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Abstract Stationary points embedded in the derivatives are often critical for a model to be interpretable and may be considered as key features of interest in many applications. We propose a semiparametric Bayesian model to efficiently infer the locations of stationary points of a nonparametric function, which also produces an estimate of the function. We use Gaussian processes as a flexible prior for the underlying function and impose derivative constraints to control the function's shape via conditioning. We develop an inferential strategy that intentionally restricts estimation to the case of at least one stationary point, bypassing possible mis-specifications in the number of stationary points and avoiding the varying dimension problem that often brings in computational complexity. We illustrate the proposed methods using simulations and then apply the method to the estimation of event-related potentials derived from electroencephalography (EEG) signals. We show how the proposed method automatically identifies characteristic components and their latencies at the individual level, which avoids the excessive averaging across subjects that is routinely done in the field to obtain smooth curves. By applying this approach to EEG data collected from younger and older adults during a speech perception task, we are able to demonstrate how the time course of speech perception processes changes with age. 

We report a series of redox-active bis(pincer) Pd( ii ) complexes in which the redox active units are based on either a diarylamido or a carbazolide framework. Compounds 1 and 2 contain two full diarylamido/bis(pincer) PNP units connected either via an Ar–O–Ar linker ( 1 ) or an Ar–Ar bond ( 2 ). Compound 3 is a fused bis(pincer) where the two PNP units share an aromatic ring. Compound 4 is built around an indolo[3,2- b ]carbazole core in which two NNN pincers share an aromatic ring similarly to 3 . These metal complexes all display two reversible oxidation waves with the Δ E values increasing in the order of 1 < 2 < 4 < 3 . The same trend in increasing electronic coupling emerges from the analysis of the IV-CT bands in the NIR portion of the optical spectra. The analysis of these compounds was further advanced by data from EPR spectroscopy, X-ray diffractometry, and DFT calculations. It is concluded that the monooxidized cations 2+–4+ belong to Class III on the Robin-Day classification of mixed-valence compounds. Compound 4 possesses enforced near-planarity that enables delocalization of the unpaired electron in 4+ across a broader conjugated system compared to 3+ .