By the well ordering principle, if there are positive integers that satisfy a given property, then there is a smallest positive integer that satisfies that property ; therefore, there is a smallest positive integer satisfying the property " not definable in under eleven words ".

By World War II, however, the practice was so widespread that during the Nuremberg trials, the charges against German Admiral Karl Dönitz for ordering unrestricted submarine warfare were dropped, notwithstanding that the activity constituted a clear violation of the Second London Naval Treaty of 1936.

By this time, Allenby had reached Mahamba and set camp there, Henry was pursuing another Boer wagon convoys and Queen-regent Labotsibeni was ordering her Impis to clear their land from the Boers.

By ordering the codestream in various ways, applications can achieve significant performance increases.

By Louis's reign, Bastille prisoners were detained using a " lettre de cachet ", " a letter under royal seal ", issued by the king and countersigned by a minister, ordering a named person to be held.

By 1855, the gold rush was over and Westervelt, like other shipbuilders, began looking for new markets to keep his yard busy, as the shipping lines that had carried almost 2. 7 million emigrants across the Atlantic by sailing vessels between 1846 – 1855, were ordering fewer ships than before.

By obliquely ordering Westmoreland to demand more forces, Wheeler was attempting to solve two pressing problems.

By some divine ordering he comprehended the meaning of this and dug there, and there was found a coffin of a man of extraordinary size, a bronze spear lying by its side, and a sword.

By May 1943, a wooden mockup of the final Maus configuration was ready and presented to Hitler, who approved it for mass production, ordering a first series of 150.

By logic in this context we mean the rules that relate set operations and ordering relations, such as de Morgan's laws.

By contrast, a translation moves every point, a reflection exchanges left-and right-handed ordering, and a glide reflection does both.

By using a zig-zag ordering to group the non-zero entries and run length encoding, the quantized matrix can be much more efficiently stored than the non-quantized version.

By March 1943, the OUN-B leadership issued secret instructions ordering their members who had joined the German police in 1941-1942, numbering between 4, 000-5, 000 trained and armed soldiers, to desert with their weapons and to join the units of the OUN-B in Volyn.

By changing the ordering of members in a structure, it is possible to change the amount of padding required to maintain alignment.

By the late-1990s, many of the showrooms had been converted to allow a more traditional approach to shopping in addition to the catalog ordering process.

By weight, gels are mostly liquid, yet they behave like solids due to a three-dimensional cross-linked network within the liquid.

Eudoxus used 27 concentric spherical solids to answer Plato's challenge: " By the assumption of what uniform and orderly motions can the apparent motions of the planets be accounted for?

By progressively converting dissolved material into solids, usually a biological floc, which is then settled out, an effluent stream of increasing purity is produced.

By taking N to be the Avogadro constant N < sub > A </ sub >, and using the relation R = N < sub > A </ sub > k < sub > B </ sub > between the gas constant R and the Boltzmann constant k < sub > B </ sub >, this provides an explanation for the Dulong – Petit law of specific heat capacities of solids, which stated that the specific heat capacity ( per unit mass ) of a solid element is inversely proportional to its atomic weight.

By understanding how matter behaves at the critical point, scientists hope to gain a better insight into a variety of physics problems ranging from phase changes in fluids to changes in the composition and magnetic properties of solids.

By adjusting the local oscillator frequency correctly, the beat frequency is in the audio range, and can be heard as a tone in the receiver's earphones whenever the transmitter signal is present.

By selecting the witness as a random string, the verifier is a probabilistic polynomial-time Turing Machine whose probability of accepting x when x is in X is large ( greater than 1 / 2, say ), but zero if x is not in X ( for RP ); of rejecting x when x is not in X is large but zero if x is in X ( for co-RP ); and of correctly accepting or rejecting x as a member of X is large, but zero of incorrectly accepting or rejecting x ( for ZPP ).

By making use of the newly-invented achromatic microscope, Ehrenberg was able to see Euglena's eyespot, which he correctly identified as a " rudimentary eye " ( although he reasoned, wrongly, that this meant the creature also had a nervous system ).

By the first few years of the 21st century the underlying cryptographic engineering was clearly not easy to deploy correctly.

By contrast, the role of extrinsic rewards and stimuli can be seen in the example of training animals by giving them treats when they perform a trick correctly.

By practicing prāyaṣcitta ( repentance ), a soul gets rid of sins, and commits no transgressions ; he who correctly practises prāyaṣcitta gains the road and the reward of the road, he wins the reward of good conduct.

By referencing a firing order diagram and knowing the direction the rotor turns, ( which can be seen by cranking the engine with the cap off ) the spark plug wires can be correctly routed.

By providing an air space in front of the eyes, light enters normally and the eye is able to focus correctly.

By now Hincks had recognized a large number of determinatives and had correctly established their readings.

By decisions of the Stoglavy Sobor, church ceremonies and duties in the whole territory of Russia were unified, and norms of church life were regulated with the purpose of increasing the educational and moral level of the clergy to ensure they would correctly fulfill their duties, such as creation of schools for preparation of priests.

By correctly positioning the plate and pressing it against one end of the array of printing wires, only the correct wires were pressed against the ribbon and then the punched card.

By 1974 the machine still didn't work correctly.

By playing this way, the opponent had less of an advantage if the contestant missed the question and the opponent answered it correctly.

By example, in El Mundo de los Aviones ( World of Planes ), Capulina is a pilot that always fail to land his plane correctly hitting on some wall and encrusting a partner in the process.

By then, the number of United States ground forces in South Vietnam had fallen to 65, 000 from an all-time high of 543, 000 in 1969 ; and the North Vietnamese correctly anticipated that the US was unwilling to commit ground troops again.

By setting VAD parameters correctly, dialers can determine whether a person or a machine answered the call, and if it's a person, transfer the call to an available agent.

By definition of, if at most errors are made during transmission of a codeword then minimum distance decoding will decode it correctly ( i. e., it decodes the received word as the codeword that was sent ).

By citing the previous research of Edward Sabine, which established a correlation between sunspots and magnetic storms, Stewart was able to correctly advance the theory that the event observed by Richard Carrington and the magnetic disturbance that was recorded at the same time were in fact connected.

By, the Braves were rumored ( correctly ) to be moving to Atlanta.

By 3 – 5 years, children usually have difficulty using words correctly.

Whenever any result is sought by its aid, the question will then arise — By what course of calculation can these results be arrived at by the machine in the shortest time?

By terms of the treaty, moreover, Alfred was to have control over the Mercian city of London and its mints — at least for the time being.

By a further act of 1541 — which was not repealed until 1845 — artificers, labourers, apprentices, servants and the like were forbidden to play bowls at any time except Christmas, and then only in their master's house and presence.

By the summer of 1944, victories in the Southwest and Central Pacific had brought the war closer to Japan, with American bombers able to strike at the Japanese main islands from air bases secured during the Mariana Islands campaign ( June — August 1944 ).

By now the rebel forces were said to have numbered 230, 000, however, this number should be treated with scepticism — Dio's account is known only from a late epitome, and ancient sources commonly exaggerate enemy numbers.

By 1914, around 100 people — men, women, and children — were living on Clipperton Island, resupplied every two months by a ship from Acapulco.

Confucius never stated whether man was born good or evil, noting that ' By nature men are similar ; by practice men are wide apart ' — implying that whether good or bad, Confucius must have perceived all men to be born with intrinsic similarities, but that man is conditioned and influenced by study and practise.

By the fourth century, however, " confessors "— people who had confessed their faith not by dying but by word and life — began to be venerated publicly.

By regulating the processing parameters — especially the gases introduced, but also including the pressure the system is operated under, the temperature of the diamond, and the method of generating plasma — many different materials that can be considered diamond can be made.

By the 1980s, several independent publishers, such as Pacific, Eclipse, First, Comico, and Fantagraphics had started releasing a wide range of styles and formats — from color superhero, detective, and science fiction comic books to black-and-white magazine-format stories of Latin American magical realism.

By connecting outward light to inward light, via an interaction point, this equation stands for the whole ' light transport ' — all the movement of light — in a scene.

By conceiving a timeless setting — " a vanished age " — and by carefully choosing names that resembled human history, Howard shrewdly avoided the problem of historical anachronisms and the need for lengthy exposition.

By adopting the name Chung-cheng (" central uprightness "), he was choosing a name very similar to the name of Sun Yat-sen, who was ( and still is ) known among Chinese as Zhongshan ( 中山 — meaning " central mountain "), thus establishing a link between the two.

By the end of this conflict, Commodore had shipped somewhere around 22 million C64s — making the C64 the best selling computer of all time.

" By the end of the following year he had taken up the ukelele and tea-chest bass and begun to participate in skiffle sessions with friends, and had started to play the piano ; meanwhile his stage presentation of numbers by both Presley and Chuck Berry — complete with gyrations in tribute to the original artists — to his local Wolf Cub group was described as " mesmerizing ... like someone from another planet.

By definition, this classifies it as both a river and a strait — a strait being a narrow passageway connecting two large bodies of water.

By 1859 — known by Ecuadorian historians as the Terrible Year — the nation was on the brink of anarchy.

By 1993 — nine months ahead of schedule — the military had cut personnel from a war-time high of 63, 000 to the level of 32, 000 required by the peace accords.