Derivatization of azaspiracid biotoxins for analysis by liquid chromatography with fluorescence detection

Azaspiracids (AZAs) are an important group of regulated lipophilic biotoxins that cause shellfish poisoning. Currently, the only widely available analytical method for quantitation of AZAs is liquid chromatography-mass spectrometry (LC-MS). Alternative methods for AZA analysis are needed for detailed characterization work required in the preparation of certified reference materials (CRMs) and by laboratories not equipped with LC-MS. Chemical derivatization of the amine and carboxyl groups on AZAs was investigated for the purpose of facilitating analysis by LC with fluorescence detection (FLD). Experiments towards chemical modification of AZA1 at the amine achieved only limited success. Derivatization of the carboxyl group, on the other hand, proved successful using the 9-anthryldiazomethane (ADAM) method previously applied to the okadaic acid (OA) group toxins. Extraction and clean-up methods were investigated for shellfish tissue samples and a post-reaction solid phase extraction procedure was developed for the AZA ADAM derivatives. Chromatographic separations were developed for the LC-FLD analysis of derivatized AZAs alone or in the presence of other derivatized toxins. This new analytical method for analysis of AZAs enabled verification of AZA1-3 concentrations in recently certified reference materials. The method demonstrated good linearity, repeatability and accuracy showing its potential as an alternative to LC-MS for measurement of AZAs.     

Symmetric Pairs of Unbounded Operators in Hilbert Space,and Their Applications in Mathematical Physics

In a previous paper, the authors introduced the idea of a symmetric pair of operators as a way to compute self-adjoint extensions of symmetric operators. In brief, a symmetric pair consists of two densely defined linear operators A and B, with \(A \subseteq B^{\star }\) and \(B \subseteq A^{\star }\). In this paper, we will show by example that symmetric pairs may be used to deduce closability of operators and sometimes even compute adjoints. In particular, we prove that the Malliavin derivative and Skorokhod integral of stochastic calculus are closable, and the closures are mutually adjoint. We also prove that the basic involutions of Tomita-Takesaki theory are closable and that their closures are mutually adjoint. Applications to functions of finite energy on infinite graphs are also discussed, wherein the Laplace operator and inclusion operator form a symmetric pair.     

Determination of surface heterogeneity profiles on graphite by finite concentration inverse gas chromatography

Inverse gas chromatography (IGC) is an established tool in the determination of the adsorption potential distribution function. This function reflects the energetic heterogeneity profile of a surface and therefore provides interesting information on the nature and population of different surface sites. IGC is shown to be a fast and accurate technique for the determination of the adsorption potential distribution function of two different graphite samples. In this paper the adsorption of acidic and basic organic vapours is studied. Unlike heterogeneity profiles determined by nitrogen measurements, experiments with polar vapours can provide additional information on the adsorption mechanism and polar sorption sites. The heterogeneity profiles of all probes used are significantly different from one another and allow discreet energy levels to be distinguished. Chemically different probes reveal different adsorption mechanisms for the graphite surface.     

Pointwise tube formulas for fractal sprays and self-similar tilings with arbitrary generators

In a previous paper by the first two authors, a tube formula for fractal sprays was obtained which also applies to a certain class of self-similar fractals. The proof of this formula uses distributional techniques and requires fairly strong conditions on the geometry of the tiling (specifically, the inner tube formula for each generator of the fractal spray is required to be polynomial). Now we extend and strengthen the tube formula by removing the conditions on the geometry of the generators, and also by giving a proof which holds pointwise, rather than distributionally. Hence, our results for fractal sprays extend to higher dimensions the pointwise tube formula for (1-dimensional) fractal strings obtained earlier by Lapidus and van Frankenhuijsen.Our pointwise tube formulas are expressed as a sum of the residues of the “tubular zeta function” of the fractal spray in Rd. This sum ranges over the complex dimensions of the spray, that is, over the poles of the geometric zeta function of the underlying fractal string and the integers 0,1,…,d. The resulting “fractal tube formulas” are applied to the important special case of self-similar tilings, but are also illustrated in other geometrically natural situations. Our tube formulas may also be seen as fractal analogues of the classical Steiner formula.     

Zur Bestimmung des Kohlenstoffes in Eisencarbureten

Ohne Zusammenfassung     

Tube Formulas and Complex Dimensions of Self-Similar Tilings

We use the self-similar tilings constructed in (Pearse in Indiana Univ. Math J. 56(6):3151–3169, 2007) to define a generating function for the geometry of a self-similar set in Euclidean space. This tubularzeta function encodes scaling and curvature properties related to the complement of the fractal set, and the associated system of mappings. This allows one to obtain the complex dimensions of the self-similar tiling as the poles of the tubularzeta function and hence develop a tube formula for self-similar tilings in ℝ^d. The resulting power series in εis a fractal extension of Steiner’s classical tube formula for convex bodies K⊆ℝ ^d . Our sum has coefficients related to the curvatures of the tiling, and contains terms for each integer i=0,1,…,d−1, just as Steiner’s does. However, our formula also contains a term for each complex dimension. This provides further justification for the term “complex dimension”. It also extends several aspects of the theory of fractal strings to higher dimensions and sheds new light on the tube formula for fractals strings obtained in (Lapidus and van Frankenhuijsen in Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings, 2006).     

A Hilbert Space Approach to Effective Resistance Metric

A resistance network is a connected graph (G, c). The conductance function c _xy weights the edges, which are then interpreted as conductors of possibly varying strengths. The Dirichlet energy form E{\mathcal E} produces a Hilbert space structure (which we call the energy space H_E{{\mathcal H}_{\mathcal E}}) on the space of functions of finite energy. We use the reproducing kernel {v _x } constructed in a previous work to analyze the effective resistance R, which is a natural metric for such a network. It is known that when (G, c) supports nonconstant harmonic functions of finite energy, the effective resistance metric is not unique. The two most natural choices for R(x, y) are the “free resistance” R ^F , and the “wired resistance” R ^W . We define R ^F and R ^W in terms of the functions v _x (and certain projections of them). This provides a way to express R ^F and R ^W as norms of certain operators, and explain R ^F ≠  R ^W in terms of Neumann versus Dirichlet boundary conditions. We show that the metric space (G, R ^F ) embeds isometrically into H_E{{\mathcal H}_{\mathcal E}}, and the metric space (G, R ^W ) embeds isometrically into the closure of the space of finitely supported functions; a subspace of H_E{{\mathcal H}_{\mathcal E}}. Typically, R ^F and R ^W are computed as limits of restrictions to finite subnetworks. A third formulation R ^tr is given in terms of the trace of the Dirichlet form E{\mathcal E} to finite subnetworks, and is related to R ^F by a probabilistic argument.