Although some problems in statistical physics can be solved analytically using approximations and expansions, most current research utilizes the large processing power of modern computers to simulate or approximate solutions.

Although no elementary function exists for the error function, as can be proven by the Risch algorithm, the Gaussian integral can be solved analytically through the tools of calculus.

Although the sheath equation must generally be integrated numerically, we can find an approximate solution analytically by neglecting the term.

Although convective heat transfer can be derived analytically through dimensional analysis, exact analysis of the boundary layer, approximate integral analysis of the boundary layer and analogies between energy and momentum transfer, these analytic approaches may not offer practical solutions to all problems when there are no mathematical models applicable.

Although analytically simpler than < sup > 230 </ sup > Th /< sup > 238 </ sup > U dating, in practice < sup > 234 </ sup > U /< sup > 238 </ sup > U dating is almost never used as unlike < sup > 230 </ sup > Th /< sup > 238 </ sup > U dating it requires prior knowledge of the < sup > 234 </ sup > U /< sup > 238 </ sup > U ratio at the time the material under study was formed.

Although unproven, many suspect the South African government of responsibility for his death.

Although it has been conjectured that she is the same person as the Mary mother of John Mark, this is generally considered to be unproven.

Although gay radicals used pressure to force the decision, Kaiser notes that this had been an issue of some debate for many years in the psychiatric community, and that one of the chief obstacles to normalizing homosexuality was that therapists were profiting from offering dubious, unproven " cures ".

: Although this idea was unproven for a time, recent research has indicated that bowhead whales recently killed still had harpoons in their bodies from about 1890, which, along with analysis of amino acids, has indicated a maximum life span, stated as " the 211 year-old bowhead could have been from 177 to 245 years old ".

Although such an exchange of funds remains unproven, Bush met with Bongo 10 months later in the Oval Office.

Although many of her biographers have given credence to this, it is unproven, and discounted by some historians.

Although her claims of alleged harassment and physical assault were never verified and remain unproven, the University nevertheless suspended the Student Government Association altogether, and did not reinstitute it until 1996.

Although these figures must of course be taken with a pinch of salt ( in the sense that they are based upon data whose reliability is unproven and debatable to say the least ), a comparison between the two years ( 1873 and 1935 ) reveals

Although Bulleid built two prototype electric locomotives in 1941 these were as yet unproven, and freight haulage would be undertaken by steam traction for the foreseeable future.

Although quarks also carry color charge, hadrons must have zero total color charge because of a phenomenon called color confinement.

Although string theory had its origins in the study of quark confinement and not of quantum gravity, it was soon discovered that the string spectrum contains the graviton, and that " condensation " of certain vibration modes of strings is equivalent to a modification of the original background.

Although Odo remained in confinement for the rest of William's reign, his lands were not confiscated.

Although Stewart maintained her innocence, she was found guilty and sentenced on July 16, 2004 to five months in prison, five months of home confinement, and two years probation for lying about a stock sale, conspiracy, and obstruction of justice.

Although North Vietnam was a signatory of the Third Geneva Convention of 1949, which demanded " decent and humane treatment " of prisoners of war, severe torture methods were employed, such as rope bindings, irons, beatings, and prolonged solitary confinement.

Although no evidence has come to light to support his connection with the plot, he was fined £ 6, 000 and kept in close confinement.

Although milk yields are often lower in MIRG systems, net farm income per cow is often greater as compared to confinement operations.

Although the Auburn system was favored in the United States, Eastern State's radial floor plan and system of solitary confinement was the model for over 300 prisons worldwide.

Although the above equations were derived using simplifying assumptions, the implications are clear ; the energy of the quantum dots are dependent on their size due to the quantum confinement effects, which dominate below the critical size leading to changes in the optical properties.

Although Junior has survived cancer and possible prison time, the toll of a series of ' mini-strokes ' and the confinement of house arrest has since left him confused, depressed, and borderline senile, giving reason to gradually retire him as the official head of the DiMeo Crime Family and increasingly dependent on family care and support.

Although reformatory movements such as Jainism allowed women to be admitted to the religious order, by and large, the women in India faced confinement and restrictions.

Although the novel draws on Talmudic tradition to create the framework for Elisha's life, Steinberg himself wrote that his novel " springs from historical data without any effort at rigid conformity or literal confinement to them.

Although he is perfectly willing to cooperate with Scotland Yard, Holmes has nothing but contempt for the intelligence and mentality of the police.

Although we continue to pay our conversational devotions to `` free private enterprise '', `` individual initiative '', `` the democratic way '', `` government of the people '', `` competition of the marketplace '', etc., we live rather comfortably in a society in which economic competition is diminishing in large areas, bureaucracy is corroding representative government, technology is weakening the citizen's confidence in his own power to make decisions, and the threat of war is driving him economically and physically into the ground ''.

Although it is constantly made to look foolish ( too simple to come in out of the rain, people say, who have found in the innocent an impediment ), it does not mind looking foolish because it is not concerned with how it looks.

Although open to the general public it is not overcrowded ; ;

Although we have no measures of its strength or intensity, the heritage of the doctrine of inalienable rights is retained.

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 Patchen has given previous evidence of an interest in jazz, the musical group that he works with, the Chamber Jazz Sextet, is often ignored by jazz critics.

Although the false glamour surrounding bourbon or other whisky commercials is possibly no more fatuous than the pseudo-sophistication with which TV soft-drinks are downed or toothpaste applied, there is a sad difference between enticing a viewer into sipping Oopsie-Cola and gulling him into downing bourbon.

Although the United States and the U.S.S.R. have been arguing whether there shall be four, five or six top assistants, the most important element in the situation is not the number of deputies but the manner in which these deputies are to do their work.

`` Although it is not the best of which he is capable '', said Shelley as he closed the book, `` it is still poetry of a high order ''.

Although Sam Rayburn affects a gruff exterior in many instances, nevertheless he is fundamentally a man of warm heart and gentle disposition.

Although a similar situs for tangible property is mentioned in the statute, this is cancelled out by the provision that definite kinds of property `` and all other tangible property '' situated or being in any town is taxable where the property is situated.

Although much of the Industrial Division's promotional effort is devoted to securing new locations and expansions by major industries, small business is also afforded considerable attention.

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 economic and personal circumstances vary widely among those now choosing apartments, Leo J. Pantas, vice president of a hardware manufacturing company, pointed out recently that many apartment seekers seem to have one characteristic in common: a desire for greater convenience and freedom from the problems involved in maintaining a house.

Although Doubleday achieved minor fame as a competent combat general with experience in many important Civil War battles, he is more widely remembered as the supposed inventor of the game of baseball, in Elihu Phinney's cow pasture in Cooperstown, New York, in 1839.

Although this classification was widely accepted, further infrageneric classification was ( and still is ) needed to differentiate this widely diverse group.

Although this concept remains a guiding principle of the profession, it is a concept that has been widely critiqued within the conservation profession and is now considered by many to be " a fuzzy concept ".

Although his paper was widely cited, a random selection of 60 of these citations revealed that 29 of the papers were direct rebuttals or criticisms of Jensen's arguments, 8 cited the paper as an " example of controversy ," 8 used it as a background reference.

Although this document was subsequently adopted by International Organization for Standardization ( ISO ) and subsequent revisions published by ISO have been adopted by ANSI, the name ANSI C ( rather than ISO C ) is still more widely used.

Although they were previously considered the sister group to the remaining crustaceans, it is now widely accepted that crustaceans form a paraphyletic group, and Branchiopoda are thought to be sister to a clade comprising Xenocarida ( Remipedia and Cephalocarida ) and Hexapoda ( insects and their relatives ).

Although BCD is not as widely used as in the past, decimal fixed-point and floating-point formats are still important and continue to be used in financial, commercial, and industrial computing, where subtle conversion and rounding errors that are inherent to floating point binary representations cannot be tolerated.

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 the Royal Navy is widely regarded as having been vital for the rise of the British Empire, and British dominance of the world, the British Army played an important role in the colonisation of India and other regions.

Although it is widely accepted that Descartes explained the problem cogently, few later philosophers have been happy with his solution, and his ideas about the pineal gland have especially been ridiculed.

Although CCDs are not the only technology to allow for light detection, CCD image sensors are widely used in professional, medical, and scientific applications where high-quality image data is required.

Although such programs are widely available and are sometimes seen as clever toys for the non-musician, some professional musicians have given them attention also.

Although widely used in naval warfare, and in anti-air guns, both the British and Americans feared unexploded proximity fuses would be reverse engineered leading to them limiting its use in continental battles.

Although many were initially wary of the concept, it was because people were able to witness the procedure's consistently positive results, within their own community of ordinary citizens, that it became widely utilized and supported.

Although the idea expressed in Cogito ergo sum is widely attributed to Descartes, he was not the first to mention it.

Although the concept of consilience in Whewell's sense was widely discussed by philosophers of science, the term was unfamiliar to the broader public until the end of the 20th century, when it was revived in Consilience: The Unity of Knowledge, a 1998 book by the humanist biologist Edward Osborne Wilson, as an attempt to bridge the culture gap between the sciences and the humanities that was the subject of C. P. Snow's The Two Cultures and the Scientific Revolution ( 1959 ).

Although these guidelines have been widely implemented, opponents argue that even when a diagnostic criteria set is accepted across different cultures, it does not necessarily indicate that the underlying constructs have any validity within those cultures ; even reliable application can only demonstrate consistency, not legitimacy.

Although Jones is especially remembered for his work on the phonetics and phonology of English, he ranged far more widely.

Although not widely employed for esterifications, salts of carboxylate anions can be alkylating agent with alkyl halides to give esters.

Although many of the descriptions in this book were written in an exaggerated manner or were plainly inventive fiction or 3rd-source misinterpretation, his notes are widely accepted as a useful guide to the cultural aspects and lifestyle of 17th-century Ottoman Empire.

Although political power during Suiko's reign is widely viewed as having been wielded by Prince Shōtoku and Soga no Umako, Suiko was far from powerless.

Although several designs ( such as the Convair flying car ) have flown, none have enjoyed commercial success, and those that have flown are not widely known about by the general public.

Although not widely fielded until World War I, early machine guns were being used by militaries in the second half of the 19th century.