On the Tanaka formula for the derivative of self-intersection local time of fractional Brownian motion

The derivative of self-intersection local time (DSLT) for Brownian motion was introduced by Rosen (2005) and subsequently used by others to study the L²$L^2$ and L³$L^3$ moduli of continuity of Brownian local time. A version of the DSLT for fractional Brownian motion (fBm) was introduced in Yan et al. (2008); however, the definition given there presents difficulties, since it is motivated by an incorrect application of the fractional Itô formula. To rectify this, we introduce a modified DSLT for fBm and prove existence using an explicit Wiener chaos expansion. We will then argue that our modification is the natural version of the DSLT by rigorously proving the corresponding Tanaka formula. This formula corrects a formal identity given in both Rosen (2005) and Yan et al. (2008). In the course of this endeavor we prove a Fubini theorem for integrals with respect to fBm. The Fubini theorem may be of independent interest, as it generalizes (to Hida distributions) similar results previously seen in the literature. As a further byproduct of our investigation, we also provide a small correction to an important technical second-moment bound for fBm which has appeared in the literature many times.     

Asymptotically liberating sequences of random unitary matrices

A fundamental result of free probability theory due to Voiculescu and subsequently refined by many authors states that conjugation by independent Haar-distributed random unitary matrices delivers asymptotic freeness. In this paper we exhibit many other systems of random unitary matrices that, when used for conjugation, lead to freeness. We do so by first proving a general result asserting “asymptotic liberation” under quite mild conditions, and then we explain how to specialize these general results in a striking way by exploiting Hadamard matrices. In particular, we recover and generalize results of the second-named author and of Tulino, Caire, Shamai and Verdú.     

Nevanlinna–Pick interpolation on distinguished varieties in the bidisk

This article treats Nevanlinna–Pick interpolation in the setting of a special class of algebraic curves called distinguished varieties. An interpolation theorem, along with additional operator theoretic results, is given using a family of reproducing kernels naturally associated to the variety. The examples of the Neil parabola and doubly connected domains are discussed.     

Semicontinuous viscosity solutions for quasiconvex Hamiltonians

The main theorem connecting convex Hamiltonians and semicontinuous viscosity solutions due to Barron and Jensen is extended to quasiconvex Hamiltonians. Some applications are indicated.     

Cup and cap products in intersection (co)homology

We construct cup and cap products in intersection (co)homology with field coefficients. The existence of the cap product allows us to give a new proof of Poincaré duality in intersection (co)homology which is similar in spirit to the usual proof for ordinary (co)homology of manifolds.     

Correlated atomic force microscopy and fluorescence lifetime imaging of live bacterial cells

We report on imaging living bacterial cells by using a correlated tapping-mode atomic force microscopy (AFM) and confocal fluorescence lifetime imaging microscopy (FLIM). For optimal imaging of Gram-negative Shewanella oneidensis MR-1 cells, we explored different methods of bacterial sample preparation, such as spreading the cells on poly-L-lysine coated surfaces or agarose gel coated surfaces. We have found that the agarose gel containing 99% ammonium acetate buffer can provide sufficient local aqueous environment for single bacterial cells. Furthermore, the cell surface topography can be characterized by tapping-mode in-air AFM imaging for the single bacterial cells that are partially embedded. Using in-air rather than under-water AFM imaging of the living cells significantly enhanced the contrast and signal-to-noise ratio of the AFM images. Near-field AFM-tip-enhanced fluorescence lifetime imaging (AFM-FLIM) holds high promise on obtaining fluorescence images beyond optical diffraction limited spatial resolution. We have previously demonstrated near-field AFM-FLIM imaging of polymer beads beyond diffraction limited spatial resolution. Here, as the first step of applying AFM-FLIM on imaging bacterial living cells, we demonstrated a correlated and consecutive AFM topographic imaging, fluorescence intensity imaging, and FLIM imaging of living bacterial cells to characterize cell polarity.     

Post‐Synthetic Ligand Exchange in Zirconium‐Based Metal–Organic Frameworks: Beware of The Defects!

Post‐synthetic ligand exchange in the prototypical zirconium‐based metal–organic framework (MOF) UiO‐66 was investigated by in situ solution ¹H NMR spectroscopy. Samples of UiO‐66 having different degrees of defectivity were exchanged using solutions of several terephthalic acid analogues in a range of conditions. Linker exchange only occurred in defect‐free UiO‐66, whereas monocarboxylates grafted at defect sites were found to be preferentially exchanged with respect to terephthalic acid over the whole range of conditions investigated. A 1:1 exchange ratio between the terephthalic acid analogue and modulator was observed, providing evidence that the defects had missing‐cluster nature. Ex situ characterisation of the MOF powders after exchange corroborated these findings and showed that the physical‐chemical properties of the MOF depend on whether the functionalisation occurs at defective sites or on the framework.     

Synthesis and characterization of immobilized PAMAM dendrons

This communication describes the synthesis and characterization of immobilized PAMAM dendrons onto a surface modified silicon wafer substrate (functionalized using plasma polymerized PAA) using a "growing from" strategy.