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picture info	Double Factorial In mathematics, the double factorial or semifactorial of a number , denoted by , is the product of all the integers from 1 up to that have the same parity (odd or even) as . That is, :n!! = \prod_^ (n-2k) = n (n-2) (n-4) \cdots. For even , the double factorial is :n!! = \prod_^\frac (2k) = n(n-2)(n-4)\cdots 4\cdot 2 \,, and for odd it is :n!! = \prod_^\frac (2k-1) = n(n-2)(n-4)\cdots 3\cdot 1 \,. For example, . The zero double factorial as an empty product. The sequence of double factorials for even = starts as : 1, 2, 8, 48, 384, 3840, 46080, 645120,... The sequence of double factorials for odd = starts as : 1, 3, 15, 105, 945, 10395, 135135,... The term odd factorial is sometimes used for the double factorial of an odd number. History and usage In a 1902 paper, the physicist Arthur Schuster wrote: states that the double factorial was originally introduced in order to simplify the expression of certain trigonometric integrals that arise in the derivation of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Exclamation Mark The exclamation mark, , or exclamation point (American English), is a punctuation mark usually used after an interjection or exclamation to indicate strong feelings or to show emphasis. The exclamation mark often marks the end of a sentence, for example: "Watch out!". Similarly, a bare exclamation mark (with nothing before or after) is often used in warning signs. The exclamation mark is often used in writing to make a character seem as though they are shouting and/or excited/surprised. Other uses include: * In mathematics, it denotes the factorial operation. * Several computer languages use at the beginning of an expression to denote logical negation. For example, means "the logical negation of A", also called "not A". This usage has spread to ordinary language (e.g., "!clue" means no-clue or clueless). * Some languages use to denote a click consonant. History Graphically, the exclamation mark is represented by variations on the theme of a full stop point with a vertical l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	!! (chess) When annotating chess games, commentators frequently use widely recognized annotation symbols. Question marks and exclamation points that denote a move as bad or good are ubiquitous in chess literature. Some publications intended for an international audience, such as the '' Chess Informant'', have a wide range of additional symbols that transcend language barriers. The common symbols for evaluating the merits of a move are "??", "?", "?!", "!?", "!", and "!!". The chosen symbol is appended to the text describing the move (e.g. Re7? or Kh1!?); see Algebraic chess notation. Use of these annotation symbols is subjective, as different annotators use the same symbols differently. Moreover, an annotator's use of symbols is often influenced by the player's strength: for example, a positional misjudgment that an annotator might give a "??" if played by a strong grandmaster might pass unremarked if played by a novice. Annotators' use of punctuation may also be influenced by the res ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	General Punctuation (Unicode Block) General Punctuation is a Unicode block containing punctuation, spacing, and formatting characters for use with all scripts and writing systems. Included are the defined-width spaces, joining formats, directional formats, smart quotes, archaic and novel punctuation such as the interrobang, and invisible mathematical operators. Additional punctuation characters are in the Supplemental Punctuation block and sprinkled in dozens of other Unicode blocks. Block Several characters in this block are usually not rendered with a directly visible glyph. Ten whitespace characters U+2002 through U+200B (fixed ''en'' or ''em, em, em, em, em, figure'' and ''punctuation space'', variable ''thin'' or ''em'' and ''hair space'', fixed ''zero-width space'') and U+205F (''math medium'' or '' em space'') differ by horizontal width, while U+2000 and U+2001 (''en'' and ''em quad'') are effectively aliases of U+2002 and U+2003, respectively; another two, U+202F and U+2060 (ill-termed ''word joiner'') ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Retroflex Click The retroflex clicks are a family of click consonants known only from the Central !Kung dialects of Namibia. They are sub-apical retroflex and should not be confused with the more widespread postalveolar clicks, which are sometimes mistakenly called "retroflex" (for example in Unicode) due to their concave tongue shape. There is no official symbol in the International Phonetic Alphabet that represents the forward articulation of these sounds, and the expected symbol is rarely seen. In the literature they are typically written with the ''ad hoc'' digraph , the convention since Doke identified them as retroflex in 1926. (Doke's proposed symbol, , did not catch on, nor did Vedder's and Anders' . For a while Amanda Miller, who noted a lateral fricated release (as had Vedder), transcribed them .) The Damin ritual jargon of Australia may have had a voiced nasal click, transcribed by Hale & Nash as , though it's not known if it was phonemically distinct. However, Damin presumably h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	! (other) ! is a punctuation mark, called an exclamation mark ( 33 in ASCII), exclamation point, ecphoneme, or bang. ! or exclamation point may also refer to: Mathematics and computers Factorial, a mathematical function * Derangement, a related mathematical function Negation, in logic and some programming languages Uniqueness quantification, in mathematics and logic * ! (CONFIG.SYS directive), usage for unconditional execution of directives in FreeDOS configuration files Music * ''!'' (The Dismemberment Plan album), released in 1995 * ''!'' (Donnie Vie album), released in 2016 * "!" (The Song Formerly Known As), a single on the 1997 album ''Unit'' by Regurgitator * ''Exclamation Mark'' (album), a 2011 album by Jay Chou ''Exclamation Point'', a 2010 LP by DA! ''!'' (Trippie Redd album), released in 2019 ** ''!'' (Trippie Redd song), that album's title track * ''!'' (Cláudia Pascoal album), released in 2020 Other ǃ, the IPA symbol for postalveolar click in speech An ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Double-negation Translation In proof theory, a discipline within mathematical logic, double-negation translation, sometimes called negative translation, is a general approach for embedding classical logic into intuitionistic logic, typically by translating formulas to formulas which are classically equivalent but intuitionistically inequivalent. Particular instances of double-negation translation include Glivenko's translation for propositional logic, and the Gödel–Gentzen translation and Kuroda's translation for first-order logic. Propositional logic The easiest double-negation translation to describe comes from Glivenko's theorem, proved by Valery Glivenko in 1929. It maps each classical formula φ to its double negation ¬¬φ. Glivenko's theorem states: :If φ is a propositional formula, then φ is a classical tautology if and only if ¬¬φ is an intuitionistic tautology. Glivenko's theorem implies the more general statement: :If ''T'' is a set of propositional formulas and φ a propositional formu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]