On PAC extensions and scaled trace forms

Any non-degenerate quadratic form over a Hilbertian field (e.g., a number field) is isomorphic to a scaled trace form. In this work we extend this result to more general fields, in particular, prosolvable and prime-to-p extensions of a Hilbertian field. The proofs are based on the theory of PAC extensions.     

Temperature-dependent solid-state electron transport through bacteriorhodopsin: experimental evidence for multiple transport paths through proteins

Electron transport (ETp) across bacteriorhodopsin (bR), a natural proton pump protein, in the solid state (dry) monolayer configuration, was studied as a function of temperature. Transport changes from thermally activated at T > 200 K to temperature independent at <130 K, similar to what we have observed earlier for BSA and apo-azurin. The relatively large activation energy and high temperature stability leads to conditions where bR transports remarkably high current densities above room temperature. Severing the chemical bond between the protein and the retinal polyene only slightly affected the main electron transport via bR. Another thermally activated transport path opens upon retinal oxime production, instead of or in addition to the natural retinal. Transport through either or both of these paths occurs on a background of a general temperature-independent transport. These results lead us to propose a generalized mechanism for ETp across proteins, in which tunneling and hopping coexist and dominate in different temperature regimes.     

Deterministic vs non-deterministic graph property testing

A graph property \(\mathcal{P}\) is said to be testable if one can check whether a graph is close or far from satisfying \(\mathcal{P}\) using few random local inspections. Property \(\mathcal{P}\) is said to be non-deterministically testable if one can supply a “certificate” to the fact that a graph satisfies \(\mathcal{P}\) so that once the certificate is given its correctness can be tested. The notion of non-deterministic testing of graph properties was recently introduced by Lovász and Vesztergombi [9], who proved that (somewhat surprisingly) a graph property is testable if and only if it is non-deterministically testable. Their proof used graph limits, and so it did not supply any explicit bounds. They thus asked if one can obtain a proof of their result which will supply such bounds. We answer their question positively by proving their result using Szemerédi’s regularity lemma. An interesting aspect of our proof is that it highlights the fact that the regularity lemma can be interpreted as saying that all graphs can be approximated by finitely many “template” graphs.     

Irreducible values of polynomials

Schinzel's Hypothesis H is a general conjecture in number theory on prime values of polynomials that generalizes, e.g., the twin prime conjecture and Dirichlet's theorem on primes in arithmetic progression. We prove a quantitative arithmetic analog of this conjecture for polynomial rings over pseudo algebraically closed fields. This implies results over large finite fields via model theory. A main tool in the proof is an irreducibility theorem à la Hilbert.     

A Simple Homemade Reaction Station for use in Parallel Solution-Phase Synthesis. Optimization of a Regioselective One-Step Deprotective o-formylation Reaction Mediated by the Vilsmeier-Haack Reagent POCl3⋅DMF

Summary We report herein the fabrication of a simple and price-affordable portable reaction station for use in parallel solution-phase synthesis. This homemade device uses currently available laboratory components and equipment. Specifically designed to fit standard magnetic hotplates/stirrers, it can simultaneously hold up to 24 heated and magnetically stirred glass reactors of both 10 and 50 mL capacities. Glass reactors are connected by flexible 16-gauge metal needles to a central gas manifold equipped with an inlet/outlet for vacuum and inert gases. Reaction temperatures can be optimally varied from −78 ^∘C to 150 ^∘C. Using a statistical screening DOE method, this parallel array reactor station has been successfully operated to optimize the one-step deprotective O-formylation of a sterically hindered bis-O-tert-butyldiphenylsilyl (O-TBDPS) aromatic diol. The latter transformation was mediated by the Vilsmeier-Haack reagent POCl₃⋅DMF using a range of Lewis acid and metal salt promoters, including their binary combinations.     

On Distributions of Functionals of Anomalous Diffusion Paths

Functionals of Brownian motion have diverse applications in physics, mathematics, and other fields. The probability density function (PDF) of Brownian functionals satisfies the Feynman-Kac formula, which is a Schrödinger equation in imaginary time. In recent years there is a growing interest in particular functionals of non-Brownian motion, or anomalous diffusion, but no equation existed for their PDF. Here, we derive a fractional generalization of the Feynman-Kac equation for functionals of anomalous paths based on sub-diffusive continuous-time random walk. We also derive a backward equation and a generalization to Lévy flights. Solutions are presented for a wide number of applications including the occupation time in half space and in an interval, the first passage time, the maximal displacement, and the hitting probability. We briefly discuss other fractional Schrödinger equations that recently appeared in the literature.     

Heat-Treated Aerogel as a Catalyst for the Oxygen Reduction Reaction

Aerogels are fascinating materials that can be used for a wide range of applications, one of which is electrocatalysis of the important oxygen reduction reaction. In their inorganic form, aerogels can have ultrahigh catalytic site density, high surface area, and tunable physical properties and chemical structures—important features in heterogeneous catalysis. Herein, we report on the synthesis and electrocatalytic properties of an iron–porphyrin aerogel. 5,10,15,20-(Tetra-4-aminophenyl)porphyrin (H₂TAPP) and Fe^II were used as building blocks of the aerogel, which was later heat-treated at 600 °C to enhance electronic conductivity and catalytic activity, while preserving its macrostructure. The resulting material has a very high concentration of atomically dispersed catalytic sites (9.7×10²⁰ sites g⁻¹) capable of catalyzing the oxygen reduction reaction in alkaline solution (E_onset=0.92 V vs. RHE, TOF=0.25 e⁻ site⁻¹ s⁻¹ at 0.80 V vs. RHE).     

Heat‐Treated Aerogel as a Catalyst for the Oxygen Reduction Reaction

Aerogels are fascinating materials that can be used for a wide range of applications, one of which is electrocatalysis of the important oxygen reduction reaction. In their inorganic form, aerogels can have ultrahigh catalytic site density, high surface area, and tunable physical properties and chemical structures—important features in heterogeneous catalysis. Herein, we report on the synthesis and electrocatalytic properties of an iron–porphyrin aerogel. 5,10,15,20‐(Tetra‐4‐aminophenyl)porphyrin (H₂TAPP) and Fe^II were used as building blocks of the aerogel, which was later heat‐treated at 600 °C to enhance electronic conductivity and catalytic activity, while preserving its macrostructure. The resulting material has a very high concentration of atomically dispersed catalytic sites (9.7×10²⁰ sites g^?1) capable of catalyzing the oxygen reduction reaction in alkaline solution (E_onset=0.92 V vs. RHE, TOF=0.25 e^? site^?1 s^?1 at 0.80 V vs. RHE).