Lifan (China)

This paper explains the vibration evaluation methods for floors supporting sensitive equipment in the second edition of the AISC Design Guide 11, Vibrations of Steel-Framed Structural Systems Due to Human Activity. It presents new experimentally verified equations that are simplified for practical implementation. Sensitive equipment tolerance limits are stated in numerous forms, including peak acceleration and various spectral acceleration and velocity. For this reason, response predictions are given in terms of all tolerance limit forms that are commonly used. The most severe responses of lower frequency floors to walking are due to resonant build-ups whereas the most severe responses of higher frequency floors are due to a series of individual footsteps impulses, so equations are given for both. Because the sensitive equipment tolerance limit is objective, each equation includes a calibration factor that results in a high probability that the predicted response will exceed the actual response.

The main goal of this paper is to put some known results in a common perspective and to simplify their proofs. We start with a simple proof of a result from (Vereshchagin, 2002) saying that $\limsup_n\KS(x|n)$ (here $\KS(x|n)$ is conditional (plain) Kolmogorov complexity of $x$ when $n$ is known) equals $\KS^{\mathbf{0'}(x)$, the plain Kolmogorov complexity with $\mathbf{0'$-oracle. Then we use the same argument to prove similar results for prefix complexity (and also improve results of (Muchnik, 1987) about limit frequencies), a priori probability on binary tree and measure of effectively open sets. As a by-product, we get a criterion of $\mathbf{0'}$ Martin-L\"of randomness (called also 2-randomness) proved in (Miller, 2004): a sequence $\omega$ is 2-random if and only if there exists $c$ such that any prefix $x$ of $\omega$ is a prefix of some string $y$ such that $\KS(y)\ge |y|-c$. (In the 1960ies this property was suggested in (Kolmogorov, 1968) as one of possible randomness definitions; its equivalence to 2-randomness was shown in (Miller, 2004) while proving another 2-randomness criterion (see also (Nies et al. 2005)): $\omega$ is 2-random if and only if $\KS(x)\ge |x|-c$ for some $c$ and infinitely many prefixes $x$ of $\omega$. Finally, we show that the low-basis theorem can be used to get alternative proofs for these results and to improve the result about effectively open sets; this stronger version implies the 2-randomness criterion mentioned in the previous sentence.