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).     

Modeling Thioredoxin Reductase-Like Activity with Cyclic Selenenyl Sulfides: Participation of an NH⋅⋅⋅Se Hydrogen Bond through Stabilization of the Mixed Se−S Intermediate

At the redox-active center of thioredoxin reductase (TrxR), a selenenyl sulfide (Se−S) bond is formed between Cys497 and Sec498, which is activated into the thiolselenolate state ([SH,Se⁻]) by reacting with a nearby dithiol motif ([SH^Cys59,SH^Cys64]) present in the other subunit. This process is achieved through two reversible steps: an attack of a cysteinyl thiol of Cys59 at the Se atom of the Se−S bond and a subsequent attack of a remaining thiol at the S atom of the generated mixed Se−S intermediate. However, it is not clear how the kinetically unfavorable second step progresses smoothly in the catalytic cycle. A model study that used synthetic selenenyl sulfides, which mimic the active site structure of human TrxR comprising Cys497, Sec498, and His472, suggested that His472 can play a key role by forming a hydrogen bond with the Se atom of the mixed Se−S intermediate to facilitate the second step. In addition, the selenenyl sulfides exhibited a defensive ability against H₂O₂-induced oxidative stress in cultured cells, which suggests the possibility for medicinal applications to control the redox balance in cells.     

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.     

Complexation Constants of Lanthanide Ions with Poly(Methacrylic Acid) and its Copolymers

Complexation of some lanthanide ions with poly(methacrylic acid) and its copolymers was studied by potentiometric titration. Poly [methacrylic acid-co-oligo(ethylene oxide)methacrylate] and poly(methacrylic acid-co-acrylamide) formed tris-carboxylate coordinate lanthanide complexes with large overall complexation constants, while poly-(methacrylic acid)s and copolymer with higher content of the methacrylic acid residue formed bis-coordinate ones. It was concluded that the comonomer residues in the copolymer chains decreased the steric hindrance for the complexation and/or acted as co-coordinating groups of the carboxylic group to lanthanide ions. Very large positive and favorable entropy changes were observed for the complexation with poly(methacrylic acid) and its copolymers. This contribution of thermodynamic parameters to the complexation was contrary to that for the analogous monomeric methacrylic acid complex and is assumed to be induced by dehydration of the polymers through the lanthanide ion complexation.     

Biosynthetic Conclusions from the Functional Dissection of Oxygenases for Biosynthesis of Actinorhodin and Related Streptomyces Antibiotics

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Highlights? ActVA-ORF5 is a flavin-dependent monooxygenase required for actinorhodin biosynthesis ? ActVA-ORF5 and its three close homologs were functionally dissected ? ActVA-ORF5 and Gra-21 are bifunctional at C-6/C-8, while Med-7 acts only for C-6 ? AlnT exhibits different regiospecificity for oxidation of tricyclic substrates