Anatomy of Malicious Singularities

As well known, the b-boundaries of the closed Friedman world model and of Schwarzschild solution consist of a single point. We study this phenomenon in a broader context of differential and structured spaces. We show that it is an equivalence relation $\rho$, defined on the Cauchy completed total space $\bar{E}$ of the frame bundle over a given space-time, that is responsible for this pathology. A singularity is called malicious if the equivalence class $[p_0]$ related to the singularity remains in close contact with all other equivalence classes, i.e., if $p_0 \in \mathrm{cl}[p]$ for every $p \in E$. We formulate conditions for which such a situation occurs. The differential structure of any space-time with malicious singularities consists only of constant functions which means that, from the topological point of view, everything collapses to a single point. It was noncommutative geometry that was especially devised to deal with such situations. A noncommutative algebra on $\bar{E}$, which turns out to be a von Neumann algebra of random operators, allows us to study probabilistic properties (in a generalized sense) of malicious singularities. Our main result is that, in the noncommutative regime, even the strongest singularities are probabilistically irrelevant.Comment: 16 pages in LaTe

Measurements of farfield sound generation from a flow-excited cavity

Results of 1/3-octave-band spectral measurements of internal pressures and the external acoustic field of a tangentially blown rectangular cavity are compared. Proposed mechanisms for sound generation are reviewed, and spectra and directivity plots of cavity noise are presented. Directivity plots show a slightly modified monopole pattern. Frequencies of cavity response are calculated using existing predictions and are compared with those obtained experimentally. The effect of modifying the upstream boundary layer on the noise was investigated, and its effectiveness was found to be a function of cavity geometry and flow velocity