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For an essentially normal operator$T$, it is shown that there exists a unilateral shift of multiplicity$m$in$C$^{$*$}$(T)$if and only if γ($T$)≠0 and γ$(T)/m$. As application, we prove that the essential commutant of a unilateral shift and that of a bilateral shift are not isomorphic as$C$^{$*$}-algebras. Finally, we construct a natural$C$^{$*$}-algebra ε + ε_{*}on the Bergman space$L$_{$a$}^{$2$}($B$_{$n$}), and show that its essential commutant is generated by Toeplitz operators with symmetric continuous symbols and all compact operators.

In an infinite sequence of independent identically distributed continuous random variables we study the number of strings of two subsequent records interrupted by a given number of non-records. By embedding in a marked Poisson process we prove that these counts are independent and Poisson distributed. Also the distribution of the number of uninterrupted strings of records is considered.

Here we propose a novel method to compute Teichmller shape space based shape index to study brain morphometry. Such a shape index is intrinsic, and invariant under conformal transformations, rigid motions and scaling. We conformally map a genus-zero open boundary surface to the Poincar disk with the Yamabe flow method. The shape indices that we compute are the lengths of a special set of geodesics under hyperbolic metric. Tests on longitudinal brain imaging data were used to demonstrate the stability of the derived feature vectors. In leave-one-out validation tests, we achieved 100% accurate classification (versus only 68% accuracy for volume measures) in distinguishing 11 HIV/AIDS individuals from 8 healthy control subjects, based on Teichmller coordinates for lateral ventricular surfaces extracted from their 3D MRI scans.

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