Although counting the number of elements in a set is a rather broad mathematical problem, many of the problems that arise in applications have a relatively simple combinatorial description.

Although these works were not actually composed with a computer in the modern sense, it uses a rudimentary form of the random combinatorial techniques sometimes used in computer-generated composition.

Although brute-force guessing is of course possible, a better weapon is the understanding of the various combinatorial forms that entries can take for various pairings of clues and entry lengths.

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 many other chemists have repeated this advice, IUPAC and most chemistry texts still favour the usage of allotrope and allotropy for elements only.

Although modern analytical chemistry is dominated by sophisticated instrumentation, the roots of analytical chemistry and some of the principles used in modern instruments are from traditional techniques many of which are still used today.

Although pre-mixed concentrated nutrient solutions are generally purchased from commercial nutrient manufacturers by hydroponic hobbyists and small commercial growers, several tools exists to help anyone prepare their own solutions without extensive knowledge about chemistry.

Although very useful in the foundation of modern chemistry, the law of definite proportions is not universally true.

Although this appears a challenging problem given current resources, many tools will be available to help future researchers: Moore's Law predicts further increases in computer power, semiconductor fabrication techniques continue to approach the nanoscale, and researchers grow ever more skilled at using proteins, ribosomes and DNA to perform novel chemistry.

Although this difference is trivial in bulk chemistry calculations, it can result in complete failure in situations where the behavior of individual molecules matters, such as in mass spectrometry and particle physics ( where the mixture of isotopes does not act as an average ).

Although the mathematical basis of quantum chemistry had been laid by Schrödinger in 1926, it is generally accepted that the first true calculation in quantum chemistry was that of the German physicists Walter Heitler and Fritz London on the hydrogen ( H < sub > 2 </ sub >) molecule in 1927.

Although nothing " improper " was displayed, the obviously much closer chemistry between Steed and Peel constantly suggests intimacy between the two.

Although undeniably a cornerstone of organic chemistry, Hückel's concepts were undeservedly unrecognized for two decades.

Although Rowling has said that Gilderoy Lockhart is her only character that she " deliberately based on a real person ", Snape was reportedly based, at least in part, on John Nettleship, who taught Rowling chemistry and employed her mother as an assistant at Wyedean School near Chepstow.

Although his research clearly has its roots in the alchemical tradition, Boyle is largely regarded today as the first modern chemist, and therefore one of the founders of modern chemistry, and one of the pioneers of modern experimental scientific method.

Although Freud and Jung remain hugely influential, mainstream psychology has made a determined effort to rid itself of the most mystical of these concepts in an attempt to appear more like the hard sciences of chemistry and physics.

Although their exteriors are subjected to intense solar radiation that can alter their chemistry, their interiors contain pristine material essentially untouched since the planetesimal was formed.

Although he initially majored in English, he switched to chemistry under the influence of Arthur Pardee, head of that department.

Although she was only supposed to have a three week role in the company, Heyman was impressed by the chemistry between Psaltis and Storm and offered her a contract, which she accepted.

Although they turned out to be wrong, these conjectures catalyzed further measurement of atomic weights, a great benefit to chemistry.

Although one could argue that even the earliest analytical experiments in chemistry involved a form of chemometrics, the field is generally recognized to have emerged in the 1970s as computers became increasingly exploited for scientific investigation.

Although at that time there was an American science society ( American Association for the Advancement of Science ), the growth of chemistry prompted those assembled, including William H. Nichols, under the direction of Professor Charles F. Chandler of the Columbia School of Mines, to found the American Chemical Society, which would focus more directly on theoretical and applied chemistry.

Although he never received his Ph. D., he studied chemistry in Germany at the Academy of Mines ( 1859 ) of Freiberg, and served as an assistant to Robert Bunsen at Heidelberg, and then with Wurtz in Paris ( 1861 ).

Although to a lesser level than Hiram Lodge and Mr. Weatherbee, he is sometimes the victim of Archie's ever-present accidents ( in particular when Archie blows up the school's chemistry lab ).

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

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 wood has been changed to coal, much of it still retains its original cell structure.

Although today's trucks are as fast as passenger cars, a truck driver has to be a sensible person and guard against hogging the road.

Although the matter has not been unequivocally demonstrated, the available data show that micelles in themselves do not contribute significantly to the detergency process.

Although a somewhat technical subject, it has important political implications as the above discussion of the voting system indicated.

Although the pause in the advance of general business activity this year has thus far been quite modest, it is hard to escape the conclusion that the softening process will continue into the first quarter of 1961 and possibly somewhat longer.

Although refrigeration has served to extend the storage life of these products, substantially increased consumption might be possible if areas remote from the seacoast could be served adequately.

Although science has given us more effective materials, preparations from anise, castorbean, colchicum, nux vomica, mustard, fennel, and stramonium are familiar to many for the relief of human ailments.

Although progress has been made in America's system of libraries it still falls short of what is required if we are to maintain the standards that are needed for an informed America.

Although I have emphasized the barriers which an aroused nationalism has raised against relations between Christians and non-Christians in Asia, the fact is that this development has also widened the gulf between certain Afro-Asian religions themselves.

Although the axiom of countable choice in particular is commonly used in constructive mathematics, its use has also been questioned.

Although argon is a noble gas, it has been found to have the capability of forming some compounds.

Although the Indian and Greek concepts of the atom were based purely on philosophy, modern science has retained the name coined by Democritus.

Although panic attacks are not experienced by every person who has anxiety, they are a common symptom.

Although a cease-fire has held since 1994, the conflict with Azerbaijan over Nagorno-Karabakh has not been resolved.

Although MacDougall's results varied considerably from " 21 grams ", for some people this figure has become synonymous with the measure of a soul's mass.

Although, as of 2009, none of the extrasolar planets detected by ground-based astrometry has been verified in subsequent studies, astrometry is expected to be more accurate in space missions that are not affected by the distorting effects of the Earth's atmosphere.

Although hydrogen-like orbitals are still used as pedagogical tools, the advent of computers has made STOs preferable for atoms and diatomic molecules since combinations of STOs can replace the nodes in hydrogen-like atomic orbital.

Although the sign still exists, the " of " has now been replaced by the URL of their website.

Although the Caiman has not been studied in-depth, scientists have learned that their mating cycles ( previously thought to be spontaneous or year-round ) are linked to the rainfall cycles and the river levels, which increases chances of survival for their offspring.

Although Albert has received relatively little recognition in German history, his dissolution of the Teutonic State caused the founding of the Duchy of Prussia ( and also the Hohenzollern dynasty ), which would eventually become arguably the most powerful German state and instrumental in uniting the whole of Germany.

Although the family ( Amaranthaceae ) is distinctive, the genus has few distinguishing characters among the 70 species included.