Hyperconvex hulls in catergories of quasi-metric spaces

Includes bibliographical references.Isbell showed that every metric space has an injective hull, that is, every metric space has a “minimal” hyperconvex metric superspace. Dress then showed that the hyperconvex hull is a tight extension. In analogy to Isbell’s theory Kemajou et al. proved that each T₀-quasi-metric space X has a q-hyperconvex hull QX , which is joincompact if X is joincompact. They called a T₀-quasi-metric space q-hyperconvex if and only if it is injective in the category of T₀-quasi-metric spaces and non-expansive maps. Agyingi et al. generalized results due to Dress on tight extensions of metric spaces to the category of T₀-quasi-metric spaces and non-expansive maps. In this dissertation, we shall study tight extensions (called uq-tight extensions in the following) in the categories of T₀-quasi-metric spaces and T₀-ultra-quasimetric spaces. We show in particular that most of the results stay the same as we move from T₀-quasi-metric spaces to T₀-ultra-quasi-metric spaces. We shall show that these extensions are maximal among the uq-tight extensions of the space in question. In the second part of the dissertation we shall study the q-hyperconvex hull by viewing it as a space of minimal function pairs. We will also consider supseparability of the space of minimal function pairs. Furthermore we study a special subcollection of bicomplete supseparable quasi-metric spaces: bicomplete supseparable ultra-quasi-metric spaces. We will show the existence and uniqueness (up to isometry) of a Urysohn Γ-ultra-quasi-metric space, for an arbitrary countable set Γ of non-negative real numbers including 0

The tax-inducible actin-bundling protein fascin is crucial for release and cell-to-cell transmission of human T-cell leukemia virus type 1 (HTLV-1)

The delta-retrovirus Human T-cell leukemia virus type 1 (HTLV-1) preferentially infects CD4(+) T-cells via cell-to-cell transmission. Viruses are transmitted by polarized budding and by transfer of viral biofilms at the virological synapse (VS). Formation of the VS requires the viral Tax protein and polarization of the host cytoskeleton, however, molecular mechanisms of HTLV-1 cell-to-cell transmission remain incompletely understood. Recently, we could show Tax-dependent upregulation of the actin-bundling protein Fascin (FSCN-1) in HTLV-1-infected T-cells. Here, we report that Fascin contributes to HTLV-1 transmission. Using single-cycle replication-dependent HTLV-1 reporter vectors, we found that repression of endogenous Fascin by short hairpin RNAs and by Fascin-specific nanobodies impaired gag p19 release and cell-to-cell transmission in 293T cells. In Jurkat T-cells, Tax-induced Fascin expression enhanced virus release and Fascin-dependently augmented cell-to-cell transmission to Raji/CD4(+) B-cells. Repression of Fascin in HTLV-1-infected T-cells diminished virus release and gag p19 transfer to co-cultured T-cells. Spotting the mechanism, flow cytometry and automatic image analysis showed that Tax-induced T-cell conjugate formation occurred Fascin-independently. However, adhesion of HTLV-1-infected MT-2 cells in co-culture with Jurkat T-cells was reduced upon knockdown of Fascin, suggesting that Fascin contributes to dissemination of infected T-cells. Imaging of chronically infected MS9 T-cells in co-culture with Jurkat T-cells revealed that Fascin's localization at tight cell-cell contacts is accompanied by gag polarization suggesting that Fascin directly affects the distribution of gag to budding sites, and therefore, indirectly viral transmission. In detail, we found gag clusters that are interspersed with Fascin clusters, suggesting that Fascin makes room for gag in viral biofilms. Moreover, we observed short, Fascin-containing membrane extensions surrounding gag clusters and clutching uninfected T-cells. Finally, we detected Fascin and gag in long-distance cellular protrusions. Taken together, we show for the first time that HTLV-1 usurps the host cell factor Fascin to foster virus release and cell-to-cell transmission

Optimality of Universal Bayesian Sequence Prediction for General Loss and Alphabet

Various optimality properties of universal sequence predictors based on Bayes-mixtures in general, and Solomonoff's prediction scheme in particular, will be studied. The probability of observing $x_t$ at time $t$, given past observations $x_1...x_{t-1}$ can be computed with the chain rule if the true generating distribution $\mu$ of the sequences $x_1x_2x_3...$ is known. If $\mu$ is unknown, but known to belong to a countable or continuous class \M one can base ones prediction on the Bayes-mixture $\xi$ defined as a $w_\nu$-weighted sum or integral of distributions \nu\in\M. The cumulative expected loss of the Bayes-optimal universal prediction scheme based on $\xi$ is shown to be close to the loss of the Bayes-optimal, but infeasible prediction scheme based on $\mu$. We show that the bounds are tight and that no other predictor can lead to significantly smaller bounds. Furthermore, for various performance measures, we show Pareto-optimality of $\xi$ and give an Occam's razor argument that the choice $w_\nu\sim 2^{-K(\nu)}$ for the weights is optimal, where $K(\nu)$ is the length of the shortest program describing $\nu$. The results are applied to games of chance, defined as a sequence of bets, observations, and rewards. The prediction schemes (and bounds) are compared to the popular predictors based on expert advice. Extensions to infinite alphabets, partial, delayed and probabilistic prediction, classification, and more active systems are briefly discussed.Comment: 34 page

Vector-valued optimal Lipschitz extensions

Consider a bounded open set $U$ in $R^n$ and a Lipschitz function g from the boundary of $U$ to $R^m$. Does this function always have a canonical optimal Lipschitz extension to all of $U$? We propose a notion of optimal Lipschitz extension and address existence and uniqueness in some special cases. In the case $n=m=2$, we show that smooth solutions have two phases: in one they are conformal and in the other they are variants of infinity harmonic functions called infinity harmonic fans. We also prove existence and uniqueness for the extension problem on finite graphs.Comment: 24 pages, 10 figure