Stable Methods for Evaluating the Points cos(i{pi}/n)

A variety of alternative methods for evaluating cos(i/n) fori= 0, 1,...,n are compared theoretically and experimentally,and the best ones selected on the basis of accuracy and thecomputational efficiency. The error analysis involved demonstrateshow certain mathematically equivalent recurence relations canexhibit very different computational behaviour.     

Gauge invariance and renormalization

The renormalization of Abelian and non-Abelian local gauge theories is discussed. It is recalled that whereas Abelian gauge theories are invariant to local c-number gauge transformations δA_μ(x) = ?_μ,…, with □Λ = 0, and to the operator gauge transformation δA_μ(x) = ?_μφ(x), …, δφ(x) = α^?1?·A(x), with □φ = 0, non-Abelian gauge theories are invariant only to the operator gauge transformations $\text{δA}{}_{\mathrm{\mu }}\text{(x) ～}\textcolor{red}{}{}_{\mathrm{\mu }}\text{C(x)}$, …, introduced by Becchi, Rouet and Stora, where

_μ is the covariant derivative matrix and C is the vector of ghost fields. The renormalization of these gauge transformation is discussed in a formal way, assuming that a gauge-invariant regularization is present. The naive renormalized local non-Abelian c-number gauge transformation $\text{δA}{}_{\mathrm{\mu }}\text{(x) = (Z}{}_1\text{/Z}{}_3\text{)gA}{}_{\mathrm{\mu }}\text{(x) ×}\text{Λ}\text{(x)+?}{}_{\mathrm{\mu }}\text{Λ}\text{(x)}$, …, is never a symmetry transformation and is never finite in perturbation theory. Only for $\text{Λ}\text{(x) = (Z}{}_3\text{/Z}{}_1\text{)L}$ with L finite constants or for $\text{Λ}\text{(x) = Ω}\text{z}\text{?}{}_3\text{C(x)}$ with Ω a finite constant does it become a finite symmetry transformation, where $\text{z}\text{?}{}_3$ is the ghost field renormalization constant. The renormalized non-Abelian Ward-Takahashi (Slavnov-Taylor) identities are consequences of the invariance of the renormalized gauge theory to this formation. It is also shown how the symmetry generators are renormalized, how photons appear as Goldstone bosons, how the (non-multiplicatively renormalizable) composite operator A_μ × C is renormalized, and how an Abelian c-number gauge symmetry may be reinstated in the exact solution of many asymptotically fr ee non-Abelian gauge theories.     

Structure and cooperativity of the hydrogen bonds in sodium dihydrogen triacetate

Geometry and energetics of low energy conformers of sodium dihydrogen triacetate (SDHTA) and its anion are studied using density functional theory (DFT) at the Becke, Lee‐Yang‐Parr hybrid functional (BLYP) and Becke, three‐parameter, Lee‐Yang‐Parr hybrid functional (B3LYP) levels. For both cases, two structures of comparable energy are found, which have different symmetry with respect to the two hydrogen bonds (HBs). DFT‐based Born–Oppenheimer molecular dynamics simulations are performed for SDHTA, which show that both structures are visited at room temperature conditions. The trajectory analysis further reveals that the two HBs behave anticooperative, that is, on average elongation of one HB is accompanied by a compression of the other one. This is in accord with nuclear magnetic resonance (NMR) experimental studies for a similar counter ion–dihydrogen triacetate complex. © 2012 Wiley Periodicals, Inc.     

Hard Thermal Loops in the n-Dimensional ? 3 Theory

We derive a closed-form result for the leading thermal contributions which appear in the n-dimensional ? ³ theory at high temperature. These contributions become local only in the long wavelength and in the static limits, being given by different expressions in these two limits.     

Synthesis of Stereochemically Flexible Cyclic Biphenylbisphosphinite Ligands: Control of the Dynamics and Selectivity

Novel l -tartaric acid based stereodynamic biphenylbisphosphinite ligands with anilino linkers were prepared as cyclic diamide compounds. The influence of the constitutional substitution pattern of the amino group at the aniline moiety on the diastereoselective formation of the ligand was investigated. While the meta-anilino bridged ligands showed a highly dynamic behavior and no discrimination between (R_ax) and (S_ax) configuration, l -tartaric acid substitution on ortho-anilino bridged ligands leads to diastereoselective self-alignment of the central biphenol axis, which controls the stereoselectivity of the ligand and catalyst. The experimental findings were corroborated by theoretical calculations. In the rhodium-catalyzed enantioselective hydrogenation of methyl 2-acetamidoacrylate (MAA), an enantiomeric ratio of 75 : 25 (R/S) was obtained for the ortho-substituted ligand.     

Montecrinanes A–C: Triterpenes with an Unprecedented Rearranged Tetracyclic Skeleton from Celastrus vulcanicola. Insights into Triterpenoid Biosynthesis Based on DFT Calculations

Three new triterpenoids with an unprecedented 6/6/6/6‐fused tetracyclic carbon skeleton, montecrinanes A–C ( 1 – 3 ), were isolated from the root bark of Celastrus vulcanicola, along with known D:B‐friedobaccharanes ( 4 – 6 ), and lupane‐type triterpenes ( 7 – 12 ). The stereostructures of the new metabolites were elucidated based on spectroscopic (1D and 2D NMR) and spectrometric (HR‐EIMS and HR‐ESIMS) techniques. Their absolute configurations were determined by both NMR spectroscopy, with (R)‐(?)‐α‐methoxyphenylacetic acid as a chiral derivatizing agent, and biogenetic considerations. Biogenetic pathways for montecrinane and D:B‐friedobaccharane skeletons were proposed and studied by DFT methods. The theoretical results support the energetic feasibility of the putative biogenetic pathways, in which the 1,2‐methyl shift from the secondary baccharenyl cation represents a novel and key reaction step for a new montecrinane skeleton.     

Autonomously Propelled Motors for Value‐Added Product Synthesis and Purification

A proof‐of‐concept design for autonomous, self‐propelling motors towards value‐added product synthesis and separation is presented. The hybrid motor design consists of two distinct functional blocks. The first, a sodium borohydride (NaBH₄) granule, serves both as a reaction prerequisite for the reduction of vanillin and also as a localized solid‐state fuel in the reaction mixture. The second capping functional block consisting of a graphene–polymer composite serves as a hydrophobic matrix to attract the reaction product vanillyl alcohol (VA), resulting in facile separation of this edible value‐added product. These autonomously propelled motors were fabricated at a length scale down to 400 μm, and once introduced in the reaction environment showed rapid bubble‐propulsion followed by high‐purity separation of the reaction product (VA) by the virtue of the graphene–polymer cap acting as a mesoporous sponge. The concept has excellent potential towards the synthesis/isolation of industrially important compounds, affinity‐based product separation, pollutant remediation (such as heavy metal chelation/adsorption), as well as localized fuel‐gradients as an alternative to external fuel dependency.     

Macromodeling of high-speed interconnects by positive interpolation of vertical segments

In this paper a novel macromodeling scheme is presented to model the per unit of length (p.u.l.) parameters of uniform transmission lines. In particular, it is focused on single on-chip interconnects, because their p.u.l. parameters are influenced by the presence of semiconductor (s) and as such exhibit a strong frequency-dependency, making the modeling process harder. Starting from a set of very accurate tabulated data samples, obtained by two-dimensional electromagnetic modeling, rational models for the four p.u.l. parameters are constructed. The novelty of the approach lies in the fact that the rational models are positive by construction and that a controllable accuracy is obtained. These models can then further be used to construct multivariate models, e.g., for variability analysis. Here, the novel scheme is applied to an on-chip inverted embedded microstrip line, of which the signal integrity behavior is assessed in both the frequency and the time domain, demonstrating the applicability of the macromodels.