Analysis of Discrete L^2 Projection on Polynomial Spaces with Random Evaluations

We analyze the problem of approximating a multivariate function by discrete least-squares projection on a polynomial space starting from random, noise-free observations. An area of possible application of such technique is uncertainty quantification for computational models. We prove an optimal convergence estimate, up to a logarithmic factor, in the univariate case, when the observation points are sampled in a bounded domain from a probability density function bounded away from zero and bounded from above, provided the number of samples scales quadratically with the dimension of the polynomial space. Optimality is meant in the sense that the weighted $L^2$ norm of the error committed by the random discrete projection is bounded with high probability from above by the best $L^\infty $ error achievable in the given polynomial space, up to logarithmic factors. Several numerical tests are presented in both the univariate and multivariate cases, confirming our theoretical estimates. The numerical tests also clarify how the convergence rate depends on the number of sampling points, on the polynomial degree, and on the smoothness of the target function.     

Inductive topological Hausdorff dimensions and fibers of generic continuous functions

In an earlier paper Buczolich, Elekes and the author introduced a new concept of dimension for metric spaces, the so called topological Hausdorff dimension. They proved that it is precisely the right notion to describe the Hausdorff dimension of the level sets of the generic real-valued continuous function (in the sense of Baire category) defined on a compact metric space $K$ . The goal of this paper is to determine the Hausdorff dimension of the fibers of the generic continuous function from $K$ to $\mathbb {R}^n$ . In order to do so, we define the $n$ th inductive topological Hausdorff dimension, $\dim _{t^nH} K$ . Let $\dim _H K,\,\dim _t K$ and $C_n(K)$ denote the Hausdorff and topological dimension of $K$ and the Banach space of the continuous functions from $K$ to $\mathbb {R}^n$ . We show that $\sup _{y\in \mathbb {R}^n} \dim _{H}f^{-1}(y) = \dim _{t^nH} K -n$ for the generic $f \in C_n(K)$ , provided that $\dim _t K\ge n$ , otherwise every fiber is finite. In order to prove the above theorem we give some equivalent definitions for the inductive topological Hausdorff dimensions, which can be interesting in their own right. Here we use techniques coming from the theory of topological dimension. We show that the supremum is actually attained on the left hand side of the above equation. We characterize those compact metric spaces $K$ for which $\dim _{H} f^{-1}(y)=\dim _{t^nH}K-n$ for the generic $f\in C_n(K)$ and the generic $y\in f(K)$ . We also generalize a result of Kirchheim by showing that if $K$ is self-similar and $\dim _t K\ge n$ then $\dim _{H} f^{-1}(y)=\dim _{t^nH}K-n$ for the generic $f\in C_n(K)$ for every $y\in {{\mathrm{int}}}f(K)$ .     

Elimination of autofluorescence in fluorescence correlation spectroscopy using the AzaDiOxaTriAngulenium (ADOTA) fluorophore in combination with time-correlated single-photon counting (TCSPC)

Fluorescence correlation spectroscopy (FCS) is a frequently applied technique that allows for the precise and sensitive analysis of molecular diffusion and interactions. However, the potential of FCS for in vitro or ex vivo studies has not been fully realized due in part to artifacts originating from autofluorescence (fluorescence of inherent components and fixative-induced fluorescence). Here, we propose the azadioxatriangulenium (ADOTA) dye as a solution to this problem. The lifetime of the ADOTA probe, about 19.4 ns, is much longer than most components of autofluorescence. Thus, it can be easily separated by time-correlated single-photon counting methods. Here, we demonstrate the suppression of autofluorescence in FCS using ADOTA-labeled hyaluronan macromolecules (HAs) with Rhodamine 123 added to simulate diffusing fluorescent background components. The emission spectrum and decay rate of Rhodamine 123 overlap with the usual sources of autofluorescence, and its diffusion behavior is well known. We show that the contributions from Rhodamine 123 can be eliminated by time gating or by fluorescence lifetime correlation spectroscopy (FLCS). While the pairing of ADOTA and time gating is an effective strategy for the removal of autofluorescence from fluorescence imaging, the loss of photons leads to erroneous concentration values with FCS. On the other hand, FLCS eliminates autofluorescence without such errors. We then show that both time gating and FLCS may be used successfully with ADOTA-labeled HA to detect the presence of hyaluronidase, the overexpression of which has been observed in many types of cancer.     

Synthesis of (3R)-Amino-(2S)-Hydroxy Amino Acids for Inhibition of Methionine Aminopeptidase-1

(3R)-amino-(2S)-hydroxy heptanoic acid (AHHpA) was synthesized via asymmetric aminohydroxylation and coupled with di-and tripeptides to form three compounds that inhibit MetAP-1 with IC₅₀ values in the low micromolar range.     

From ubicomp to ubiex(pectations)

Based on Weiser’s notion of ubicomp this paper examines the idea of ubiquitous social expectations. It examines this in the context of networked mediation technologies. A central function of these technologies is to allow people to perpetually be in contact with one another. As these technologies gain a critical mass in society we increasingly expect them of one another. Thus, these mediation artifacts are not simple devices that facilitate our individual lives, they are the basis of reciprocal expectations. As this process continues, we increasingly do not have the option of opting out, rather we are, to one degree or another compelled to have them.     

Verhalten der Nitrosothioglycols?ure gegen Eisenchlorid

Ohne Zusammenfassung     

Measurement of the core diameter of multimode graded-index fibers: A comparison of transmitted near-field and index profiling techniques

The core diameters of six graded-index fiber from four different fiber manufacturers were compared using the transmitted near-field (TNF), the refracted near-field (RNF), and the transverse-interferometric (TI) measurement methods. This study was part of an effort to develop a standardized, industry-wide definition of core diameter and to determine the precision of interlaboratory core-diameter measurements using different measurement techniques. For fibers with smooth index-of-refraction profiles, all three methods were in good agreement (< 1.0-mum difference). Substantial differences between the transmitted near field and the two profiling methods (RNF and TI) were observed for fibers having step structure near the core-cladding boundary. In an attempt to resolve these differences, splice-loss measurements were used as an indicator of diameter differences. These experiments suggested that curve-fitting routines should be applied to the two profiling methods. A comparison of the curve-fitted profile data with measured transmitted near-field data at points 2 percent above the baseline produced values for the diameters which agreed to within 1 μm for all of the fibers measured.     

Complex formation processes of terminally protected peptides containing two or three histidyl residues. Characterization of the mixed metal complexes of peptides

Copper(ii), nickel(ii) and zinc(ii) complexes of the peptides Ac-HVVH-NH(2) and Ac-HAAHVVH-NH(2) have been studied by potentiometric, UV-vis, CD, EPR and NMR spectroscopic measurements. Both tetra and heptapeptides can form relatively stable macrochelates with copper(ii), nickel(ii) and zinc(ii) ions, in which the ligands are coordinated via the side-chain imidazole functions. Formation of the macrochelates slightly suppresses, but cannot prevent the copper(ii) and nickel(ii) ion promoted deprotonation and coordination of the amide functionalities. The overall stoichiometry of the major species is [MH(-3)L](-) with a 4N (= N(-),N(-),N(-),N(im)) coordination mode. In the case of Ac-HAAHVVH-NH(2), coordination isomers of this species can exist with a preference for copper(ii) or nickel(ii) binding at the internal histidyl residue. In the copper(ii)-Ac-HAAHVVH-NH(2) system, the presence of the two anchoring sites results in the formation of dinuclear complexes. The existence of these species requires the involvement of amide functions in metal binding. Both equilibrium and spectroscopic data support the fact that the copper(ii) ions of the dinuclear species are independent from each other providing a good chance for the formation of various mixed metal complexes. It was found that zinc(ii) is not able to significantly alter the copper(ii) binding of the heptapeptide, but it can occupy the uncoordinated histidyl sites. The formation of the copper(ii)-nickel(ii) mixed species was obtained in alkaline solutions and CD spectra suggest the statistical distribution of the two metal ions among the histidyl residues. The binding of HAAHVVH to palladium(ii) is exclusive below pH 8 and the mixed metal species of palladium(ii) and copper(ii) ions are formed only in slightly basic solutions.