While at university, Gauss independently rediscovered several important theorems ; his breakthrough occurred in 1796 when he showed that any regular polygon with a number of sides which is a Fermat prime ( and, consequently, those polygons with any number of sides which is the product of distinct Fermat primes and a power of 2 ) can be constructed by compass and straightedge.

While neither of these theorems can be proven in a completely effective manner, each one can be effectively obtained from the other.

While the original proofs of Hadamard and de la Vallée-Poussin are long and elaborate, and later proofs have introduced various simplifications through the use of Tauberian theorems but remained difficult to digest, a surprisingly short proof was discovered in 1980 by American mathematician Donald J. Newman.

While other students spend their summers working or becoming teenage parents, Escalante's students learn complex theorems and formulas.

While the first three meanings have some relevance for gamblers, they are also general theorems with wide application and many related results in probability and statistics.

While recognising the primary importance of logical proof, Gauss also illustrates many theorems with numerical examples.

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 the NFL results constitute, in a strict sense, full employment theorems for optimization professionals, it is important not to take the term literally.

While there is a meaningful notion of a C < sup > k </ sub > atlas, there is no distinct notion of a C < sup > k </ sub > manifold other than C < sup > 0 </ sub > ( continuous maps: a topological manifold ) and C < sup >∞</ sub > ( smooth maps: a smooth manifold ), because every C < sup > k </ sub >- structure with k > 0, there is a unique C < sup > k </ sub >- equivalent C < sup >∞</ sub >- structure ( every C < sup > k </ sup >- structure is uniquely smoothable ) – a result of Whitney ( and further, two C < sup > k </ sup > atlases that are equivalent to a single C < sup >∞</ sup > atlas are equivalent as C < sup > k </ sup > atlases, so two distinct C < sup > k </ sup > atlases do not collide ); see Differential structure: Existence and uniqueness theorems for details.

While Stephen Hawking was initially a believer in the Theory of Everything, after considering Gödel's Incompleteness Theorem, he has concluded that one is not obtainable, and has stated so publicly in his lecture " Gödel and the End of Physics " ( 2002 ).

While at Princeton, he came very close to discovering the incompleteness of Principia Mathematica, which Kurt Gödel proved in 1931.

While Gödel never published anything bearing on his independent discovery of undefinability, he did describe it in a 1931 letter to John von Neumann.

While von Neumann – Bernays – Gödel set theory restricts the bound variables in the schematic formula appearing in the axiom schema of Class Comprehension to range over sets alone, Morse – Kelley set theory allows these bound variables to range over proper classes as well as sets.

While von Neumann – Bernays – Gödel set theory is a conservative extension of Zermelo – Fraenkel set theory ( ZFC, the canonical set theory ) in the sense that a statement in the language of ZFC is provable in NBG if and only if it is provable in ZFC, Morse – Kelley set theory is a proper extension of ZFC.

While white is the coolest summer shade, there are lots of pastel hues along with tintable fabrics that will blend with any wardrobe color.

While it is easy enough to ridicule Hawkins' pronouncement in Pleas Of The Crown from a metaphysical point of view, the concept of the `` oneness '' of a married couple may reflect an abiding belief that the communion between husband and wife is such that their actions are not always to be regarded by the criminal law as if there were no marriage.

While there are now allegations of the withholding of `` favorable evidence developed at the hearing '' and a denial of a `` full and fair hearing '', no such claim was made by petitioner at any stage of the administrative process.

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 individual sportsmen are aware of this situation, too many of our political, social, educational and even religious leaders too often forget it.

While some suppression and some denial are not only necessary but healthy, the worker's clinical knowledge must determine how these defenses are being used, what healthy shifts in defensive adaptation are indicated, and when efforts at bringing about change can be most effectively timed.

While there are many different possibilities for the timing of casework intervention, the experiments recently reported from a variety of traditional settings all point up the importance of an immediate response to the client's initial need for help.

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 this influence is a complex matter, depending upon personality factors in the individual as well as upon his social-class experience, there probably are some general statements about social-class background and educational policy that can be made with a fair degree of truth.

While costs on this order are sometimes separately charged for in residential and commercial rates, in the form of a mere `` service charge '', they are more frequently wholly or partly covered by a minimum charge which entitles the consumer to a very small amount of gas or electricity with no further payment.

While ATP appears to be necessary for the occurrence of contraction, its presence and enzymatic hydrolysis of it by the muscle protein myosin are not the only criteria for contraction.

While there are still many bugs to be ironed out, the technique is fast developing.

While nowadays we recognize the fact that there are many causes for bleeding at the nose, not long ago a nosebleed was simply that, and treatment had little variation.

Many years later ( on August 3, 1915 ), Lucy Upton wrote Winslow's daughter soon to be graduated from Smith College: `` While I love botany which, after dabbling in for years, I studied according to the methods of that day exactly forty years ago in a summer school, it must be fascinating to take up zoology in the way you are doing.

While patent suits are still among the most complex and expensive forms of litigation, these rules have saved litigants uncounted sums of money.

While details are still to be worked out, Ratcliff said he expects to tell home folks in Dallas why he thinks Berry's proposed constitutional amendment should be rejected.

While availability of mortgage money has been a factor in encouraging apartment construction, the generally high level of prosperity in the past few years plus rising consumer income are among the factors that have encouraged builders to concentrate in the apartment-building field.

While it must be said that these same Protestants have built some new churches during this period, and that religious population shifts have emptied churches, a principal reason for this phenomenon of redundancy is that fewer Protestants are going to church.

While we are filling outer space with scientific successes, for many the `` inner '' space of their soul is an aching void.

While there is general agreement on these six families, there are some points of disagreement among linguists who study Afroasiatic.

While most alphabets have letters composed of lines ( linear writing ), there are also exceptions such as the alphabets used in Braille, fingerspelling, and Morse code.

While five minutes ago the place had presented a scene of easy revelry, with Gyp Carmer a prominent figure, it was now as somnolent and dull as the day before payday.

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 it had long been known in general, that `` the public is always wrong '', the use of odd-lot indices now puts the adage on a statistical basis.

While Crittenden was away in Richmond, Zollicoffer moved his forces to the north bank of the upper Cumberland River near Mill Springs ( now Nancy, Kentucky ), putting the river to his back and his forces into a trap.

While down there, along with the dead, he is shown the place where the wrongly convicted reside, the fields of sorrow where those who committed suicide and now regret it reside, including Aeneas ' former lover, the warriors and shades, Tartarus ( where the titans and powerful non-mortal enemies of the Olympians reside ) where he can hear the groans of the imprisoned, the palace of Pluto, and the fields of Elysium where the descendants of the divine and bravest heroes reside.

While Grothendieck lived in Chambon, he attended the Collège Cévenol ( now known as the Le Collège-Lycée Cévenol International ), a unique secondary school founded in 1938 by local Protestant pacifists and anti-war activists.

While BASE jumps had been made prior to that time, the El Capitan activity was the effective birth of what is now called BASE jumping.

While once the province of high-end print shops, dye-sublimation printers are now increasingly used as dedicated consumer photo printers.

While following the tradition that she lived in Italy, he comments wryly that there are now many more temptresses like her to lead men astray.

While Centaurus now has a high southern latitude, at the dawn of civilization it was an equatorial constellation.

While much of the urban areas were now under the control of the KMT, much of the countryside remained under the influence of weakened yet undefeated warlords and Communists.

While raising Chicago, and at first improving the health of the city, the untreated sewage and industrial waste now flowed into the Chicago River, then into Lake Michigan, polluting the primary source of fresh water for the city.

While tormented with sorrow, sixteen-year-old Catherine was now faced with her parents ' wish that she marry Bonaventura's widower.

While Mike, Mark, Zonker, BD, and Boopsie were all now graduates, BD and Boopsie were living in Malibu, where BD was a third-string quarterback for the Los Angeles Rams, and Boopsie was making a living from walk-on and cameo roles.

While traditionally accepted as the genuine words of Moses delivered on the eve of the occupation of Canaan, a broad consensus of modern scholars now see its origins in traditions from Israel ( the northern kingdom ) brought south to the Kingdom of Judah in the wake of the Assyrian destruction of Samaria ( 8th century BCE ) and then adapted to a program of nationalist reform in the time of King Josiah ( late 7th century ), with the final form of the modern book emerging in the milieu of the return from the Babylonian exile during the late 6th century.

While we now see with our own eyes that such operations were a habit which is prevalent among all civilized people of the west "

While this method, now known as the triboelectric effect, is capable of lifting light objects and even generating sparks, it is extremely inefficient.

While many mathematicians before Galois gave consideration to what are now known as groups, it was Galois who was the first to use the word group ( in French groupe ) in a sense close to the technical sense that is understood today, making him among the founders of the branch of algebra known as group theory.

While the van der Waals equation is commonly referenced in text-books and papers for historical reasons, it is now obsolete.

While it is now used to apply to groups of people fascinated with any subject, the term has its roots in those with an enthusiastic appreciation for sports.

While rare now, some late 19th – early 20th century authors used kanji as furigana for loanwords written in katakana.

While old-fashioned, this expression is still used in French, often mockingly, but the English gallicism to flirt has made its way and has now become an anglicism.

While most of his proposals-" to abandon the gold standard, let international exchange rates float, use federal surpluses and deficits as macroeconomic policy tools that could counter cyclical trends, and establish bureaus of economic statistics ( including a consumer price index ) in order to facilitate this effort "-are now conventional practice, his critique of fractional-reserve banking still " remains outside the bounds of conventional wisdom ".

While he had been in the vanguard with The Byrds and the Burrito Brothers, Parsons was now perceived as being too authentic and traditional in an era dominated by the stylings of the Eagles, whose sound Parsons disdained ( although he did maintain cordial relations with Bernie Leadon, a former member of the Eagles who left the group in 1975 ).