kidd likely would have been able to find the deposition Palmer gave when he was captured in Rhode Island two years earlier.

Given that a natural language such as English contains, at any given time, a finite number of words, any comprehensive list of definitions must either be circular or rely upon primitive notions.

Given that the dispute with Pizarro over Cuzco had kept intensifying, Almagro spent a great deal of time and money equipping a company of 500 men for a new exploration south of Peru.

Given the magnitude of the disaster, reconstruction and economic recovery will remain the primary focus of the Salvadoran Government for some time to come.

Given the limited budget for schools, it is claimed that a voucher system would weaken public schools while at the same time not necessarily providing enough money for people to attend private schools.

Given the state at some initial time ( t = 0 ), we can solve it to obtain the state at any subsequent time.

Given these two assumptions, the coordinates of the same event ( a point in space and time ) described in two inertial reference frames are related by a Galilean transformation.

Given the repeated failures and frustrations of workers ' revolutions and movements, Marx also sought to understand capitalism, and spent a great deal of time in the reading room of the British Museum studying and reflecting on the works of political economists and on economic data.

Given two secondary stations, the time difference ( TD ) between the primary and first secondary identifies one curve, and the time difference between the primary and second secondary identifies another curve, the intersections of which will determine a geographic point in relation to the position of the three stations.

Given that the cost of replacing an executive can run over 100 % of his or her annual salary, any investment of time and energy in re-recruitment will likely pay for itself many times over if it helps a business retain just a handful of key players that would have otherwise left.

Given fixed resources, it is often better to spend more time collecting additional training data and more informative features than it is to spend extra time tuning the learning algorithms.

Given enough time and repeated exposure to the antibiotic, a population of antibiotic-resistant bacteria will emerge.

Given printing practices at the time ( which included type-setting from dictation ), no two editions turned out to be identical, and it is relatively rare to find even two copies that are exactly the same.

Given time t, the source produces ωt oscillations.

Given prevailing views on homosexuality at the time, circulation of the diaries helped undermine support for clemency for Casement.

Given their wide historical distribution, red wolves probably utilized a large suite of habitat types at one time.

Given that by this time it was common for castles to be built in stone, and that many barons had expanded or refortified their castles, this was not an easy task.

Given the medieval medicine's main theory of humorism, spices and herbs were indispensable to balance " humors " in food, a daily basis for good health at a time of recurrent pandemics.

Given the number of homes, it seems likely that no more than fifty people lived in Skara Brae at any given time.

" Given enough time, these formations can meet and fuse to create columns of calcium carbonate.