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One can then prove that this smoothed sum is asymptotic to − + 1 / 12 + CN 2, where C is a constant that depends on f. The constant term of the asymptotic expansion does not depend on f : it is necessarily the same value given by analytic continuation, − + 1 / 12 .
For example, to form the "pea pattern" one reads the previous term and counts all instances of each digit, listed in order of their first appearance, not just those occurring in a consecutive block. So beginning with the seed 1, the pea pattern proceeds 1, 11 ("one 1"), 21 ("two 1s"), 1211 ("one 2 and one 1"), 3112 ("three 1s and one 2 ...
Suppose we drive a transition in an atomic system to an excited state |e , which can decay into either the ground state |g or a long-lived state |d . If the probability to decay (the branching fraction) into the |g state is p {\displaystyle p} , then the probability to decay into the other state |d would be 1 − p {\displaystyle 1 ...
In British military practice, Mark ("Mk") designations were given in Roman numerals (replaced by Arabic numerals in 1944) to reflect variants of or production changes to service weaponry, either on their own or as part of numerical ("No.") designations; in the Lee-Enfield rifle series for example, the SMLE rifles were produced to Mk I, Mk III, and Mk V specification (with the latter two later ...
The carbonate buffer system is a series of reactions that uses carbonate as a buffer to convert into bicarbonate. [12] The carbonate buffer reaction helps maintain a constant H+ concentration in the ocean because it consumes hydrogen ions, [ 13 ] and thereby maintains a constant pH. [ 12 ]
A long ton, also called the weight ton (W/T), [1] imperial ton, or displacement ton, is equal to: . 2,240 pounds (1,016.0 kilograms; 1.0160 metric tons) exactly 12% more than the 2,000 pounds of the North American short ton, being 20 long hundredweight (112 lb) rather than 20 short hundredweight (100 lb)
The symmetry of a carbon dioxide molecule is linear and centrosymmetric at its equilibrium geometry. The length of the carbon–oxygen bond in carbon dioxide is 116.3 pm, noticeably shorter than the roughly 140 pm length of a typical single C–O bond, and shorter than most other C–O multiply bonded functional groups such as carbonyls. [19]
Magnesium hydroxychloride [1] is the traditional term for several chemical compounds of magnesium, chlorine, oxygen, and hydrogen whose general formula xMgO·yMgCl 2 ·zH 2 O, for various values of x, y, and z; or, equivalently, Mg x+y (OH) 2x Cl 2y (H 2 O) z−x.