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In mathematics, the following inequality is known as Titu's lemma, Bergström's inequality, Engel's form or Sedrakyan's inequality, respectively, referring to the article About the applications of one useful inequality of Nairi Sedrakyan published in 1997, [1] to the book Problem-solving strategies of Arthur Engel published in 1998 and to the book Mathematical Olympiad Treasures of Titu ...
where (,) denotes the modulus of continuity of function with the step . An even more general result of four authors can be formulated as the following Jackson theorem. Theorem 3 : For every natural number n {\displaystyle n} , if f {\displaystyle f} is 2 π {\displaystyle 2\pi } -periodic continuous function, there exists a trigonometric ...
Symbolab is an answer engine [1] that provides step-by-step solutions to mathematical problems in a range of subjects. [2] It was originally developed by Israeli start-up company EqsQuest Ltd., under whom it was released for public use in 2011. In 2020, the company was acquired by American educational technology website Course Hero. [3] [4]
Then S N is identically equal to zero, hence E[S N] = 0, but E[X 1] = 1 / 2 and E[N] = 1 / 2 and therefore Wald's equation does not hold. Indeed, the assumptions , , and are satisfied, however, the equation in assumption holds for all n ∈ except for n = 1.
In mathematics, an inequation is a statement that an inequality holds between two values. [ 1 ] [ 2 ] It is usually written in the form of a pair of expressions denoting the values in question, with a relational sign between them indicating the specific inequality relation.
The infinite complete binary tree T is an infinite tree where one vertex (called the root) has two neighbors and every other vertex has three neighbors. The second moment method can be used to show that at every parameter p ∈ (1/2, 1] with positive probability the connected component of the root in the percolation subgraph of T is infinite.
In probability theory, Hoeffding's lemma is an inequality that bounds the moment-generating function of any bounded random variable, [1] implying that such variables are subgaussian. It is named after the Finnish–American mathematical statistician Wassily Hoeffding. The proof of Hoeffding's lemma uses Taylor's theorem and Jensen's inequality.
The union of the tree and the matching is a cycle, with no possible shortcuts, and with weight approximately 3n/2. However, the optimal solution uses the edges of weight 1 + ε together with two weight-1 edges incident to the endpoints of the path, and has total weight (1 + ε)(n − 2) + 2, close to n for small values of ε. Hence we obtain an ...