Critical Sets of Elliptic Equations

Given a solution u to a linear, homogeneous, second‐order elliptic equation with Lipschitz coefficients, we introduce techniques for giving improved estimates of the critical set ??(u)u {x :|δu|(x) = 0}, as well as the first estimates on the effective critical set ??_r(u), which roughly consists of points x such that the gradient of u is small somewhere on B_r(x) compared to the nonconstancy of u. The results are new even for harmonic functions on ?ⁿ. Given such a u, the standard first‐order stratification {l^k} of u separates points x based on the degrees of symmetry of the leading‐order polynomial of u‐u(x). In this paper we give a quantitative stratification of u, which separates points based on the number of almost symmetries of approximate leading‐order polynomials of u at various scales. We prove effective estimates on the volume of the tubular neighborhood of each , which lead directly to (n‐2 + ?)‐Minkowski type estimates for the critical set of u. With some additional regularity assumptions on the coefficients of the equation, we refine the estimate to give new proofs of the uniform (n‐2)‐Hausdorff measure estimate on the critical set and singular sets of u.© 2014 Wiley Periodicals, Inc.     

Ternary Sn-Ti-O Electrocatalyst Boosts the Stability and Energy Efficiency of CO2 Reduction

Simultaneously improving energy efficiency (EE) and material stability in electrochemical CO₂ conversion remains an unsolved challenge. Among a series of ternary Sn-Ti-O electrocatalysts, 3D ordered mesoporous (3DOM) Sn_0.3Ti_0.7O₂ achieves a trade-off between active-site exposure and structural stability, demonstrating up to 71.5 % half-cell EE over 200 hours, and a 94.5 % Faradaic efficiency for CO at an overpotential as low as 430 mV. DFT and X-ray absorption fine structure analyses reveal an electron density reconfiguration in the Sn-Ti-O system. A downshift of the orbital band center of Sn and a charge depletion of Ti collectively facilitate the dissociative adsorption of the desired intermediate COOH* for CO formation. It is also beneficial in maintaining a local alkaline environment to suppress H₂ and formate formation, and in stabilizing oxygen atoms to prolong durability. These findings provide a new strategy in materials design for efficient CO₂ conversion and beyond.     

Asymmetric Hydrogenation of Unfunctionalized Tetrasubstituted Acyclic Olefins

Asymmetric hydrogenation has evolved as one of the most powerful tools to construct stereocenters. However, the asymmetric hydrogenation of unfunctionalized tetrasubstituted acyclic olefins remains the pinnacle of asymmetric synthesis and an unsolved challenge. We report herein the discovery of an iridium catalyst for the first, generally applicable, highly enantio‐ and diastereoselective hydrogenation of such olefins and the mechanistic insights of the reaction. The power of this chemistry is demonstrated by the successful hydrogenation of a wide variety of electronically and sterically diverse olefins in excellent yield and high enantio‐ and diastereoselectivity.     

Effects of Roxadustat on Erythropoietin Production in the Rat Body

Anemia is a major complication of chronic renal failure. To treat this anemia, prolylhydroxylase domain enzyme (PHD) inhibitors as well as erythropoiesis-stimulating agents (ESAs) have been used. Although PHD inhibitors rapidly stimulate erythropoietin (Epo) production, the precise sites of Epo production following the administration of these drugs have not been identified. We developed a novel method for the detection of the Epo protein that employs deglycosylation-coupled Western blotting. With protein deglycosylation, tissue Epo contents can be quantified over an extremely wide range. Using this method, we examined the effects of the PHD inhibitor, Roxadustat (ROX), and severe hypoxia on Epo production in various tissues in rats. We observed that ROX increased Epo mRNA expression in both the kidneys and liver. However, Epo protein was detected in the kidneys but not in the liver. Epo protein was also detected in the salivary glands, spleen, epididymis and ovaries. However, both PHD inhibitors (ROX) and severe hypoxia increased the Epo protein abundance only in the kidneys. These data show that, while Epo is produced in many tissues, PHD inhibitors as well as severe hypoxia regulate Epo production only in the kidneys.     

Determination of biogenic amines in alcoholic beverages by ion chromatography with suppressed conductivity detection and integrated pulsed amperometric detection

The determination of biogenic amines in alcoholic beverages is important to assess the potential risks associated with the consumption of high concentrations of these compounds. In addition, product storage conditions and the length of storage can cause the formation of biogenic amines that reduce product quality. We report a new method using cation-exchange chromatography with either suppressed conductivity, integrated pulsed amperometry, UV, or a combination of these detection techniques to determine biogenic amines in alcoholic beverages. The main objective was to provide a direct comparison between IPAD and suppressed conductivity detection for determining biogenic amines in alcoholic beverages. Suppressed conductivity is the simplest detection approach for determining putrescine, cadaverine, histamine, agmatine, phenylethylamine, spermidine, and spermine with good sensitivity (0.004-0.08 mg/l) and was used to evaluate the influence of storage time and conditions on the evolution of biogenic amines in alcoholic beverages. Integrated pulsed amperometric detection (IPAD) detects more biogenic amines than suppressed conductivity detection, enabling the detection of dopamine, tyramine, and serotonin. Tyramine was simultaneously determined by UV detection and IPAD to provide confirmation and ensure the accuracy of the analytical results. The linearity of biogenic amine responses was within 0.1-20 mg/l and peak area precisions were 0.24-4.97% for IPAD, suppressed conductivity-IPAD, and UV detection. The sensitivity for the 10 biogenic amines using the 3 detection techniques varied considerably from 0.004-1.1 mg/l and recoveries were within 85-122%.     

Entropy, independent sets and antichains: A new approach to Dedekind's problem

For -regular, -vertex bipartite graphs with bipartition , a precise bound is given for the sum over independent sets of the quantity . (In other language, this is bounding the partition function for certain instances of the hard-core model.) This result is then extended to graded partially ordered sets, which in particular provides a simple proof of a well-known bound for Dedekind's Problem given by Kleitman and Markowsky in 1975.

     

Broken symmetries and pattern formation in two-frequency forced faraday waves

We exploit the approximate (broken) symmetries of time translation, time reversal, and Hamiltonian structure to obtain general scaling laws governing the process of pattern formation in weakly damped Faraday waves. Using explicit parameter symmetries we determine, for the case of two-frequency forcing, how the strength of observed three-wave interactions depends on the frequency ratio and on the relative phase of the two driving terms. These symmetry-based predictions are verified for numerically calculated coefficients, and help explain the results of recent experiments.     

Synthetic gene network for entraining and amplifying cellular oscillations

We present a model for a synthetic gene oscillator and consider the coupling of the oscillator to a periodic process that is intrinsic to the cell. We investigate the synchronization properties of the coupled system, and show how the oscillator can be constructed to yield a significant amplification of cellular oscillations. We reduce the driven oscillator equations to a normal form, and analytically determine the amplification as a function of the strength of the cellular oscillations. The ability to couple naturally occurring genetic oscillations to a synthetically designed network could lead to possible strategies for entraining and/or amplifying oscillations in cellular protein levels.