Although α and − α produce the same differential equation, it is conventional to define different Bessel functions for these two orders ( e. g., so that the Bessel functions are mostly smooth functions of α ).

Although not usually stated in polynomial form, this example is equivalent to the polynomial equation 4xyz = yzn + xzn + xyn = n ( yz + xz + xy ).</ td >

Although the law can be used to tell whether any quadratic equation modulo a prime number has a solution, it does not provide any help at all for actually finding the solution.

Although there are several approaches to integrating other illumination effects such as specular and glossy reflections, radiosity-based methods are generally not used to solve the complete rendering equation.

Although energy generated by stimulated emission is always at the exact frequency of the field which has stimulated it, the above rate equation refers only to excitation at the particular optical frequency corresponding to the energy of the transition.

Although Dirac did not at first fully appreciate what his own equation was telling him, his resolute faith in the logic of mathematics as a means to physical reasoning, his explanation of spin as a consequence of the union of quantum mechanics and relativity, and the eventual discovery of the positron, represents one of the great triumphs of theoretical physics, fully on a par with the work of Newton, Maxwell, and Einstein before him.

Although Dirac's original intentions were satisfied, his equation had far deeper implications for the structure of matter, and introduced new mathematical classes of objects that are now essential elements of fundamental physics.

Although it is not a successful relativistic generalization of the Schrödinger equation, this equation is resurrected in the context of quantum field theory, where it is known as the Klein – Gordon equation, and describes a spinless particle field ( e. g. pi meson ).

Although not recognized as such by Dirac at the time the equation was formulated, in hindsight the introduction of this geometric algebra represents an enormous stride forward in the development of quantum theory.

Although physical modelling was not a new concept in acoustics and synthesis, having been implemented using finite difference approximations of the wave equation by Hiller and Ruiz in 1971, it was not until the development of the Karplus-Strong algorithm, the subsequent refinement and generalization of the algorithm into the extremely efficient digital waveguide synthesis by Julius O. Smith III and others, and the increase in DSP power in the late 1980s that commercial implementations became feasible.

Although the birefringence Δn may vary slightly due to dispersion, this is negligible compared to the variation in phase difference according to the wavelength of the light due to the fixed path difference ( λ < sub > 0 </ sub > in the denominator in the above equation ).

Although the original motivation was to solve the heat equation, it later became obvious that the same techniques could be applied to a wide array of mathematical and physical problems, and especially those involving linear differential equations with constant coefficients, for which the eigensolutions are sinusoids.

Although Richard Feynman noted that a simplistic interpretation of the relativistic Dirac equation runs into problems with electron orbitals at Z > 1 / α = 137, suggesting that neutral atoms cannot exist beyond untriseptium, and that a periodic table of elements based on electron orbitals therefore breaks down at this point, a more rigorous analysis calculates the limit to be Z ≈ 173, but also that this limit would not actually spell the end of the periodic table.

Although the stereotyped equation of wealth with aristocracy may have applied early in Priene's history, in the 4th century BCE it was a democracy.

Although Lagrange only sought to describe classical mechanics, the action principle that is used to derive the Lagrange equation was later recognized to be applicable to quantum mechanics as well.

Although it turned out that the Dirac equation describes the spinning electron, the Klein – Gordon equation correctly describes the spinless pion.

Although the equation is usually referred to as the " Liouville equation ", this equation was in fact first published by Gibbs in 1902.

Although the cubical model of the atom was soon abandoned in favor of the quantum mechanical model based on the Schrödinger equation, and is therefore now principally of historical interest, it represented an important step towards the understanding of the chemical bond.

Although the book covered a large amount of topics, from energetics of evolution ( see below ) to the physical nature of consciousness, the author is primarily known today for the Lotka – Volterra equation of population dynamics.

Although it was dark as usual I could see that the hall had only recently contained a great many people.

Although New Orleans was not to learn of it for a spell, she also was a sadist, a nymphomaniac and unobtrusively mad -- the perpetrator of some of the worst crimes against humanity ever committed on American soil.

Although Faulkner was the heir in his own family to this tradition, he did not have Stark Young's inclination to romanticize and sentimentalize the planter society.

Although his tender nights were not the ones I dreamed of, nor was it for yachts, sports cars, tall drinks, and swimming pools, nor yet for money or what money buys that I burned, I too was burning and watching myself burn.

Although after much trouble he did manage to get it back, he discovered there was no trade to be had.

Although he did not attend any celebrated schools or universities, he was a master of Greek and Hebrew and could read the Bible in the original.

Although the fort was evacuated in the face of the force of Cornwallis, Morgan and his men did have a chance to take another swing at the redcoats.

Although because of the important achievements of nineteenth century scholars in the field of textual criticism the advance is not so striking as it was in the case of archaeology and place-names, the editorial principles laid down by Stevenson in his great edition of Asser and in his Crawford Charters were a distinct improvement upon those of his predecessors and remain unimproved upon today.

Although Mr. Brown was not himself its inventor ( it was a French idea ), it is typical that his intuition first conceived the importance of mass producing this basic tool for general use.

Although it was a mere fragment of the victim's remains, it was enough.

Although there were only four fluids tested, it was apparent that there were two distinct types.

Although the tape was run for over 1 hr., a steady state was not reached, and it was concluded that the reason for this was that the back pressure of the manometer was built up from the material fed from between the blocks and this was available at a very slow rate.

-- Although there was some variation in results which must be attributed either to trace impurities or to variation in wall effects, the photochemical exchange in the gas phase was sufficiently reproducible so that it seemed meaningful to compare the reaction rates in different series of reaction tubes for the purpose of obtaining information on the effect of chlorine concentration and of carbon tetrachloride concentration on the reaction rate.

Although the Taylor Scale was designed as a group testing device, in this study it was individually administered by psychologically trained workers who established rapport and assisted the children in reading the items.

Although there is no question but that the process of washing fabrics involves a number of phenomena which are related together in an extremely complicated way and that these phenomena and their interrelations are not well understood at the present, this section attempts to present briefly an up-to-date picture of the physical chemistry of washing either fabrics or hard surfaces.

Although BBSing survives only as a niche hobby in most parts of the world, it is still an extremely popular form of communication for Taiwanese youth ( see PTT Bulletin Board System ).

Although cumin seeds contain a relatively large percentage of iron, extremely large quantities of cumin would need to be consumed for it to serve as a significant dietary source ( see nutrition data ).

Although the isosahedral structure is extremely common among viruses because of size differences slight variations exist between virions.

Although mesoclisis is extremely formal in Brazilian Portuguese and tends to be circumscribed in lesser formal registers by avoiding synthetical future / conditional verb forms, European Portuguese still allows clitic object pronouns to surface as mesoclitics in colloquial situations:

Although extremely intelligent, Ed is still a child, and looks up to the crew of the Bebop as members of her family.

Although his political career ended in disgrace, he remained extremely influential through his works, especially as philosophical advocate and practitioner of the scientific method during the scientific revolution.

John Landis recalled that " Although he was extremely ill he told me he could not die until he voted for Obama for President and he did.

Although the oil discoveries of the 1960s have brought it immense petroleum wealth, at the time of its independence it was an extremely poor desert state whose only important physical asset appeared to be its strategic location at the midpoint of Africa's northern rim.

Although Mazda's concept was for the MX-5 to be an inexpensive sports car, the market proved extremely eager for it ; the car became an overnight novelty, resulting in many dealers placing customers on lists for pre-order due to the demand exceeding the anticipated supply.

Although the Pagan movement is extremely disparate in its beliefs and practices, a number of sociologists and religious studies scholars involved in Pagan studies have highlighted commonalities shared within many, if not all, pagan groups.

Although Osman did not physically participate in the battle, the victory at Bursa proved to be extremely vital for the Ottomans as the city served as a staging ground against the Byzantines in Constantinople, and as a newly adorned capital for Osman's son, Orhan.

Although their differences often resulted in extremely heated on-air exchanges and confrontations, Cowell says he played a major role in convincing Abdul not to leave the show.

Although nearly flat surfaces are not lacking in this subregion, the rolling terrain is extremely uneven.

Although it is computationally easy for the intended recipient to generate the public and private keys, to decrypt the message using the private key, and easy for the sender to encrypt the message using the public key, it is extremely difficult ( or effectively impossible ) for anyone to derive the private key, based only on their knowledge of the public key.

Although Admiral Aziz Mirza ( a life-long friend of Musharraf shared the dorm with the admiral in 1960s and graduated together from the academy ) was appointed by Prime minister Nawaz Sharif, Mirza remained extremely supportive to Musharraf's coup and was also a close friend of Musharraf since 1971 when both participate in an joint operations against Indian Army.

Although encountering the flaw was extremely rare in practice ( Byte magazine estimated that 1 in 9 billion floating point divides with random parameters would produce inaccurate results ),< ref > both the flaw and Intel's initial handling of the matter were heavily criticized.

Although on the surface an entertaining escapist romance, alert contemporary readers would have quickly recognized the political subtext of Ivanhoe, which appeared immediately after the English Parliament, fearful of French-style revolution in the aftermath of Waterloo, had passed the Habeas Corpus Suspension acts of 1817 and 1818 and other extremely repressive measures and when traditional English Charter rights versus revolutionary human rights was a topic of discussion.

Although he continued to be extremely popular and widely read, both at home and abroad, Scott's critical reputation declined in the last half of the nineteenth century as serious writers turned from romanticism to realism, and Scott began to be regarded as an author suitable for children.

Although extremely unpopular at the time, it was the Sheffield trade that brought in pitcher Trevor Hoffman, who was virtually unknown to Padres fans.

Although Hodge's suggestion that Mount Multnomah is a supervolcano was rejected long ago, the term " supervolcano " was popularised by the BBC popular science television program Horizon in 2000 to refer to eruptions that produce extremely large amounts of ejecta.

Although the rules are extremely simple, there is enormous scope for scientific play.

Although the specific impulse of an electric thruster itself optionally could range up to where mass drivers merge into particle accelerators with fractional-lightspeed exhaust velocity for tiny particles, trying to use extreme exhaust velocity to accelerate a far slower spacecraft could be suboptimally low thrust when the energy available from a spacecraft's reactor or power source is limited ( a lesser analogue of feeding onboard power to a row of spotlights, photons being an example of an extremely low momentum to energy ratio ).

Although this is only about, when comparing this speed with that of higher organisms in terms of number of lengths moved per second, it is extremely fast.