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In mathematics, specifically in the theory of Markovian stochastic processes in probability theory, the Chapman–Kolmogorov equation (CKE) is an identity relating the joint probability distributions of different sets of coordinates on a stochastic process.
For any integer n, n ≡ 1 (mod 2) if and only if 3n + 1 ≡ 4 (mod 6). Equivalently, n − 1 / 3 ≡ 1 (mod 2) if and only if n ≡ 4 (mod 6). Conjecturally, this inverse relation forms a tree except for the 1–2–4 loop (the inverse of the 4–2–1 loop of the unaltered function f defined in the Statement of the problem section of ...
where f (2k−1) is the (2k − 1)th derivative of f and B 2k is the (2k)th Bernoulli number: B 2 = 1 / 6 , B 4 = − + 1 / 30 , and so on. Setting f ( x ) = x , the first derivative of f is 1, and every other term vanishes, so [ 15 ]
A necessary (but not sufficient) condition for solvability is that n is not divisible by 4 or by a prime of form 4k + 3. [note 3] Thus, for example, x 2 − 3 y 2 = −1 is never solvable, but x 2 − 5 y 2 = −1 may be. [27] The first few numbers n for which x 2 − n y 2 = −1 is solvable are with only one trivial solution: 1
An example of such a rearrangement is the shift of substituents on tropilidenes (1,3,5-cycloheptatrienes). When heated, the pi-system goes through an electrocyclic ring closing to form bicycle[4,1,0]heptadiene (norcaradiene). Thereafter follows a [1,5] alkyl shift and an electrocyclic ring opening. norcaradiene rearrangement
Only two Super Bowls have ever gone to overtime. One was last year's Chiefs vs. 49ers game. Here's what to know about OT in the Super Bowl.
Fosse was born in Chicago, Illinois, on June 23, 1927, to a Norwegian-American father, Cyril Kingsley Fosse, a traveling salesman for Hershey, [2] and an Irish-American mother, Sarah Alice "Sadie" (née Stanton) Fosse. He was the fifth of six children. [3] [4] [5] He was drawn to dance and took lessons.
[4] The musical did not recreate the musical numbers as originally presented but instead had primarily black-and-white costumes (including the all-important hats), set against a simple setting. [ 5 ] Reviewer Ben Brantley of The New York Times describes the show: "The ghost of the man being celebrated -- blurred and fleeting but definitely ...