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Similarly, a function g defined on domain D and having the same codomain (K, ≤) is an upper bound of f, if g(x) ≥ f (x) for each x in D. The function g is further said to be an upper bound of a set of functions, if it is an upper bound of each function in that set.
The following bounds are known for the Chebyshev functions: (in these formulas p k is the k th prime number; p 1 = 2, ... Upper bounds exist for both ...
The main objective of interval arithmetic is to provide a simple way of calculating upper and lower bounds of a function's range in one or more variables. These endpoints are not necessarily the true supremum or infimum of a range since the precise calculation of those values can be difficult or impossible; the bounds only need to contain the function's range as a subset.
In probability theory, a Chernoff bound is an exponentially decreasing upper bound on the tail of a random variable based on its moment generating function.The minimum of all such exponential bounds forms the Chernoff or Chernoff-Cramér bound, which may decay faster than exponential (e.g. sub-Gaussian).
In the graph at right the top line y = n − 1 is an upper bound valid for all n other than one, and attained if and only if n is a prime number. A simple lower bound is φ ( n ) ≥ n / 2 {\displaystyle \varphi (n)\geq {\sqrt {n/2}}} , which is rather loose: in fact, the lower limit of the graph is proportional to n / log log n .
By the boundedness theorem, f is bounded from above, hence, by the Dedekind-completeness of the real numbers, the least upper bound (supremum) M of f exists. It is necessary to find a point d in [a, b] such that M = f(d). Let n be a natural number. As M is the least upper bound, M – 1/n is not an upper bound for f.
Therefore, if one can compute or obtain an upper bound on -Selmer rank of , then one would be able to bound the Mordell-Weil rank on average as well. In Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves , [ 7 ] Bhargava and Shankar computed the 2-Selmer rank of elliptic curves on average.
where () is the density function of the standard normal distribution, and the bounds become increasingly tight for large x. Using the substitution v = u 2 /2, the upper bound is derived as follows: