During this period, his visual style simplified ( owing in part to the constraints of the Hollywood studio system ) and his worldview became increasingly pessimistic, culminating in the cold, geometric style of his last American films, While the City Sleeps ( 1956 ) and Beyond a Reasonable Doubt ( 1956 ).

While for p equal to 0 we assume that it's equal to the geometric mean ( which is, in fact, the limit of means with exponents approaching zero ):

While earlier Greek proofs were largely geometric demonstrations, the development of arithmetic and algebra by Islamic mathematicians allowed more general proofs that no longer depended on geometry.

While Euclid was the originator of what we now understand as the published geometric proof, Pythagoras created a closed community and suppressed results ; he is even said to have drowned a student in a barrel for revealing the existence of irrational numbers.

While seldom used explicitly, the geometric view of the complex numbers is implicitly based on its structure of a Euclidean vector space of dimension 2, where the inner product of complex numbers and is given by ; then for a complex number its absolute value || coincides with its Euclidean norm, and its argument with the angle turning from 1 to.

While contrary to normal physical intuitions, such omnipresence is logically possible by way of the classic geometric point or its equivalent, in that such a point is, by definition, within all of space without taking up any space.

While the theory has developed in its own right, a fairly recent trend has sought to parallel the commutative development by building the theory of certain classes of noncommutative rings in a geometric fashion as if they were rings of functions on ( non-existent ) ' noncommutative spaces '.

While many users of lower doses of 2C-E have reported that the compound produces mainly closed-eye visuals, users of higher doses have compared its open-eye visuals with those produced by LSD ; a significant proportion of these comparisons have favored the visual effects of 2C-E in terms both of their geometric complexity and in the variety of colors seen during an experience.

While operation of a phased array is readily understood as a completely geometric technique, the fact that a synthetic aperture system gathers its data as it ( or its target ) moves at some speed means that phases which varied with the distance traveled originally varied with time, hence constituted temporal frequencies.

While centered square numbers, like figurate numbers in general, have few if any direct practical applications, they are sometimes studied in recreational mathematics for their elegant geometric and arithmetic properties.

While this seems at first to be a weakness in the concept of the word metric, it can be exploited to prove theorems about geometric properties of groups, as is done in geometric group theory.

While this ' geometric ' description related to the kinematic solution ( using the Bragg condition ) is very powerful and useful for orientation and texture analysis, it only describes the geometry of the crystalline lattice and ignores many physical processes involved within the diffracting material.

While at the beginning of the nineteenth century Italy had very little mathematical standing, largely influence by his work, the end of the century found Italy in the lead along geometric lines.

While the interpretations that have been given are inferences only, they gain support from such comments as the following, which was made by one of the Kohnstamm-negative subjects who did not, on the first trial, perceive the tilt illusion.

While accounts of the progress of the tsunami came in from various points in the Pacific ( Midway reported it was covered with nine feet of water ), the Hawaiian station made its calculations and notified the military services and the police that the first big wave would arrive at Honolulu at 23:30 Greenwich time.

While Poirot's actual death and funeral occurred in " Curtain ", years after his retirement from active investigation, it was not the first time Hastings attended the funeral of his best friend.

While the engine installation crew works on the second car, the first car can be moved to the hood station and fitted with a hood, then to the wheels station and be fitted with wheels.

While Renaissance artists sought nature to find their style, the Mannerists looked first for a style and found a manner. In Mannerist paintings, compositions can have no focal point, space can be ambiguous, figures can be characterized by an athletic bending and twisting with distortions, exaggerations, an elastic elongation of the limbs, bizarre posturing on one hand, graceful posturing on the other hand, and a rendering of the heads as uniformly small and oval.

While Parsons pursued his own solo career and took many members of the Project on the road for the first time in a successful worldwide tour, Woolfson went on to produce musical plays influenced by the Project's music.

While anagramming is certainly a recreation first, there are ways in which anagrams are put to use, and these can be more serious, or at least not quite frivolous and formless.

While Stradivari's first known violin states that he was a pupil of Amati, the validity of his statement is questioned.

124-126 .</ ref > While other philosophers also recognized such transitions in states of matter, Anaximenes was the first to associate the quality pairs hot / dry and cold / wet with the density of a single material and add a quantitative dimension to the Milesian monistic system .< ref > Kirk, G. S., J. E.

While Aristotle likewise identifies the first two characteristics, St. Thomas conceives of the third as an appropriation from principles developed by neo-Platonic and Augustinian thinkers.

While touring Scandinavia he first joined forces with guitarist and singer Peter Thorup, together forming the band New Church, who were one of the support bands at the Rolling Stones Free Concert in Hyde Park, London, on 5 July 1969.

While ascribing the divine attributes and divinity to each person of the Trinity, thus avoiding subordinationism, the first half of the Athanasian Creed also stresses the unity of the three persons in the one Godhead, thus avoiding a theology of tritheism.

While it is common for scholars to state that Alexander was the first to write a commentary on the Sentences of Peter Lombard, it is not quite accurate.

While the early explorers used horses to cross the outback, the first woman to make the journey riding a horse was Anna Hingley, who rode from Broome to Cairns in 2006.

While Tacitus called it Mare Suebicum after the Germanic people of the Suebi, the first to name it also as the Baltic Sea ( Mare Balticum ) was eleventh century German chronicler Adam of Bremen.

While it is not recorded as a " steal ", in a practical sense a batter can be said to " steal first base " by successfully running to first base ( without being tagged or thrown out ) in rare circumstances following an uncaught third strike ; the rarely-seen play avoids an " out " and gains a baserunner.

While the attack on Taviers went in the Earl of Orkney launched his first line of English across the Petite Gheete in a determined attack against the barricaded villages of Offus and Autre-Eglise on the Allied right.

While protons and neutrons combined to form the first atomic nuclei only a few minutes after the Big Bang, it would take thousands of years for electrons to combine with them and create electrically neutral atoms.

While Bonaparte was sailing to Malta, the Royal Navy re-entered the Mediterranean for the first time in over a year.

While on his first cross-country tour, he appeared on the Grand Ole Opry and defied that conservative show's ban on using drums of any sort.

While in Rome for Otto's imperial coronation, Bruno met Saint Adalbert of Prague, the first Apostle of the Prussians, killed a year later, which inspired Bruno to write a biography of St Adalbert when he reached the recently Christianized and consolidated Kingdom of Hungary himself.

While the first telegraph company was the Toronto, Hamilton and Niagara Electro-Magnetic Telegraph Company, founded in 1846, it was the Montreal Telegraph Company, controlled by Hugh Allan and founded a year later, that dominated in Canada during the technology's early years.

While in the first five years following World War II, massive emigration from these states to the West occurred, restrictions implemented thereafter stopped most East-West migration, except that under limited bilateral and other agreements.

While David Kessler and Peter Temin consider ancient Rome to be the largest city before 19th century London and the first to have exceeded a population of over 1 million, George Modelski considers medieval Baghdad, with an estimated population of 1. 2 million at its peak, to be the largest city before 19th century London and the first with a population of over one million.

While the final combat of the campaign was being worked out at Jonesborough, Thomas, on Sherman's instructions, ordered Slocum, now commanding the Twentieth Corps, to make an effort to occupy Atlanta if he could do so without exposing his bridgehead to a counterattack.

While 2-year-olds are still gaining most of the attention at the track, green horses are starting to go a bit, and Jimmy Cruise has several that can really make it.

While there may be several such industries to which the model of this paper is applicable, the authors make particular claim of relevance to the explanation of the course of wages and prices in the steel industry of the United States since World War 2.

While the two herbicides that make up Agent Orange, 2, 4-D and 2, 4, 5-T, remain toxic over a short period — a scale of days or weeks — they quickly degrade.

While the adapter was only made to work on the two-port version, modifications can be made to the four-port to make it line-compatible.

While most foods are harvested during the autumn, foods particularly associated with the season include pumpkins ( which are integral parts of both Thanksgiving and Halloween ) and apples, which are used to make the seasonal beverage apple cider.

While some purists consider luck not to be a desirable component of a game, others counter that elements of luck can make for far more diverse and multi-faceted strategies, as concepts such as expected value and risk management must be considered.

While the TV show went into reruns, Mike Judge went on to make movies: he directed such films as Extract, Idiocracy, and Office Space, which found favor with moviegoers and later became cult classics.

While many of the modifications in the newer editions make minor alterations to dynamics and articulation, both editions change the orchestral lead-in to the final statement of the choral theme in the fourth movement ( IV: m525 m542 ).

While STS-51-F's primary payload was the Spacelab-2 laboratory module, the payload which received the most publicity was the Carbonated Beverage Dispenser Evaluation, which was an experiment in which both Coca-Cola and Pepsi tried to make their carbonated drinks available to astronauts.

While Eddy's Manual established limited executive functions under the rule of law in place of a traditional hierarchy, the controversial 1991 publication of a book by Bliss Knapp led the then Board of Directors to make the unusual affidavit during a suit over Knapp's estate that neither acts by it violating the Manual, nor acts refraining from required action, constituted violations of the Manual.

Former Browns QB Otto Graham ( left, with head coach Paul Brown ), who led the Browns to 4 AAFC and 3 NFL Championships, and is a Pro Football Hall of Fame member. While the championship losses sowed bitterness among Cleveland fans who had grown accustomed to winning, the team continued to make progress.

The power of making by-laws was “ tacitly annexed to corporations by the very act of their establishment .” While they must not directly contradict the overarching laws of the land, the central or local government cannot be expected to regulate toward the peculiar circumstances of a given body, and so “ they are invested with authority to make regulations for the management of their own interests and affairs .”

While his system was less " scientific " than that of J. P. Lesley, it's other key features – notation, specificity, and versatility – make it deserving of the praise it has received.

While not on Earth, Xavier's natural telepathic abilities have reached across space to make universal mental contact with multiple alien races.

( While the use of Hindu-Arabic numerals is occasionally seen, such as in the game Pop-O-Matic Trouble, the ambigrammatic properties of the digits ( for instance, 6 can look like a 9 upside down ) make such dice uncommon.

While the exact definition and what is included as extreme sport is debatable, some attempted to make classification for extreme sports.

While Muir wanted nature preserved for the sake of pure beauty, Roosevelt subscribed to Pinchot's formulation, " to make the forest produce the largest amount of whatever crop or service will be most useful, and keep on producing it for generation after generation of men and trees.

While on location in Ireland for The Young Racers in 1963, Corman, ever alert for an opportunity to produce a decent movie on a shoestring budget, persuaded Coppola to make a low-budget horror movie with funds left over from that movie.

While " glasnost " is associated with freedom of speech, the main goal of this policy was to make the country's management transparent and open to debate, thus circumventing the narrow circle of apparatchiks who previously exercised complete control of the economy.

While ill, Marlow continued to make the repairs to the steamer: " I toiled wearily in a wretched scrap-heap — unless I had the shakes too bad to stand.

While the classic bosal-style hackamore is usually used to start young horses, other designs, such as various bitless bridles and the mechanical hackamore are often seen on mature horses with dental issues that make bit use painful, horses with certain training problems, and on horses with mouth or tongue injuries.

While Constantinople experienced a succession of councils alternately approving and condemning doctrine concerning hesychasm considered as identified with Palamism ( the last of the five senses in which, according to Kallistos Ware, the term is used ), the Western Church held no council in which to make a pronouncement on the issue, and the word " hesychasm " does not appear in the Enchiridion Symbolorum et Definitionum ( Handbook of Creeds and Definitions ), the collection of Roman Catholic teachings originally compiled by Heinrich Joseph Dominicus Denzinger.

While the earlier era was characterized by a dramatic use of the major and minor scale systems, impressionist music tended to make more use of dissonance.