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  2. Upper and lower bounds - Wikipedia

    en.wikipedia.org/wiki/Upper_and_lower_bounds

    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.

  3. Variational method (quantum mechanics) - Wikipedia

    en.wikipedia.org/wiki/Variational_method...

    The expectation value of the total Hamiltonian H (including the term V ee) in the state described by ψ 0 will be an upper bound for its ground state energy. V ee is −5E 1 /2 = 34 eV, so H is 8E 1 − 5E 1 /2 = −75 eV. A tighter upper bound can be found by using a better trial wavefunction with 'tunable' parameters.

  4. Extreme value theorem - Wikipedia

    en.wikipedia.org/wiki/Extreme_value_theorem

    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.

  5. Estimation lemma - Wikipedia

    en.wikipedia.org/wiki/Estimation_lemma

    In mathematics the estimation lemma, also known as the ML inequality, gives an upper bound for a contour integral.If f is a complex-valued, continuous function on the contour Γ and if its absolute value | f (z) | is bounded by a constant M for all z on Γ, then

  6. Geometrical properties of polynomial roots - Wikipedia

    en.wikipedia.org/wiki/Geometrical_properties_of...

    Applied to Cauchy's bound, this gives the upper bound + = for the real roots of a polynomial with real coefficients. If this bound is not greater than 1, this means that all nonzero coefficients have the same sign, and that there is no positive root.

  7. Infimum and supremum - Wikipedia

    en.wikipedia.org/wiki/Infimum_and_supremum

    Similarly, the set of integers has the least-upper-bound property; if is a nonempty subset of and there is some number such that every element of is less than or equal to , then there is a least upper bound for , an integer that is an upper bound for and is less than or equal to every other upper bound for .

  8. Euler's totient function - Wikipedia

    en.wikipedia.org/wiki/Euler's_totient_function

    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 ⁠ .

  9. Q-function - Wikipedia

    en.wikipedia.org/wiki/Q-function

    The Q-function is not an elementary function. However, it can be upper and lower bounded as, [6] [7] ... , the best upper bound is given by = and = with maximum ...