On the Convergence Complexity of Gibbs Samplers for a Family of Simple Bayesian Random Effects Models

The emergence of big data has led to so-called convergence complexity analysis, which is the study of how Markov chain Monte Carlo (MCMC) algorithms behave as the sample size, n, and/or the number of parameters, p, in the underlying data set increase. This type of analysis is often quite challenging, in part because existing results for fixed n and p are simply not sharp enough to yield good asymptotic results. One of the first convergence complexity results for an MCMC algorithm on a continuous state space is due to Yang and Rosenthal (2019), who established a mixing time result for a Gibbs sampler (for a simple Bayesian random effects model) that was introduced and studied by Rosenthal (Stat Comput 6:269–275, 1996). The asymptotic behavior of the spectral gap of this Gibbs sampler is, however, still unknown. We use a recently developed simulation technique (Qin et al. Electron J Stat 13:1790–1812, 2019) to provide substantial numerical evidence that the gap is bounded away from 0 as n → ∞. We also establish a pair of rigorous convergence complexity results for two different Gibbs samplers associated with a generalization of the random effects model considered by Rosenthal (Stat Comput 6:269–275, 1996). Our results show that, under a strong growth condition, the spectral gaps of these Gibbs samplers converge to 1 as the sample size increases.

     

Oxime functionalization strategy for iodinated poly(epsilon‐caprolactone) X‐ray opaque materials

Since two of the most common technologies for imaging the human body are X‐ray radiography and computed tomography (CT), researchers are focused on developing biodegradable and biocompatible polymeric molecules as an alternative to the traditional small molecule contrast agents. This report highlights the synthesis of novel biodegradable iodinated poly(ε‐caprolactone) copolymers by oxime “Click” ligation reactions. A series of ketone‐bearing materials are built by tin (II)‐mediated ring‐opening polymerization followed by a postpolymerization deprotection step. The intended X‐ray opacity is imparted through acid‐catalyzed oxime postpolymerization modification of the resultant polymers with an iodinated hydroxylamine. All small molecules and polymeric materials are characterized using proton nuclear magnetic resonance (NMR) for purity, functional group stoichiometry, and number‐averaged molecular weight calculations. Additionally, the polymers are evaluated with gel permeation chromatography (GPC) to determine polymer sample polydispersity and general molecular weight distribution shapes and by differential scanning calorimetry (DSC) and thermogravimetric analysis (TGA) for thermal properties. © 2015 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2015 , 53, 2421–2430     

Semi‐synthesis and NMR spectral assignments of flavonoid and chalcone derivatives

Previous investigations of the aerial parts of the Australian plant Eremophila microtheca and Syzygium tierneyanum resulted in the isolation of the antimicrobial flavonoid jaceosidin ( 4 ) and 2′,6′‐dihydroxy‐4′‐methoxy‐3′,5′‐dimethyl chalcone ( 7 ), respectively. In this current study, compounds 4 and 7 were derivatized by acetylation, pivaloylation, and methylation reactions. The final products, 5,7,4′‐triacetoxy jaceosidin ( 10 ), 5,7,4′‐tripivaloyloxy jaceosidin ( 11 ), 5,7,4′‐trimethoxy jaceosidin ( 12 ), 2′,6′‐diacetoxy‐4′‐methoxy‐3′,5′‐dimethyl chalcone ( 13 ), 2′‐hydroxy‐4′‐methoxy‐6′‐pivaloyloxy‐3′,5′‐dimethyl chalcone ( 14 ), and 2′‐hydroxy‐4′,6′‐dimethoxy‐3′,5′‐dimethyl chalcone ( 15 ) were all fully characterized by NMR and MS. Derivatives 10 and 13 have been previously reported but were only partially characterized. This is the first reported synthesis of 11 and 14 . The natural products and their derivatives were evaluated for their antibacterial and antifungal properties, and the natural product, jaceosidin ( 4 ) and the acetylated derivative, 5,7,4′‐triacetoxy jaceosidin ( 10 ), showed modest antibacterial activity (32–128 µg/ml) against Staphylococcus aureus strains. Copyright © 2016 John Wiley & Sons, Ltd.     

Unequivocal 13C NMR assignment of cyclohexadienyl rings in bromotyrosine‐derived metabolites from marine sponges

Bromotyrosine‐derived compounds are commonly isolated from Verongida sponges and are a major class of marine natural products. Here we report on the unequivocal ¹³C NMR assignment of the brominated carbons at positions C‐2 and C‐4 of the cyclohexadiene ring, two carbons whose resonances are often incorrectly assigned. Interpretation of HMBC data acquired for a series of known bromotyrosine analogues, which included ianthesine E (1), aerothionin (2), 11‐hydroxyaerothionin (3), and 11,19‐dideoxyfistularin‐3 (4), allowed us to unequivocally assign the carbons in question, C‐2 and C‐4, through the observance of unique HMBC correlations from the C‐1 hydroxyl proton. Here we present the complete 2D NMR data sets recorded in DMSO‐d₆ for 2–4 that were used to confirm the assignment and establish the working model. Using this model, a survey of the literature revealed that many members of this structure class had been wrongly assigned. This paper serves to reassign those compounds whose ¹³C NMR assignment at positions C‐2 and C‐4 of the cyclohexadiene ring should be reversed. Copyright © 2012 John Wiley & Sons, Ltd.     

Use of a force plate versus armature accelerometer for measuring frequency response functions

Experimental Techniques -