Cohomological Kernels for Cyclic by Cyclic Semi-Direct Product Extensions

Let $F$ be a field and $E$ an extension of $F$ with $[E:F]=d$ where the characteristic of $F$ is zero or prime to $d$. We assume $\mu_{d^2}\subset F$ where $\mu_{d^2}$ are the $d^2$th roots of unity. This paper studies the problem of determining the cohomological kernel $H^n(E/F):=\ker(H^n(F,\mu_d) \rightarrow H^n(E,\mu_d))$ (Galois cohomology with coefficients in the $d$th roots of unity) when the Galois closure of $E$ is a semi-direct product of cyclic groups. The main result is a six-term exact sequence determining the kernel as the middle map and is based on tools of Positelski. When $n=2$ this kernel is the relative Brauer group ${\rm Br}(E/F)$, the classes of central simple algebras in the Brauer group of $F$ split in the field $E$. The work of Aravire and Jacob which calculated the groups $H^n_{p^m}(E/F)$ in the case of semidirect products of cyclic groups in characteristic $p$, provides motivation for this work

Nickel-Catalyzed Negishi Arylations of Propargylic Bromides: A Mechanistic Investigation

Although nickel-catalyzed stereoconvergent couplings of racemic alkyl electrophiles are emerging as a powerful tool in organic chemistry, to date there have been no systematic mechanistic studies of such processes. Herein, we examine the pathway for enantioselective Negishi arylations of secondary propargylic bromides, and we provide evidence for an unanticipated radical chain pathway wherein oxidative addition of the C–Br bond occurs through a bimetallic mechanism. In particular, we have crystallographically characterized a diamagnetic arylnickel(II) complex, [(i-Pr-pybox)NiIIPh]BAr^F_4, and furnished support for [(i-Pr-pybox)NiIIPh]^+ being the predominant nickel-containing species formed under the catalyzed conditions as well as a key player in the cross-coupling mechanism. On the other hand, our observations do not require a role for an organonickel(I) intermediate (e.g., (i-Pr-pybox)NiIPh), which has previously been suggested to be an intermediate in nickel-catalyzed cross-couplings, oxidatively adding alkyl electrophiles through a monometallic pathway

Anodic deposition of a robust iridium-based water-oxidation catalyst from organometallic precursors

Artificial photosynthesis, modeled on natural light-driven oxidation of water in Photosystem II, holds promise as a sustainable source of reducing equivalents for producing fuels. Few robust water-oxidation catalysts capable of mediating this difficult four-electron, four-proton reaction have yet been described. We report a new method for generating an amorphous electrodeposited material, principally consisting of iridium and oxygen, which is a robust and long-lived catalyst for water oxidation, when driven electrochemically. The catalyst material is generated by a simple anodic deposition from Cp*Ir aqua or hydroxo complexes in aqueous solution. This work suggests that organometallic precursors may be useful in electrodeposition of inorganic heterogeneous catalysts

SBOL Visual: A Graphical Language for Genetic Designs

Synthetic Biology Open Language (SBOL) Visual is a graphical standard for genetic engineering. It consists of symbols representing DNA subsequences, including regulatory elements and DNA assembly features. These symbols can be used to draw illustrations for communication and instruction, and as image assets for computer-aided design. SBOL Visual is a community standard, freely available for personal, academic, and commercial use (Creative Commons CC0 license). We provide prototypical symbol images that have been used in scientific publications and software tools. We encourage users to use and modify them freely, and to join the SBOL Visual community: http://www.sbolstandard.org/visual

Cohomological Kernels for Cyclic, Dihedral and Other Extensions

Let $F$ be a field and $E$ an extension of $F$ with $[E:F]=d$ where the characteristic of $F$ is zero or prime to $d$. We assume $\mu_{d^2}\subset F$ where $\mu_{d^2}$ are the $d^2$th roots of unity. This thesis studies the problem of determining the cohomological kernel $H^n(E/F):=\ker(H^n(F,\mu_d) \rightarrow H^n(E,\mu_d))$ (Galois cohomology with coefficients in the $d$th roots of unity) when the Galois closure of $E$ is a semi-direct product of cyclic groups. The main result is a six-term exact sequence determining the kernel as the middle map and is based on tools of Positelski \cite{Positselski}. When $n=2$ this kernel is the relative Brauer group ${\rm Br}(E/F)$, the classes of central simple algebras in the Brauer group of $F$ split in the field $E$. In the case where $E$ has degree $d$ and the Galois closure of $E$, \tE has Galois group {\rm Gal}(\tE/F) a dihedral group of degree $2d$, then work of Rowen and Saltman (1982) \cite{RowenSaltman} shows every division algebra $D$ of index $d$ split by $E$ is cyclic over $F$ (that is, $D$ has a cyclic maximal subfield.) This work, along with work of Aravire and Jacob (2008, 2018) \cite{AJ08} \cite{AJ18} which calculated the groups $H^n_{p^m}(E/F)$ in the case of semi-direct products of cyclic groups in characteristic $p$, provides motivation for this work

Cohomological Kernels for Cyclic, Dihedral and Other Extensions

Let $F$ be a field and $E$ an extension of $F$ with $[E:F]=d$ where the characteristic of $F$ is zero or prime to $d$. We assume $\mu_{d^2}\subset F$ where $\mu_{d^2}$ are the $d^2$th roots of unity. This thesis studies the problem of determining the cohomological kernel $H^n(E/F):=\ker(H^n(F,\mu_d) \rightarrow H^n(E,\mu_d))$ (Galois cohomology with coefficients in the $d$th roots of unity) when the Galois closure of $E$ is a semi-direct product of cyclic groups. The main result is a six-term exact sequence determining the kernel as the middle map and is based on tools of Positelski \cite{Positselski}. When $n=2$ this kernel is the relative Brauer group ${\rm Br}(E/F)$, the classes of central simple algebras in the Brauer group of $F$ split in the field $E$. In the case where $E$ has degree $d$ and the Galois closure of $E$, \tE has Galois group {\rm Gal}(\tE/F) a dihedral group of degree $2d$, then work of Rowen and Saltman (1982) \cite{RowenSaltman} shows every division algebra $D$ of index $d$ split by $E$ is cyclic over $F$ (that is, $D$ has a cyclic maximal subfield.) This work, along with work of Aravire and Jacob (2008, 2018) \cite{AJ08} \cite{AJ18} which calculated the groups $H^n_{p^m}(E/F)$ in the case of semi-direct products of cyclic groups in characteristic $p$, provides motivation for this work

Cohomological Kernels of Elementary Abelian Degree p2p^2 Extensions

Let $p$ be an odd prime, $F$ a field with a primitive $p^2$th root of unity, and $E=F(\sqrt[p]{b_1},\sqrt[p]{b_2})$ an elementary abelian extension of degree $p^2$. This paper studies the cohomological kernel $H^n(E/F,{\mathbb Z}/p{\mathbb Z}):={\rm ker}(H^n(F,{\mathbb Z}/p{\mathbb Z})\rightarrow H^n(E,{\mathbb Z}/p{\mathbb Z}))$ for all $n$. When $p=3$, using tools of Positselski, a six-term exact sequence is given that is analogous to the $p=2$ case. As an application the quotient $H^n(E/F,{\mathbb Z}/3{\mathbb Z})/{\rm Dec}^n(E/F,{\mathbb Z}/3{\mathbb Z})$ where ${\rm Dec}^n(E/F,{\mathbb Z}/3{\mathbb Z})$ is the "expected kernel'' is described. This quotient group is of interest because computations of Tignol [T] that show when $n=2$ nontrivial elements give rise to indecomposible division algebras of exponent $3$ and index $9$

Mechanisms and Site Selectivity of (Het)Ar–X Oxidative Addition to Pd(0) Are Controlled by Frontier Molecular Orbital Symmetry

We report how the reaction mechanism and site-selectivity of 2-halopyridine oxidative addition to L2Pd(0) are both controlled by frontier molecular orbital symmetry. Comparing oxidative addition rates for pairs of 2-chloro-3-EDG-pyridines / 2-chloro-5-EDG-pyridines (EDG = electron-donating group: NH2, OMe and F) to Pd(PCy3)2 reveals the 3-EDG isomers undergo oxidative addition ~100 times faster than their 5-EDG counterparts (∆ΔG‡OA = 10.4-11.6 kJ mol-1). Experimental and computational mechanistic studies reveal that the LUMO symmetries of the substrates control the oxidative addition mechanism. For the 3-EDG derivatives, high LUMO orbital coefficients at the reactive C2 position, and antibonding LUMO symmetry through the C2=N bond of the pyridine lead to a nucleophilic displacement oxida-tive addition mechanism. Conversely, the LUMO of the 5-EDG derivatives has a node through the C5–C2 plane, lead-ing to minimal orbital contribution at the reactive carbon. The higher energy LUMO+1 has substantial density at C2, but minimal orbital density at the nitrogen. This leads to 5-EDG substrates undergoing a 3-centered insertion oxida-tive addition mechanism. These orbital symmetry effects also control site-selectivity for multihalogenated pyridines, which we investigate for both electron-donating and electron-withdrawing substituents. Incorporating simple fron-tier orbital based molecular descriptors to a quantitative multivariate linear model for oxidative addition leads to im-proved prediction accuracy for both relative rates and site-selectivity of substituted 2-halopyridine oxidative addition to L2Pd(0)

Enolate addition to bicyclobutanes enables expedient access to 2-oxo-bicyclohexane scaffolds

We report the synthesis of 2-oxo-bicyclo[2.1.1]hexanes (2-oxo-BCHs) from bicyclobutanes (BCBs) and readily available enolate precursors. We propose this reaction proceeds via initial enolate addition to the bicyclobutane, followed by an intramolecular acyl substitution by the resulting enolate intermediate. Glycine-derived enolates directly give protected 2-oxo-3-amino-BCH derivatives that can be further functionalized. Arylacetate derivatives are also suitable enolate precursors, giving 2-oxo-3-aryl-BCH scaffolds from readily available starting materials