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Page "Combinatorics" ¶ 11
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Although and counting
Although the metre is now defined as the path length travelled by light in a given time, the practical laboratory length measurements in metres are determined by counting the number of wavelengths of laser light of one of the standard types that fit into the length, and converting the selected unit of wavelength to metres.
Although the counting process is complicated, voting is clear and most voters get at least one of their preferences elected.
Although it's possible to augment simple reference counts in a variety of ways, often a better solution can be found by performing reference counting in a fundamentally different way.
Although the Samaritans stopped counting for the Jubilee some hundreds of years ago, according to a recent report an effort is underway to determine the date when counting ceased in order to resume.
Although the NORC study was not primarily intended as a determination of which candidate " really won ", analysis of the results, given the hand counting of machine-uncountable ballots due to various types of voter error, indicated that they would lead to differing results, reported in the newspapers which funded the recount, such as The Miami Herald ( The Miami Herald Report: Democracy Held Hostage ) or the Washington Post.
" Although he clarified his statement as merely a poor choice of words, critics of Diebold and / or the Republican party interpreted this as at minimum an indication of a conflict of interest, at worst implying a risk to the fair counting of ballots.
Although the language is long extinct it is arguable that traces of its vocabulary persisted into the modern era in the form of " counting scores " and in a handful of dialectal words.
Although the number of registrations has topped 100, 000, a design flaw in the counting process means that it counts users who abandoned the signup process mid way ; the actual number of posters is smaller, clocking in at around 38, 700 as of October 2008.
Although it is a crossover film, it is primarily a Rugrats movie as the main plot focuses on those characters, and the characters from The Wild Thornberrys are supporting characters, not counting Nigel Thornberry ( voiced by Tim Curry ) who is in fact a main character.
* Vote-rigging: Although the counting at the local ballot boxes was conducted in public, with the ballot papers held up and the scores marked on boards, it was at the later stages where irregularities were frequently reported.
" Although Caesar says " at night " he specifically does not say " sunset " so we do not know how much the Gauls differed from others in methods of counting from midnight.
Although consistency in counting customers is probably more important than formulating a precise definition, a definition is needed nonetheless.
Although these sets certainly describe different things, they have the same counting sequence, namely the Catalan numbers.
Although this switch from counting rods to the abacus allowed for reduced computation times, it may have also led to the stagnation and decline of Chinese mathematics.
Although, for the United Kingdom, it was released on the same day, with both Part I and Part II counting as the song release.
Although the Codex Belli itself doesn't specify any rules for counting hits, two variants have become widely accepted at German events.
Although decimal up to 10, in most dialects the scheme then changes to counting in ( sub -) base 5.
Although not used as counting device, many Jews touch the knots on the tzitzits attached to their tallit ( prayer shawl ) at specific points in their prayers.

Although and number
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 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 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 a large number of coatings systems, particularly at low thicknesses fail cohesively by the cutting mechanism, frequently a second type of cohesive failure may also take place.
Although Brooklyn College does not yet have a junior-year-abroad program, a good number of students spend summers in Europe.
Although the primary mathematical properties of the middle number at the center of the Lo Shu, and the interrelation of all the other numbers to it, might seem enough to account for the deep fascination which the Lo Shu held for the Old Chinese philosophers, this was actually only a beginning of wonders.
Although England decisively won the Ashes 4 – 1, Bodyline caused such a furore in Australia that diplomats had to intervene to prevent serious harm to Anglo-Australian relations, and the MCC eventually changed the Laws of cricket to curtail the number of leg side fielders.
Although the vocalists varied, a small number of musicians worked with the Alan Parsons Project regularly.
Although a number of different broadcast television systems were in use worldwide, the same principles of operation apply.
Although gold has an atomic number of 79, immediately after Rutherford's paper appeared in 1911 Antonius Van den Broek made the intuitive suggestion that atomic number is nuclear charge.
Although it is commonly believed that assassins were under the influence of hashish during their killings or during their indoctrination, there is debate as to whether these claims have merit, with many Eastern writers and an increasing number of western academics coming to believe that drug-taking was not the key feature behind the name.
Although initial studies suggested that antioxidant supplements might promote health, later large clinical trials with a limited number of antioxidants detected no benefit and even suggested that excess supplementation with certain putative antioxidants may be harmful.
Although 64 is the highest number depicted on the doubling cube, the stakes may rise to 128, 256, and so on.
Although passengers rarely have cause to refer to either document, all travel is subject to the National Rail Conditions of Carriage and all tickets are valid subject to the rules set out in a number of so-called technical manuals, which are centrally produced for the network.
Although BJU never reapplied for federal tax-exempt status, and it continues to pay federal taxes, a number of its ancillaries, including Bob Jones Academy and the BJU Museum & Gallery, are tax-exempt entities.
Although all the nuclei of all atoms belonging to one element will have the same number of protons, they may not necessarily have the same number of neutrons ; such atoms are termed isotopes.
Although not known with certainty the total number of species in the family, conservative estimates indicate approximately 2. 600 species worldwide, distributed in 109 genera.
Although Attlee's second government was less radical than the first, it oversaw the passage of a number of reforms relating to issues such as industry in development areas, the restoration of land which had been devastated by ironstone pollution, and river pollution.
Although Confucius claimed that he never invented anything but was only transmitting ancient knowledge ( see Analects VII, 1 ), he did produce a number of new ideas.
Although a " bijection " seems a more advanced concept than a number, the usual development of mathematics in terms of set theory defines functions before numbers, as they are based on much simpler sets.
Although Wegener's theory was formed independently and was more complete than those of his predecessors, Wegener later credited a number of past authors with similar ideas:

Although and elements
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 most of Chaplin's films are characterised as comedies, most of them also employ strong elements of drama and even tragedy.
Although all known chemical matter is composed of these elements, chemical matter itself constitutes only about 15 % of the matter in the universe.
Although earlier precursors to this presentation exist, its invention is generally credited to Russian chemist Dmitri Mendeleev in 1869, who intended the table to illustrate recurring trends in the properties of the elements.
Although not formalised and acknowledged as a mythos per se, Lovecraft did correspond with contemporary writers ( Clark Ashton Smith, Robert E. Howard, Robert Bloch, Frank Belknap Long, Henry Kuttner, and Fritz Lieber – a group referred to as the " Lovecraft Circle ") – and shared story elements: Robert E. Howard's character Friedrich Von Junzt reads Lovecraft's Necronomicon in the short story " The Children of the Night " ( 1931 ), and in turn Lovecraft mentions Howard's Unaussprechlichen Kulten in the stories " Out of the Aeons " ( 1935 ) and " The Shadow Out of Time " ( 1936 ).
Although much of Chinese philosophy begins in the Warring States period, elements of Chinese philosophy have existed for several thousand years ; some can be found in the Yi Jing ( the Book of Changes ), an ancient compendium of divination, which dates back to at least 672 BCE.
Although all group 16 elements of the periodic table, including oxygen, are defined as chalcogens, oxygen and oxides are usually distinguished from chalcogens and chalcogenides.
Although every map element serves some purpose, convention only dictates inclusion of some elements, while others are considered optional.
Although " Oedipus's enquiry is based on supernatural, pre-rational methods that are evident in most narratives of crime until the development of Enlightenment thought in the seventeenth and eighteenth centuries " it has " all of the central characteristics and formal elements of the detective story, including a mystery sur-rounding a murder, a closed circle of suspects, and the gradual uncovering of a hidden past.
" Although not expressed as strongly, Glyn Jones believed that he and Thomas ' friendship cooled in the later years as he had not ' rejected enough ' of the elements that Thomas disliked – " Welsh nationalism and a sort of hill farm morality ".
Although various notations are used throughout the literature to denote that two elements a and b of a set are equivalent with respect to an equivalence relation R, the most common are " a ~ b " and " a ≡ b ", which are used when R is the obvious relation being referenced, and variations of " a ~< sub > R </ sub > b ", " a ≡< sub > R </ sub > b ", or " aRb ".
Although most electrical engineers will understand basic circuit theory ( that is the interactions of elements such as resistors, capacitors, diodes, transistors and inductors in a circuit ), the theories employed by engineers generally depend upon the work they do.
Although mythical elements are not so prominent in Exodus as in Genesis, the echoes of ancient legends are crucial to understanding the book's origins and purpose: for example, the story of the infant Moses's salvation from the Nile has its basis in an earlier legend of king Sargon, while the story of the parting of the Red Sea trades on Mesopotamian creation mythology.
Although the inclusion of cognitive appraisal as one of the elements is slightly controversial, since some theorists make the assumption that emotion and cognition are separate but interacting systems, the component processing model provides a sequence of events that effectively describes the coordination involved during an emotional episode.
Although the most common seven-string has a low B string, Roger McGuinn ( of The Byrds and Rickenbacker ) uses an octave G string paired with the regular G string as on a 12-string guitar, allowing him to incorporate chiming 12-string elements in standard six-string playing.
Although many of his works feature gothic elements, the first one which is considered to belong purely in the „ gothic fiction “ label is Ostrov Borngolm ( Island of Bornholm ) from 1793.
Although Magyar tribes only began to settle in the geographical area of present-day Hungary in the very end of the 9th century, some 450 years after the dissolution of the Hunnic tribal confederation, Hungarian prehistory includes Magyar origin myths, which may have preserved some elements of historical truth.
Although considered " small and primitive " by the standards of its time, it was the first working machine to contain all of the elements essential to a modern electronic computer.
Although incorporating Italian and French stylistic elements into his compositions, Purcell's legacy was a uniquely English form of Baroque music.
Although he considered the blend of gameplay elements from different genres a success, he said that Kid Icarus suffered from " frustrating " design flaws, such as its high difficulty level.
For anarchist historian George Woodcock " Although ( Pierre Joseph ) Proudhon was the first writer to call himself an anarchist, at least two predecessors outlined systems that contain all the basic elements of anarchism.
Although he frequently wrote of his longing for solitude in order to come closer to God and to extend his reflections – elements considered essential to the prophetic experience itself in his philosophy-he gave over almost all his time to caring for others.
Although musical theatre overlaps with other theatrical forms like opera and dance, it may be distinguished by the equal importance given to the music as compared with the dialogue, movement and other elements of the works.
Although the project was ultimately abandoned, some elements were integrated into other projects.

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