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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 ...
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.
This proof also shows why the bounds are quite loose in typical cases: the conditional expectation on the event where |X − μ| < kσ is thrown away, and the lower bound of k 2 σ 2 on the event |X − μ| ≥ kσ can be quite poor.
Proof. First equation: ... measures how close this estimator's variance comes to this lower bound; estimator efficiency is defined as (^) = (^) or the minimum ...
An upper bound for R(r, s) can be extracted from the proof of the theorem, and other arguments give lower bounds. (The first exponential lower bound was obtained by Paul Erdős using the probabilistic method.) However, there is a vast gap between the tightest lower bounds and the tightest upper bounds.
The upper bound is known in Quantum Mechanics as reduction criterion for density matrices; it was first proven by [12] and independently formulated by. [13] The lower bound has been obtained in [11]: Theorem A.16 that provides a Bayesian interpretation of these bounds.
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
In probability theory, Boole's inequality, also known as the union bound, says that for any finite or countable set of events, the probability that at least one of the events happens is no greater than the sum of the probabilities of the individual events. This inequality provides an upper bound on the probability of occurrence of at least one ...