1.
Exclamation mark
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The exclamation mark or exclamation point is a punctuation mark usually used after an interjection or exclamation to indicate strong feelings or high volume, and often marks the end of a sentence. Similarly, an exclamation mark is often used in warning signs. Other uses include, In mathematics it denotes the factorial operation, at the beginning of an expression to denote logical negation, e. g. A means the logical negation of A, also called not A. Graphically the exclamation mark is represented as a stop point with a vertical line above. One theory of its origin is that it is derived from a Latin exclamation of joy, the modern graphical representation is believed to have been born in the Middle Ages. Medieval copyists wrote the Latin word io at the end of a sentence to indicate joy, over time, the i moved above the o, and the o became smaller, becoming a point. The exclamation mark did not have its own dedicated key on standard manual typewriters before the 1970s, instead, one typed a period, backspaced, and typed an apostrophe. In the 1950s, secretarial dictation and typesetting manuals in America referred to the mark as bang, appeared in dialogue balloons to represent a gun being fired, although the nickname probably emerged from letterpress printing. This bang usage is behind the names of the interrobang, an unconventional character, and a shebang line. In the printing world, the mark can be called a screamer, a gasper. In hacker culture, the mark is called bang, shriek, or, in the British slang known as Commonwealth Hackish. For example, the password communicated in the spoken phrase Your password is em-nought-pee-aitch-pling-en-three is m0ph. n3, the exclamation mark is common to languages using the Latin alphabet, although usage varies slightly between languages. The exclamation mark was adopted in languages written in other scripts, such as Greek, Russian, Arabic, Hebrew, Chinese, Korean, Japanese and Devanagari. A sentence ending in an exclamation mark may be an exclamation, or an imperative, or may indicate astonishment or surprise, They were the footprints of a gigantic hound. Exclamation marks are occasionally placed mid-sentence with a similar to a comma, for dramatic effect, although this usage is obsolescent, On the walk. Informally, exclamation marks may be repeated for emphasis. The exclamation mark is used in conjunction with the question mark. This can be in protest or astonishment, a few writers replace this with a single, nonstandard punctuation mark, the interrobang, which is the combination of a question mark and an exclamation mark

2.
Double factorial
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In mathematics, the product of all the integers from 1 up to some non-negative integer n that have the same parity as n is called the double factorial or semifactorial of n and is denoted by n. = ∏ k =0 ⌈ n 2 ⌉ −1 = n ⋯ Therefore, = ∏ k =1 n 2 = n ⋯4 ⋅2, and for odd n it is n. = ∏ k =1 n +12 = n ⋯3 ⋅1, =9 ×7 ×5 ×3 ×1 =945. The double factorial should not be confused with the factorial function iterated twice, the sequence of double factorials for even n =0,2,4,6,8. Starts as 1,2,8,48,384,3840,46080,645120, the sequence of double factorials for odd n =1,3,5,7,9. Starts as 1,3,15,105,945,10395,135135, merserve states that the double factorial was originally introduced in order to simplify the expression of certain trigonometric integrals arising in the derivation of the Wallis product. Double factorials also arise in expressing the volume of a hypersphere, the term odd factorial is sometimes used for the double factorial of an odd number. For an even integer n = 2k, k ≥0. For odd n = 2k −1, k ≥1, in this expression, the first denominator equals. and cancels the unwanted even factors from the numerator. For an odd positive integer n = 2k −1, k ≥1, double factorials are motivated by the fact that they occur frequently in enumerative combinatorics and other settings. For instance, n. for odd values of n counts Perfect matchings of the complete graph Kn +1 for odd n. For instance, a graph with four vertices a, b, c. Perfect matchings may be described in several equivalent ways, including involutions without fixed points on a set of n +1 items or chord diagrams. Stirling permutations, permutations of the multiset of numbers 1,1,2,2, K, k in which each pair of equal numbers is separated only by larger numbers, where k = n + 1/2. From this recursive construction, a proof that the Stirling permutations are counted by the double permutations follows by induction, heap-ordered trees, trees with k +1 nodes labeled 0,1,2. K, such that the root of the tree has label 0, each node has a larger label than its parent. An Euler tour of the tree gives a Stirling permutation, unrooted binary trees with n + 5/2 labeled leaves. Each such tree may be formed from a tree with one leaf, by subdividing one of the n tree edges

3.
Retroflex clicks
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The retroflex clicks are a family of click consonants known only from the Central. Kung dialects of Namibia and the Damin ritual jargon of Australia. They may be sub-apical retroflex and should not be confused with the more widespread postalveolar clicks, there is no symbol in the International Phonetic Alphabet that represents the forward articulation of these sounds, but one may be derived with a rhotic diacritic from the alveolar click, ⟨ǃ˞⟩. In the literature they are written with the ad hoc digraph ⟨‼⟩. They then went largely unnoticed until ca, Miller notes that the Grootfontein retroflex clicks have a lateral release, and alternatively transcribes them ⟨ǃǁ⟩. They obey the back-vowel constraint common among retroflex consonants, basic retroflex clicks are, Features of postalveolar clicks, The basic articulation may be voiced, nasal, aspirated, glottalized, etc. The place of articulation is post-alveolar, and the shape may be subapical. The center of the tongue moves downward to create suction, clicks may be oral or nasal, which means that the airflow is either restricted to the mouth, or passes through the nose as well. It is a consonant, which means it is produced by directing the airstream over the sides of the tongue. The release of the forward closure produces the click sound, voiced and nasal clicks have a simultaneous pulmonic egressive airstream. As with other click articulations, retroflex clicks may be produced with various manners, an example is the voiced retroflex click in the Grootfontein. Kung word for water, /ǃ̬˞ ˡú/. Damin is the other language known to have had such a sound. A retroflex series claimed for Ekoka. Kung turns out to be domed palatal clicks, scott, Miller, Namaseb, Sands, & Shah,2010. Retroflex clicks in two dialects of. Xung, the fricated alveolar click of Ekoka. Kung, which was once thought to be retroflex Alveolar clicks Bilabial clicks Dental clicks Lateral clicks Palatal clicks