Continuum versus discrete networks,graph Laplacians,and reproducing kernel Hilbert spaces

Motivated by applications to machine learning, we construct a reversible and irreducible Markov chain whose state space is a certain collection of measurable sets of a chosen l.c.h. space $\mathfrak{X}$. We study the resulting network (connected undirected graph), including transience, Royden and Riesz decompositions, and kernel factorization. We describe a construction for Hilbert spaces of signed measures which comes equipped with a new notion of reproducing kernels and there is a unique solution to a regularized optimization problem involving the approximation of $L^2$ functions by functions of finite energy. The latter has applications to machine learning (for Markov random fields, for example).     

Group Theory,Computational Thinking,and Young Mathematicians

In this article, we investigate the artistic puzzle of designing mathematics experiences (MEs) to engage young children with ideas of group theory, using a combination of hands-on and computational thinking (CT) tools. We elaborate on: (1) group theory and why we chose it as a context for young mathematicians’ experiences with symmetry and transformations; (2) our ME design principles of agency, access, surprise and audience; (3) the affordances of CT that complement our design principles; and (4) three ME variations we tested in grades 3–6 classrooms. We then reflect on the ME variations based on our design principles and the affordances of CT, and consider how the MEs may be further adapted and improved.     

Novel Copolymers of Styrene. 12. Halogen Ring-Substituted Methyl 2-Cyano-3-Phenyl-2-Propenoates

Electrophilic trisubstituted ethylenes, halogen ring-substituted methyl 2-cyano-3-phenyl-2-propenoates, RPhCH=C(CN)CO₂CH₃ (where R is 3-Br-4-CH₃O, 5-Br-2-CH₃O, 2-F-5-CH₃, 2-F-6-CH₃, 4-F-3-CH₃, 4-F-3-PhO, 2-F-5-I, 2-F-6-I, 2-F₃C, 4-F₃C) were prepared and copolymerized with styrene. The monomers were synthesized by the piperidine catalyzed Knoevenagel condensation of ring-substituted benzaldehydes and methyl cyanoacetate, and characterized by CHN analysis, IR, ¹H and ¹³C-NMR. All the ethylenes were copolymerized with styrene (M₁) in solution with radical initiation (ABCN) at 70°C. The compositions of the copolymers were calculated from nitrogen analysis and the structures were analyzed by IR, ¹H and ¹³C-NMR. The order of relative reactivity (1/r ₁) for the monomers is 2-F-5-CH₃ (6.4) > 4-F-3-PhO (5.6) > 4-F₃C (4.8) > 3-Br-4-CH₃O (3.7) > 2-F-5-I (3.6) > 2-F₃C (2.2) > 2-F-6-I (2.1) > 5-Br-2-CH₃O (1.9) > 4-F-3-CH₃ (1.8) > 2-F-6-CH₃ (1.2). Relatively high T _g of the copolymers in comparison with that of polystyrene indicates a decrease in chain mobility of the copolymer due to the high dipolar character of the trisubstituted ethylene monomer unit. Decomposition of the copolymers in nitrogen occurred in two steps, first in the 200-500°C range with residue (2–21% wt), which then decomposed in the 500–800°C range.     

Mixed-Isotope Labeling with LC-IMS-MS for Characterization of Protein–Protein Interactions by Chemical Cross-Linking

Chemical cross-linking of proteins followed by proteolysis and mass spectrometric analysis of the resulting cross-linked peptides provides powerful insight into the quaternary structure of protein complexes. Mixed-isotope cross-linking (a method for distinguishing intermolecular cross-links) was coupled with liquid chromatography, ion mobility spectrometry and mass spectrometry (LC-IMS-MS) to provide an additional separation dimension to the traditional cross-linking approach. This method produced multiplet m/z peaks that are aligned in the IMS drift time dimension and serve as signatures of intermolecular cross-linked peptides. We developed an informatics tool to use the amino acid sequence information inherent in the multiplet spacing for accurate identification of the cross-linked peptides. Because of the separation of cross-linked and non-cross-linked peptides in drift time, our LC-IMS-MS approach was able to confidently detect more intermolecular cross-linked peptides than LC-MS alone.      

Analysis of polyfluoroalkyl substances and bisphenol A in dried blood spots by liquid chromatography tandem mass spectrometry

Dried blood spots (DBS), collected as part of the newborn screening program (NSP) in the USA, is a valuable resource for studies on environmental chemical exposures and associated health outcomes in newborns. Nevertheless, determination of concentrations of environmental chemicals in DBS requires assays with great sensitivity, as the typical volume of blood available on a DBS with 16-mm diameter disc is approximately 50 μL. In this study, we developed a liquid–liquid extraction and high-performance liquid chromatography/tandem mass spectrometry method for the detection of perfluorooctanesulfonate (PFOS), perfluorooctanoate (PFOA), and bisphenol A (BPA) in DBS. The method was validated for accuracy, precision, and sensitivity, by spiking of target chemicals at different levels on Whatman 903 filter cards, which is used in the collection of DBS by the NSP. Contamination arising from collection, storage, and handling of DBS is an important issue to be considered in the analysis of trace levels of environmental chemicals in DBS. For the evaluation of the magnitude of background contamination, field blanks were prepared from unspotted portions of DBS filter cards collected by the NSP. The method was applied for the measurement of PFOS, PFOA, and BPA in 192 DBS specimens provided by NSP of New York State. PFOS and PFOA were detected in 100 % of the specimens analyzed. The concentrations of PFOS and PFOA measured in DBS were similar to those reported earlier in the whole blood samples of newborns. BPA was also found in 86 % of the specimens at concentrations ranging from 0.2 to 36 ng/mL (excluding two outliers). Further studies are needed to evaluate the sources of BPA exposures and health outcomes in newborns.     

Comparison of impurities in charcoal sorbents found by neutron activation analysis

Neutron activation of gas samples in a reactor often requires a medium to retain sufficient amounts of the gas for analysis. Charcoal is commonly used to adsorb gas and hold it for activation; however, the amount of activated sodium in the charcoal after irradiation swamps most signals of interest. Neutron activation analysis was performed on several commonly available charcoal samples in an effort to determine the activation background. The results for several elements, including the dominant sodium element, are reported. It was found that ECN charcoal had the lowest elemental background, containing sodium at 2.65 ± 0.05 ppm, as well as trace levels of copper and tungsten.     

Silyl triflate-accelerated additions of catalytically generated zinc acetylides to N-phenyl nitrones

Terminal alkynes readily form zinc acetylides in the presence of iPr₂NEt and 20 mol % ZnBr₂, then attack N-phenyl nitrones activated by trimethylsilyl trifluoromethanesulfonate. Deprotection with aqueous acid yields N-hydroxyl propargylamine. Yields are generally high for nitrones derived from aromatic aldehydes. Control experiments suggest that silyl triflate has a significant accelerating effect upon the reaction.