While Ruffini and Abel established that the general quintic could not be solved, some particular quintics can be solved, such as ( x − 1 )< sup > 5 </ sup >= 0, and the precise criterion by which a given quintic or higher polynomial could be determined to be solvable or not was given by Évariste Galois, who showed that whether a polynomial was solvable or not was equivalent to whether or not the permutation group of its roots – in modern terms, its Galois group – had a certain structure – in modern terms, whether or not it was a solvable group.

While sources from antiquity agree that Alexander was confronted with the challenge of the knot, the means by which he solved the problem are disputed.

Of the package, Bob Neaverson wrote " While it certainly solved the song quota problem, one suspects that it was also partly born of The Beatles ' pioneering desire to experiment with conventional formats and packaging ".

While in terms of functional analysis such problems are typically continuous, they may suffer from numerical instability when solved with finite precision, or with errors in the data.

While the Allies were distracted, Napoleon solved his problem in characteristic fashion.

While it met with great acclaim in the short term, poorly trained operators had trouble keeping picture in synchronization with sound and, like other sound-film systems of the era, the Kinetophone had not solved the issues of insufficient amplification and unpleasant audio quality.

While a mystery climaxes when the mystery is solved ( i. e., Gosford Park ), a mystery thriller climaxes when the hero finally defeats the villain ( after reveal ), saves his own life and often the lives of others ( i. e., Oldboy ).

While the research pursued in these areas is mostly not about games, it is quite common for videogames to be used as a context in which to demonstrate the solutions and problems solved.

While no evidence she has supplied has solved murders or missing investigations on their own., her evidence had been used to corroborate theories, and in one case, included in a coroner's brief on a case Little, in reference to one case she assisted on, felt her description of what happened was " exceptional ", other officers also had been impressed by her assistance, while yet other NSW officers felt she had not helped solve any cases.

While this act solved the problem of succession, it also plunged the kingdom into a serious financial crisis.

While progress is slow, the program has reportedly advanced from barely usable to essentially useful, albeit with many issues remaining to be solved before wide acceptance can be expected.

While working for the Coast Guard and the Bureau of Foreign Control during the Prohibition era, she solved over 12, 000 rum-runners ' messages.

While the technical problems are being solved, military investigations are undertaken in order to thwart foreign espionage, especially from ' communist sympathizers ' who might be associated with socialist organizations.

While the food shortages in Virgin Land settlements were solved by 1956, the housing problem remained an issue past the end of the 1950s.

While Lutz announces publicly that the Alphabet Crimes have been solved, Foley notices some red mud at the stables, which leads him, Taggart and Rosewood to Dent's oil field, where Dent is making his final arms deal.

While GPS solved simple problems such as the Towers of Hanoi that could be sufficiently formalized, it could not solve any real-world problems because search was easily lost in the combinatorial explosion of intermediate states.

While the machine achieved a measure of popularity, its security was relatively weak ; US cryptanalyst William Friedman reported that he solved the device within 2 hours and 41 minutes.

While studying learners as they solved problems, he and his associates found that learners often use a problem solving strategy called means-ends analysis.

While the use of these special curves can easily be avoided, in 2009 it has been proven that for certain fields the discrete logarithm problem in the group of points on general elliptic curves over these fields can be solved faster than with generic methods.

While the programs were based upon actual cases, the cases had not always been solved ( as presented in the program ).

While Might and Magic 2 remained a battle-focused game, there were many puzzles to be solved, and curiosities to be discovered.

While this has solved certain problems, membranes have their own particular limitations, such as with true selectivity, clogging and, more undetectably, they provide secondary sites for other chemical reactions, which are prone to display “ false negatives ,” a condition far more severe than “ false positives ” in critical applications.

While ( 1 ) is solved much in the same way, ( 2 ) is of distinct interest because of its non-homogeneous nature.

While all real fluids are compressible, a flow problem is often considered incompressible if the density changes in the problem have a small effect on the outputs of interest.

While working for the CTC, Erlang was presented with the classic problem of determining how many circuits were needed to provide an acceptable telephone service.

While this solves the logical problem, one could argue that the philosophical problem remains.

While the CNP were not a racist organisation there was a perceived image problem relating to the similarly-styled British National Party ( BNP ).

While the perspective is compatible with Jamesian pragmatism ( above ), the notion of the transformation of embodied concepts through structural mapping makes a distinct contribution to the problem of concept formation.

While every computable number is definable, the converse is not true: the numeric representations of the Halting problem, Chaitin's constant, the truth set of first order arithmetic, and 0 < sup >#</ sup > are examples of numbers that are definable but not computable.

While a foundationalist would use Münchhausen Trilemma as a justification for demanding the validity of basic beliefs, a skeptic would see no problem with admitting the result.

While rheumatic fever since the advent of routine penicillin administration for Strep throat has become less common in developed countries, in the older generation and in much of the less-developed world, valvular disease ( including mitral valve prolapse, reinfection in the form of valvular endocarditis, and valve rupture ) from undertreated rheumatic fever continues to be a problem.

While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a long-standing problem.

While these definitions are distinct, the problem of precisely measuring rapid decomposition makes practical classification of explosives difficult.

While this gets around the infinite acceleration problem, it still would lead to closed timelike curves ( i. e., time travel ) and causality violations.

While the side-stick offers the advantages of being lighter, mechanically simpler, and unobtrusive, The Boeing Company's aerospace engineers decided that the lack of visual feedback ( none given by side-sticks ) is a significant problem, and so they designed conventional control yokes in the Boeing 777 and the Boeing 787, which entered service on 26 October 2011 with All Nippon Airways.

While this remains an unsolved problem for astronomers, it does not necessarily mean that the Lambda-CDM model is completely wrong, but rather that it requires further refinement to accurately reproduce the population of galaxies in the universe.

While some in the legal sector advocated that the National People's Congress ( NPC ) should be asked to amend the part of the Basic Law to redress the problem, the HKSAR Government decided to seek an interpretation to, rather than an amendment of, the relevant Basic Law provisions from the Standing Committee of the National People's Congress ( NPCSC ).

While the " historical " part of historical materialism does not cause a comprehension problem ( i. e., it means the present is explained by analysing the past ), the term materialism is more difficult.

While, strictly speaking, this does not conform to the ISO 9660 standard, most operating systems which can read ISO 9660 file systems have no problem with out-of-spec names.

* While the decision problem is NP-complete the optimization problem is NP-hard, its resolution is at least as difficult as the decision problem, and there is no known polynomial algorithm which can tell, given a solution, whether it is optimal ( which would mean that there is no solution with a larger, thus solving the decision problem NP-complete ).

While the more general approach of such adaptive systems is more suitable for real-world problem solving, it has far less to do with the traditional artificial intelligence connectionist models.

While it is possible to define some arbitrary, ad hoc cost function, frequently a particular cost will be used, either because it has desirable properties ( such as convexity ) or because it arises naturally from a particular formulation of the problem ( e. g., in a probabilistic formulation the posterior probability of the model can be used as an inverse cost ).

While studying the continuum hypothesis, Cohen is quoted as saying in 1985 that he " had the feeling that people thought the problem was hopeless, since there was no new way of constructing models of set theory.