Sequential optical coherence tomography and confocal imaging

We report a system capable of sequentially acquiring two en-face images of different depth resolutions. The two images are generated by use of different principles, optical coherence tomography (OCT) and confocal microscopy, and have depth resolutions, at present, of better than 20 microm and over 0.12 mm, respectively. The lower-depth-resolution image is ideal for target positioning before collection of stacks of en-face OCT images. Switching between the two types of image by flipping an opaque screen in the reference arm, coupled with self-adjusting gain operation of avalanche photodiodes in the receiver. We illustrate the usefulness of the system by imaging a leaf and an optic nerve in vivo.     

Whole-cell biodetection of halogenated organic acids

A whole-cell bacterial sensor system for short-chain halo-organic acids was constructed, using 2-chloropropionic acid (2-CPA) as a model pollutant. An Escherichia coli host was transformed with a moderate-copy plasmid containing a fusion of two foreign genetic elements: (a) a promoter-containing segment of the Pseudomonas DL-DEX (DL-2-haloacid dehalogenase) encoding gene and (b) bioluminescence (luxCDABE) genes of Photorhabdus luminescens. The resulting construct, named MT1, responded to the presence of 2-CPA by dose-dependent light emission, in a highly specific albeit a very insensitive manner. Thus, while the desired concept was successfully demonstrated, further genetic work is needed in order to make such a construct practical for environmental monitoring purposes.     

Thin points for Brownian motion

Let Θ(x,r) denote the occupation measure of the ball of radius r centered at x for Brownian motion {W_t}_0≤t≤1 in . We prove that for any analytic set E in [0,1], we have

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, where dim_P(E) is the packing dimension of E. We deduce that for any a≥1, the Hausdorff dimension of the set of “thin points” x for which

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, is almost surely 2−2/a; this is the correct scaling to obtain a nondegenerate “multifractal spectrum” for the “thin” part of Brownian occupation measure. The methods of this paper differ considerably from those of our work on Brownian thick points, due to the high degree of correlation in the present case. To prove our results, we establish general criteria for determining which deterministic sets are hit by random fractals of ‘limsup type' in the presence of long-range correlations. The hitting criteria then yield lower bounds on Hausdorff dimension. This refines previous work of Khoshnevisan, Xiao and the second author, that required decay of correlations.     

Accurate Solution to Overdetermined Linear Equations with Errors Using L1 Norm Minimization

It has been known for many years that a robust solution to an overdetermined system of linear equations Ax b is obtained by minimizing the L1 norm of the residual error. A correct solution x to the linear system can often be obtained in this way, in spite of large errors (outliers) in some elements of the (m × n) matrix A and the data vector b. This is in contrast to a least squares solution, where even one large error will typically cause a large error in x. In this paper we give necessary and sufficient conditions that the correct solution is obtained when there are some errors in A and b. Based on the sufficient condition, it is shown that if k rows of [A b] contain large errors, the correct solution is guaranteed if (m – n)/n 2k/, where > 0, is a lower bound of singular values related to A. Since m typically represents the number of measurements, this inequality shows how many data points are needed to guarantee a correct solution in the presence of large errors in some of the data. This inequality is, in fact, an upper bound, and computational results are presented, which show that the correct solution will be obtained, with high probability, for much smaller values of m – n.     

Prolongations in differential algebra

We develop the theory of higher prolongations of algebraic varieties over fields in arbitrary characteristic with commuting Hasse-Schmidt derivations. Prolongations were introduced by Buium in the context of fields of characteristic 0 with a single derivation. Inspired by work of Vojta, we give a new construction of higher prolongations in a more general context. Generalizing a result of Buium in characteristic 0, we prove that these prolongations are represented by a certain functor, which shows that they can be viewed as ‘twisted jet spaces.’ We give a new proof of a theorem of Moosa, Pillay and Scanlon that the prolongation functor and jet space functor commute. We also prove that the m-th prolongation and m-th jet space of a variety are differentially isomorphic by showing that their infinite prolongations are isomorphic as schemes.     

On Preservation Theorems for Two-Variable Logic

We show that the existential preservation theorem fails for two-variable first-order logic FO². It is known that for all k ≥ 3, FO^k does not have an existential preservation theorem, so this settles the last open case, answering a question of Andreka, van Benthem, and Németi. In contrast, we prove that the homomorphism preservation theorem holds for FO².     

Synthesis of dendritic macromolecules through divergent iterative thio‐bromo “Click” chemistry and SET‐LRP

The development of a novel nucleophilic thio‐bromo “Click” reaction, specifically base‐mediated thioetherification of thioglycerol with α‐bromoesters was reported in an earlier article. The combination of this thio‐bromo click reaction with subsequent acylation with 2‐bromopropionyl bromide provides an iterative two‐step divergent growth approach to the synthesis of a new class of poly(thioglycerol‐2‐ propionate) (PTP) dendrimers. In this article, the addition of a third step, the single‐electron transfer living radical polymerization (SET‐LRP) of methyl acrylate (MA), was shown to provides access to a three‐step “branch” and “grow” divergent approach to dendritic macromolecules wherein poly(methyl acrylate) (PMA) connects the branching subunits. This facile methodology can provide a diversity of dendritic macromolecular topologies and will ultimately provide the means to the development of self‐organizable dendritic macromolecules. © 2009 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 47: 3940–3948, 2009     

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