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Although and Dirac
Although Schrödinger himself after a year proved the equivalence of his wave-mechanics and Heisenberg's matrix mechanics, the reconciliation of the two approaches and their modern abstraction as motions in Hilbert space is generally attributed to Paul Dirac, who wrote a lucid account in his 1930 classic Principles of Quantum Mechanics.
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 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 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 for Z > 137, the wave function of the Dirac ground state is oscillatory, rather than bound, and there is no gap between the positive and negative energy spectra, as in the Klein paradox, more accurate calculations taking into account the effects of the finite size of the nucleus indicate that the binding energy first exceeds 2mc < sup > 2 </ sup > for Z > Z < sub > cr </ sub > ≈ 173.
Although there were problems, Dirac was capable of resolving them.
Although it turned out that the Dirac equation describes the spinning electron, the Klein – Gordon equation correctly describes the spinless pion.
Although named after Fermi, most of the work leading to the Golden Rule was done by Dirac who formulated an almost identical equation, including the three components of a constant, the matrix element of the perturbation and an energy difference.
Although not very common, it is also possible to derive effects of magnetic structures to charged particles starting from the Dirac equation.

Although and did
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 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 a look of alarm passed over his face, he did not arrest his movements but disappeared into the shower room just as the chambermaid emerged from number nine.
Although there were no startling developments in shotgun design this year, a number of new models and variations of existing models did hit the market.
Although Miller noted in 1907 that a difference in the pleural blood supply existed between animals, nowhere in his published works is it found that he did a comparative study of the intrapulmonary features of various mammalian lungs other than in the dog and cat ( Miller, '13 ; ;
Although Pearson disbelieved almost everything Lizzie said, and read a sinister purpose into almost everything she did, he happily accepted her statement about Bridget as the whole truth.
Although these results did not reach statistical significance, the baths are an easy treatment.
Although metics had no direct political influence many were wealthy business owners who could, and sometimes did, influence policy by not allowing their citizen employees time off to attend the assembly, as well as having the simple expedient of wealth.
Although Rodin is generally considered the progenitor of modern sculpture, he did not set out to rebel against the past.
Although Roma were unable to break into the top four during the following decade, they did achieve some measure of cup success.
Although his exact guidelines are no longer rigidly followed they did inspire this field of conservation.
Although he was highly in demand, he did not create many pieces because he was also employed as an art dealer.
Although most South African parties announced their candidate list for provincial premierships in the 2009 election, the ANC did not.
Although the Umayyads did not have a historical presence in the region ( no member of the Umayyad family was known to have ever set foot in al-Andalus before ) and there were grave concerns about young Abd al-Rahman's inexperience, several of the lower-ranking Yemenite commanders felt they had little to lose and much to gain, and agreed to support the prince.
Although his paternal great-grandfather had been a priest in the Russian Orthodox Church, and his pious mother did have him baptised, he was an atheist in later life.
Although influential for a time their popularity did not generally survive the horrors of ideological warfare, as the competing ideologies utilized their systems in their platforms.
Although this shift was an important one, it did not represent a radical break from the past so much as a small step in a broader, more gradual socio-economic movement that had been going on at least since 1907 when van de Velde had argued for a craft basis for design while Hermann Muthesius had begun implementing industrial prototypes.
Although the European film industry was then in its ascendancy, Bardot was one of the few European actresses to have the mass media's attention in the United States, an interest which she did not reciprocate by rarely, if ever, going to Hollywood.
Although the National League began its LCS MVP award in 1977, the American League did not begin its LCS MVP award until 1980.
Although the National League began its LCS MVP award in 1977, the American League did not begin its LCS MVP award till 1980.
Although Bede did not invent this method, his adoption of it, and his promulgation of it in De Temporum Ratione, his work on chronology, is the main reason why it is now so widely used.
Although he would go on to have back-to-back 17-win seasons for the Chicago Cubs in 1971 and 1972, including a no-hitter in the latter season, this did not help the Reds, who ended up losing the 1970 World Series to Robinson and the Orioles.

Although and himself
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 Grothendieck himself never published his work in this area, the proposal became the inspiration for work by other mathematicians and the source of the theory of dessins d ' enfants.
Although he himself was a blues purist, Korner criticised better-known British blues musicians during the blues boom of the late 1960s for their blind adherence to Chicago blues, as if the music came in no other form.
Although people of all creeds enjoyed tolerance and freedom of religion under his rule, he repelled the Fatimids, partly by supporting their enemies in Ifriqiya, and partly by claiming the title Caliph ( ruler of the Islamic world ) for himself.
Although neither the Nazi Party nor Hitler himself had a cohesive architectural policy before they came to power in 1933, Nazi writers like Wilhelm Frick and Alfred Rosenberg had already labeled the Bauhaus " un-German " and criticized its modernist styles, deliberately generating public controversy over issues like flat roofs.
Although Francois Bozizé has a background in FACA himself ( being its chief-of-staff from 1997 to 2001 ), he has been cautious by retaining the defence portfolio, as well as by appointing his son Jean-Francis Bozizé cabinet director in charge of running the Ministry of Defence.
Although one of his brothers became a clergyman and one of his sisters a missionary, Attlee himself is usually regarded as an agnostic.
Although not often, it may refer to the individual, biblical teachings that Calvin made himself.
Although Cantor himself defined the set in a general, abstract way, the most common modern construction is the Cantor ternary set, built by removing the middle thirds of a line segment.
Although the Christadelphian movement originated through the activities of John Thomas, he never saw himself as setting up his own disciples.
Although noting in the introduction to ' The Protestors ' that ' Some recorded herein perhaps did not have " all the truth " — so the writer has been reminded ', Eyre nevertheless claimed that the purpose of the work was to ' tell how a number of little-known individuals, groups and religious communities strove to preserve or revive the original Christianity of apostolic times ', and that ' In faith and outlook they were far closer to the early springing shoots of first century Christianity and the penetrating spiritual challenge of Jesus himself than much that has passed for the religion of the Nazarene in the last nineteen centuries '.
Although the super-soldier serum is an important part of his strength, Rogers has shown himself still sufficiently capable against stronger opponents, even when the serum has been deactivated reverting him to his pre-Captain America physique.
Although he had never seen action himself, he won the respect of front-line commanders.
Although did not think of himself as a deist, he shared so many attitudes with deists that Gay calls him " a Deist in fact, if not in name ".
Although little information survives of the battles fought, enough early victories were apparently achieved for Domitian to be back in Rome by the end of 83, where he celebrated an elaborate triumph and conferred upon himself the title of Germanicus.
Although Thomas described himself as the " Rimbaud of Cwmdonkin Drive ", he stated that the phrase " Swansea's Rimbaud " was coined by poet Roy Campbell.
Although, originally conceived as a children's programme, it quickly acquired a cult crossover following amongst many adults, including future Pythons John Cleese and Graham Chapman ( as mentioned by Cleese himself in the " Paying my ex-wife " stage performance tour, October 2010 ).
Although Arthur Tree was himself from a wealthy American family, he now had to adjust to reduced circumstances without Ethel's support.
Although in later life he prided himself on always publishing his work under his own name, in the late nineteenth century he appears to have used pseudonyms such as and in some of his published writings.
Although he did not consider himself an educator, his research and theories closely resemble those outlined by Rousseau in Emile.
Although posterity suspected him of Arianism, Eusebius had made himself indispensable by his method of authorship ; his comprehensive and careful excerpts from original sources saved his successors the painstaking labor of original research.
' Although some use an Anglicized spelling pronunciation of, that is, with the vowel of ruse, FDR himself used, with the vowel of rose.
Although Kildall preferred to leave the IBM affair in the past and to be known for his work before and afterward, he continually faced comparisons between himself and Bill Gates as well as fading memories of his contributions.
Although his work was not initially taken seriously by British critics of the Sight and Sound circle, he was venerated by French critics associated with Cahiers du cinéma, who intellectualized his work in a way Hawks himself was moderately amused by, and he was also admired by more independent British writers such as Robin Wood.

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