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A lower bound is typically described by a theorem like "for every element α of some subset of the real numbers and every rational number p/q, we have | | > ()". In some cases, "every rational number" may be replaced by "all rational numbers except a finite number of them", which amounts to multiplying φ by some constant depending on α .
Proof. First equation: ... measures how close this estimator's variance comes to this lower bound; estimator efficiency is defined as (^) = (^) or the minimum ...
A natural proof is a proof that establishes that a certain language lies outside of C and refers to a natural property that is useful against C. Razborov and Rudich give a number of examples of lower-bound proofs against classes C smaller than P/poly that can be "naturalized", i.e. converted
There is a corresponding greatest-lower-bound property; an ordered set possesses the greatest-lower-bound property if and only if it also possesses the least-upper-bound property; the least-upper-bound of the set of lower bounds of a set is the greatest-lower-bound, and the greatest-lower-bound of the set of upper bounds of a set is the least ...
The set S = {42} has 42 as both an upper bound and a lower bound; all other numbers are either an upper bound or a lower bound for that S. Every subset of the natural numbers has a lower bound since the natural numbers have a least element (0 or 1, depending on convention). An infinite subset of the natural numbers cannot be bounded from above.
However, involving the discriminant of the polynomial allows a lower bound. For square-free polynomials with integer coefficients, the discriminant is an integer, and has thus an absolute value that is not smaller than 1. This allows lower bounds for root separation that are independent from the discriminant. Mignotte's separation bound is [18 ...
The first example below describes one such result from 1947 that gives a proof of a lower bound for the Ramsey number R(r, r). First example. ... Proof. Let X be the ...
Ben-Sasson and Wigderson (1999) provided a proof method reducing lower bounds on size of Resolution refutations to lower bounds on width of Resolution refutations, which captured many generalizations of Haken's lower bound. [18] It is a long-standing open problem to derive a nontrivial lower bound for the Frege system.