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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.
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 ).
Furthermore, in the case of λ=1/p, μ=1/q and ν=1/r for integers p, q, r, then the triangle tiles the sphere, the complex plane or the upper half plane according to whether λ + μ + ν – 1 is positive, zero or negative; and the s-maps are inverse functions of automorphic functions for the triangle group 〈p, q, r〉 = Δ(p, q, r).
The function () is defined as () = (+) ... Another example requires the assumption that commutes almost surely with . ... We can find an upper bound for ...
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 Q-function is well tabulated and can be computed directly in most of the mathematical software packages such as R and those available in Python, MATLAB and Mathematica. Some values of the Q -function are given below for reference.
The LR solution as a bound [ edit ] Of particular use is the property that for any fixed set of λ ~ ⪰ 0 {\displaystyle {\tilde {\lambda }}\succeq 0} values, the optimal result to the Lagrangian relaxation problem will be no smaller than the optimal result to the original problem.
so any complex differentiable function f in an open set U ⊂ C is in fact complex analytic. All that is said for real analytic functions here holds also for complex analytic functions with the open interval I replaced by an open subset U ∈ C and a-centered intervals (a − r, a + r) replaced by c-centered disks B(c, r). In particular, the ...