A set oriented definition of finite-time Lyapunov exponents and coherent sets

The problem of phase space transport, which is of interest from both the theoretical and practical point of view, has been investigated extensively using geometric and probabilistic methods. Two important tools to study this problem that have emerged in recent years are finite-time Lyapunov exponents (FTLE) and the Perron–Frobenius operator. The FTLE measures the averaged local stretching around reference trajectories. Regions with high stretching are used to identify phase space transport barriers. One probabilistic method is to consider the spectrum of the Perron–Frobenius operator of the flow to identify almost-invariant densities. These almost-invariant densities are used to identify almost invariant sets. In this paper, a set-oriented definition of the FTLE is proposed which is applicable to phase space sets of finite size and reduces to the usual definition of FTLE in the limit of infinitesimal phase space elements. This definition offers a straightforward connection between the evolution of probability densities and finite-time stretching experienced by phase space curves. This definition also addresses some concerns with the standard computation of the FTLE. For the case of autonomous and periodic vector fields we provide a simplified method to calculate the set-oriented FTLE using the Perron–Frobenius operator. Based on the new definition of the FTLE we propose a simple definition of finite-time coherent sets applicable to vector fields of general time-dependence, which are the analogues of almost-invariant sets in autonomous and time-periodic vector fields. The coherent sets we identify will necessarily be separated from one another by ridges of high FTLE, providing a link between the framework of coherent sets and that of codimension one Lagrangian coherent structures. Our identification of coherent sets is applied to three examples.     

Central Sets and Radii of the Zero-Divisor Graphs of Commutative Rings

For a commutative ring R with identity, the zero-divisor graph, Γ(R), is the graph with vertices the nonzero zero-divisors of R and edges between distinct vertices x and y whenever xy = 0. This article gives a proof that the radius of Γ(R) is 0, 1, or 2 if R is Noetherian. The center union {0} is shown to be a union of annihilator ideals if R is Artinian. The diameter of Γ(R) can be determined once the center is identified. If R is finite, then the median is shown to be a subset of the center. A dominating set of Γ(R) is constructed using elements of the center when R is Artinian. It is shown that for a finite ring R ? ?₂ × F for some finite field F, the domination number of Γ(R) is equal to the number of distinct maximal ideals of R. Other results on the structure of Γ(R) are also presented.     

Additive‐dependent iptycene incorporation in polyanilines: Insights into the pentiptycene clipping effect and the polymerization mechanism

Despite the long history of polyaniline chemistry, backbone‐substituted polyanilines are limited. Here, we report the synthesis of pentiptycene‐incorporated polyanilines through acidic aniline oxidative polymerization with three pentiptycene derivatives, TA, DA, and TP, as nucleate additives. The reactivity of TA > DA ? TP, as evidenced by structural and property analysis of the corresponding polyaniline products, demonstrates a radical coupling mechanism and the formation of Dewar π‐complex intermediates for the chain propagation. In addition, the iptycene substituent effect on enhancing the electrochemical stability and charge storage capability of polyaniline are discussed with a clip model, namely, the threading of neighboring polyaniline chains through the U‐ and V‐shaped cavities of pentiptycene restricts lateral motions of the polymer chains and promotes interchain conductivity. Density function theory (DFT) calculations suggest a larger clipping effect for the U versus V cavities. Both the conclusion of a terminal planar p‐phenylenediamine (ppda) group being the key component of an effective nucleate and the concept of interchain clipping for enhanced electrochemical performance should facilitate the design and synthesis of novel polyanilines for electronic applications.     

Raising the bar on-bead: Efficient on-resin synthesis of α-conotoxin LvIA

α4/7-Conotoxin LvIA is an isoform-selective inhibitor of the α3β2 nicotinic acetylcholine receptor. An efficient strategy for the synthesis of this toxin is critical to advancing its utility as a probe for receptor function and as a potential pharmaceutical lead target. On-resin methods for peptide synthesis offer potential synthetic advantages; however, strategies for on-resin formation of multiple disulfides have historically been low-yielding. Here, we harness the reactivity of the Allocam protecting group and employ a 3-amino acid spacer strategy to synthesize α4/7-conotoxin LvIA via three different on-resin strategies, each of which results in an isolated yield higher than previous fully on-resin approaches.     

Scalable Biosynthesis of the Seaweed Neurochemical,Kainic Acid

Kainic acid, the flagship member of the kainoid family of natural neurochemicals, is a widely used neuropharmacological agent that helped unravel the key role of ionotropic glutamate receptors, including the kainate receptor, in the central nervous system. Worldwide shortages of this seaweed natural product in the year 2000 prompted numerous chemical syntheses, including scalable preparations with as few as six‐steps. Herein we report the discovery and characterization of the concise two‐enzyme biosynthetic pathway to kainic acid from l ‐glutamic acid and dimethylallyl pyrophosphate in red macroalgae and show that the biosynthetic genes are co‐clustered in genomes of Digenea simplex and Palmaria palmata. Moreover, we applied a key biosynthetic α‐ketoglutarate‐dependent dioxygenase enzyme in a biotransformation methodology to efficiently construct kainic acid on the gram scale. This study establishes both the feasibility of mining seaweed genomes for their biotechnological prowess.     

Synthesis and liquid crystalline behavior of bulky poly(methacrylamide)s

N‐(Bis(4‐(2‐ethylhexyloxy)phenyl)(phenyl)‐methyl)methacrylamide was synthesized and polymerized via reversible addition‐fragmentation chain‐transfer (RAFT) polymerization. The chain‐transfer agent (4‐cyano‐4‐(phenylcarbonothioylthio) pentanoic acid (CPADB)), combined with a chiral additive, and a radical initiator yielded polymers with dispersities between 1.2 and 1.4. At low concentrations, the polymers are soluble in hexanes and chloroform while at higher concentrations they swell in these solvents. Characterization of the polymers by wide‐angle X‐ray scattering (WAXS) revealed an interplanar distance of 19.0 Å. The WAXS data combined with polarized optical microscopy support a lamellar crystallization and lyotropic liquid crystalline behavior in hexanes and chloroform. © 2015 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2015 , 53, 2563–2568     

Analysis of formaldehyde formation in wastewater using on-fiber derivatization-solid-phase microextraction-gas chromatography-mass spectrometry

A method has been developed for the quantification of the formation of formaldehyde during the advanced oxidation treatment (AOT) of wastewater destined for reuse. This method uses solid-phase microextraction (SPME) with on-fiber derivatization followed by gas chromatography-mass spectrometry (GC-MS) analysis. Based on calculated method detection limits (MDL) and ambient background levels, the method reporting (MRL) limit for formaldehyde was set at 10 microg/L. Precision for formaldehyde using this technique resulted in 23% relative standard deviation (RSD), while the internal standard, acetone-d(6), was only 6%. This method was used to evaluate the formation of formaldehyde in bench scale UV-AOT experiments using natural organic matter (NOM) fortified reagent water and tertiary treated wastewater effluent. Results suggest that the formation of formaldehyde increases in both the reagent water and wastewater matrices with increasing UV exposure and hydrogen peroxide concentrations, with overall higher concentrations of formaldehyde in the wastewater samples. No appreciable amount of formaldehyde formation was observed when UV was applied in the absence of hydrogen peroxide in both matrices tested.     

Valence parity renders z(*)-type ions chemically distinct

Here we report that the odd electron z (*) -type ions formed by the electron-based peptide dissociation methods (electron capture or transfer, ECD or ETD) have distinctive chemical compositions from other common product ion types. Specifically, b-, c-, and y-type ions have an odd number of atoms with an odd valence (e.g., N and H), while z (*)-type ions contain an even number of atoms with an odd valence. This tenet, referred to as the valence parity rule, mandates that no c-type ion shall have the same chemical composition, and by extension mass, as a z (*) -type ion. By experiment we demonstrate that nearly half of all observed c- and z (*) -type product ions resulting from 226 ETD product ion spectra can be assigned to a single, correct, chemical composition and ion type by simple inspection of the m/ z peaks. The assignments provide (1) a platform to directly determine amino acid composition, (2) an input for database search algorithms, or (3) a basis for de novo sequence analysis.     

Detection of Cd, Pb, and Cu in non-pretreated natural waters and urine with thiol functionalized mesoporous silica and Nafion composite electrodes

Electrochemical sensors have great potential for environmental monitoring of toxic metal ions in waters due to their portability, field-deployability and excellent detection limits. However, electrochemical sensors employing mercury-free approaches typically suffer from binding competition for metal ions and fouling by organic substances and surfactants in natural waters, making sample pretreatments such as wet ashing necessary. In this work, we have developed mercury-free sensors by coating a composite of thiol self-assembled monolayers on mesoporous supports (SH-SAMMS™) and Nafion on glassy-carbon electrodes. With the combined benefit of SH-SAMMS™ as an outstanding metal preconcentrator and Nafion as an antifouling binder, the sensors could detect 0.5 ppb of Pb and 2.5 ppb of Cd in river water, Hanford groundwater, and seawater with a minimal amount of preconcentration time (few minutes) and without any sample pretreatment. The sensor could also detect 2.5 ppb of Cd, Pb, and Cu simultaneously. The electrodes have long service times and excellent single and inter-electrode reproducibility (5% R.S.D. after 8 consecutive measurements). Unlike SAMMS™-carbon paste electrodes, the SAMMS™-Nafion electrodes were not fouled in samples containing albumin and successfully detected Cd in human urine. Other potentially confounding factors affecting metal detection at SAMMS™-Nafion electrodes were studied, including pH effect, transport resistance of metal ions, and detection interference. With the ability to reliably detect low metal concentration ranges without sample pretreatment and fouling, SAMMS™-Nafion composite sensors have the potential to become the next-generation metal analyzers for environmental and bio-monitoring of toxic metals.