Convergence and multiplicities for the Lempert function

Given a domain $\Omega \subset \mathbb C$, the Lempert function is a functional on the space Hol (\D,\Omega) of analytic disks with values in $\Omega$, depending on a set of poles in $\Omega$. We generalize its definition to the case where poles have multiplicities given by local indicators (in the sense of Rashkovskii's work) to obtain a function which still dominates the corresponding Green function, behaves relatively well under limits, and is monotonic with respect to the indicators. In particular, this is an improvement over the previous generalization used by the same authors to find an example of a set of poles in the bidisk so that the (usual) Green and Lempert functions differ.Comment: 24 pages; many typos corrected thanks to the referee of Arkiv for Matemati

A comparison of FreeSurfer-generated data with and without manual intervention

This paper examined whether FreeSurfer - generated data differed between a fully – automated, unedited pipeline and an edited pipeline that included the application of control points to correct errors in white matter segmentation. In a sample of 30 individuals, we compared the summary statistics of surface area, white matter volumes, and cortical thickness derived from edited and unedited datasets for the 34 regions of interest (ROIs) that FreeSurfer (FS) generates. To determine whether applying control points would alter the detection of significant differences between patient and typical groups, effect sizes between edited and unedited conditions in individuals with the genetic disorder, 22q11.2 deletion syndrome (22q11DS) were compared to neurotypical controls. Analyses were conducted with data that were generated from both a 1.5 tesla and a 3 tesla scanner. For 1.5 tesla data, mean area, volume, and thickness measures did not differ significantly between edited and unedited regions, with the exception of rostral anterior cingulate thickness, lateral orbitofrontal white matter, superior parietal white matter, and precentral gyral thickness. Results were similar for surface area and white matter volumes generated from the 3 tesla scanner. For cortical thickness measures however, seven edited ROI measures, primarily in frontal and temporal regions, differed significantly from their unedited counterparts, and three additional ROI measures approached significance. Mean effect sizes for edited ROIs did not differ from most unedited ROIs for either 1.5 or 3 tesla data. Taken together, these results suggest that although the application of control points may increase the validity of intensity normalization and, ultimately, segmentation, it may not affect the final, extracted metrics that FS generates. Potential exceptions to and limitations of these conclusions are discussed

Pluricomplex Green and Lempert functions for equally weighted poles

For $\Omega$ a domain in $\mathbb C^n$, the pluricomplex Green function with poles $a_1, ...,a_N \in \Omega$ is defined as $G(z):=\sup \{u(z): u\in PSH_-(\Omega), u(x)\le \log \|x-a_j\|+C_j \text{when} x \to a_j, j=1,...,N \}$. When there is only one pole, or two poles in the unit ball, it turns out to be equal to the Lempert function defined from analytic disks into $\Omega$ by $L_S (z) :=\inf \{\sum^N_{j=1}\nu_j\log|\zeta_j|: \exists \phi\in \mathcal {O}(\mathbb D,\Omega), \phi(0)=z, \phi(\zeta_j)=a_j, j=1,...,N \}$. It is known that we always have $L_S (z) \ge G_S(z)$. In the more general case where we allow weighted poles, there is a counterexample to equality due to Carlehed and Wiegerinck, with $\Omega$ equal to the bidisk. Here we exhibit a counterexample using only four distinct equally weighted poles in the bidisk. In order to do so, we first define a more general notion of Lempert function "with multiplicities", analogous to the generalized Green functions of Lelong and Rashkovskii, then we show how in some examples this can be realized as a limit of regular Lempert functions when the poles tend to each other. Finally, from an example where $L_S (z) > G_S(z)$ in the case of multiple poles, we deduce that distinct (but close enough) equally weighted poles will provide an example of the same inequality. Open questions are pointed out about the limits of Green and Lempert functions when poles tend to each other.Comment: 25 page

Drug-Associated Changes in Amino Acid Residues in Gag p2, p7\u3csup\u3eNC\u3c/sup\u3e, and p6\u3csup\u3eGag\u3c/sup\u3e/p6\u3csup\u3ePol\u3c/sup\u3e in Human Immunodeficiency Virus Type 1 (HIV-1) Display a Dominant Effect on Replicative Fitness and Drug Response

Regions of HIV-1 gag between p2 and p6Gag/p6Pol, in addition to protease (PR), develop genetic diversity in HIV-1 infected individuals who fail to suppress virus replication by combination protease inhibitor (PI) therapy. To elucidate functional consequences for viral replication and PI susceptibility by changes in Gag that evolve in vivo during PI therapy, a panel of recombinant viruses was constructed. Residues in Gag p2/p7NC cleavage site and p7NC, combined with residues in the flap of PR, defined novel fitness determinants that restored replicative capacity to the posttherapy virus. Multiple determinants in Gag have a dominant effect on PR phenotype and increase susceptibility to inhibitors of drug-resistant or drug-sensitive PR genes. Gag determinants of drug sensitivity and replication alter the fitness landscape of the virus, and viral replicative capacity can be independent of drug sensitivity. The functional linkage between Gag and PR provides targets for novel therapeutics to inhibit drug-resistant viruses