A two weight local Tb theorem for the Hilbert transform

We obtain a two weight local Tb theorem for any elliptic and gradient elliptic fractional singular integral operator T on the real line, and any pair of locally finite positive Borel measures on the line. This includes the Hilbert transform and in a sense improves on the T1 theorem by the authors and M. Lacey.Comment: 121 pages, 3 figures, 50 pages of appendices. We correct three gaps in the treatment of the stopping form in v12: the proof of Lemma 9.3 there requires a larger size functional, a collection of pairs is missing from the decomposition at the bottom of page 149, and an error was made in the definition of restricted norm of a stopping form. Main results unchange

Theoretical Insight into the Mechanism of Gold(I)-Catalyzed Rearrangement of 2‑Propargyl 2<i>H</i>‑Azirines to Pyridines

The title reaction is investigated in detail theoretically using density functional theory. After 5-<i>endo</i>-dig cyclization by nucleophilic attack, five possible pathways are taken into account in this work: direct ring expansion followed or accompanied by proton-transfer (paths A and B, respectively), 1,3-cationic migration (path C), proton-transfer before ring expansion (path D), and processing via a gold-nitrene (path E). Results indicate that the reaction would undergo the favored sequential pathway (path A) rather than other pathways. Moreover, the concerted mechanism (path B), which is designed to account for the selectivity of product in the experiment, would be unlikely in the reaction. The selectivity of product could be explained by the hindrance of ligand (<i>t</i>-BuXPhos) and the stability of the carbocation. Moreover, the binding energy of product complexes could account for the observed reaction rate

Thermoresponsive Stability of Colloids in Butyl Acetate/Ethanol Binary Solvent Realized by Grafting Linear Acrylate Copolymers

We have developed a new class of thermoresponsive colloids that can exhibit a sharp reversible transition between dispersion and aggregation in binary BuAc/EtOH solvents based on the UCST (upper critical solution temperature)-type phase separation. This is realized by grafting linear PMMA-BA (random) copolymer onto the colloidal particles. We have selected TiO₂/PS hybrid spheres (HSs) as a model system to demonstrate our general design concept. By grafting the linear PMMA-BA copolymer onto the HS surface, with the molecular weight from 30 to 40 kDa, we found that the thermoresponsive transition between dispersion and aggregation is fast, sharp, and reversible. At high mass fractions of the HSs, we have even observed a sharp transition between dispersion and gelation (or phase separation). The transition temperature can be tuned by varying the binary solvent composition, BuAc/EtOH, and the molecular weight of the grafted linear copolymer in the range from 5 to 55 °C. One of the most important features of this work is that the thermoresponsive materials used in organic solvents are initially synthesized in water with widely applied conventional (instead of research-based) techniques, thus being well suited for industrial production. In addition, the proposed approach is rather general and applicable to realizing the thermoresponsive transition for various types of colloids and nanoparticles

Electron Transfer Controls the Photochemical Splitting of Water Mediated by a Titanocene Transition Metal Complex

In this work, accurate electronic structure calculations at the CASPT2//complete active space self-consistent field/polarizable continuum model level of theory have been employed to provide new insights into the mechanistic understanding of the photochemical splitting of water mediated by a titanocene transition metal complex. The electron transfer between the Ti metal center and the oxygen atom of the coordinated water is demonstrated to regulate the interconvertible forms by oxidation/reduction reactions and concomitant abstraction of hydrogen and hydroxyl radicals. In the presence of hydrogen radical scavenger (2,2,6,6-tetramethyl-piperidin-1-yl)­oxyl (TEMPO), the hydrogen evolution reaction triggers instantaneously the concerted electron transfer from Ti³⁺ to the hydroxyl radical, which leads to the formation of extremely stable Ti⁴-OH^- configuration, significantly lowering the barrier of O-H bond fission. As a result, the reverse electron transfer only can be launched through the UV light-driven electron promotion from the OH^-moiety to the Ti⁴⁺ center, restoring the reduction state of Ti³⁺ complex. The oxygen evolution reaction occurs effectively in the high energy of ¹<i>n</i>d state, completing the overall water splitting. The design strategy based on the proposed mechanism can be inferred to require generally the compromise between the oxidation and reduction abilities of the metal-based photocatalysts for the light-driven overall water splitting

Spin-Coated Thin Films of Polycyclic Aromatic Hydrocarbons Exhibiting High SCLC Hole Mobilities

High charge carrier mobilities have been greatly sought after in the development of cutting-edge organic electronic and optoelectronic devices. Although high field-effect transistor hole mobilities have been reported for solution-processed organic semiconductor thin films, their space-charge-limited current (SCLC) mobilities are still substantially lower. Herein, we report the synthesis and thin film SCLC hole mobilities of four polycyclic aromatic hydrocarbons, specifically, 2,5,8,17-tetra-<i>tert</i>-butyldiacenaphtho­[1,2-<i>j</i>:1′,2′-<i>l</i>]­fluoranthene (<b>1</b>), 2,5,8,11,14,17-hexa-<i>tert</i>-butyldiacenaphtho­[1,2-<i>j</i>:1′,2′-<i>l</i>]­fluoranthene (<b>2</b>), 2,9,12,15-tetra-<i>tert</i>-butylacenaphtho­[1,2-<i>j</i>]­benzo­[<i>l</i>]­fluoranthene (<b>3</b>), and 2,9,12,15-tetra-<i>tert</i>-butyl-4,5,6,7-tetraphenylacenaphtho­[1,2-<i>j</i>]­benzo­[<i>l</i>]­fluoranthene (<b>4</b>). Spin-coated pristine thin films of compounds <b>1</b>–<b>4</b> exhibit SCLC hole mobilities of 1.18 ± 0.18 × 10^–3, 3.8 ± 0.9 × 10^–5, 2.00 ± 0.87 × 10^–3, and 2.27 ± 0.67 × 10^–4 cm² V^–1 s^–1, respectively. Upon thermal annealing at 120 °C for 10 min, compounds <b>1</b>, <b>2</b>, and <b>4</b> show limited changes in their optical properties and hole mobilities. In contrast, thermal annealing of thin films of <b>3</b> leads to broadened and red-shifted absorptions and emissions as well as drastically improved SCLC hole mobilities as high as 3.4 ± 0.1 × 10^–1 cm² V^–1 s^–1, one of the highest ever recorded SCLC hole mobilities on solution-processed organic semiconductor thin films. This improvement of the hole mobilities for the thin films of <b>3</b> is attributed to its radically enhanced crystallinity and intermolecular π–π overlap. Expansion of compound <b>3</b>’s acene core with the hopes of further strengthening the π–π stacking interaction and thus further improving the SCLC mobility was unfortunately unsuccessful as the resulting molecule exhibits poor film formation properties

A colorimetric assay for vanillin detection by determination of the luminescence of o-toluidine condensates

<div><p>Vanillin (4-hydroxy-3-methoxybenzaldehyde), a food additive with rich milk flavor, is commonly used in the food, beverage and cosmetic industries. However, excessive consumption of vanillin may cause liver and kidney damage. Therefore, methods for detecting and controlling the level of vanillin in food, especially in infant powder, have important practical significance. In this study, we established a colorimetric assay for vanillin detection. The detection was performed under high-temperature and acidic conditions, which can induce the reaction of the aldehyde group of vanillin with the amino group of o-toluidine. The resulting product had a maximum absorption at 363 nm, which was quantified by a UV spectrophotometer. This assay had a limit of detection (<i>LOD</i>) of 1 pg mL⁻¹ and a linear range between 1 μg mL⁻¹ and 100 μg mL⁻¹. The average recoveries at three spiked levels were in the range from 91.1% to 101.6% with a relative standard deviation (<i>RSD</i>) of 4.62% ~ 7.27%.</p></div

Correlation analysis between vanillin concentration and absorbance.

<p>(A) Correlation between vanillin concentration and absorbance from 1 μg mL^-1 to 100 μg mL^-1 (R² = 0.9908). (B) R² = 0.92814 from blank to 10⁷ pg mL^-1.</p

Schematic illustration of the chemical structures of o-toluidine and vanillin and the condensation reaction between the two compounds.

<p>(a) A light-green color is visible before heating, and (b) a reddish-brown color is visible after heating.</p

Measurements of vanillin spiked in milk powder samples (n = 6).

<p>Measurements of vanillin spiked in milk powder samples (n = 6).</p

The absorbance and pH of 50 μg mL⁻¹ vanillin standard samples.

<p>The absorbance and pH of 50 μg mL⁻¹ vanillin standard samples.</p