Vibrational Stark Fields in Carboxylic Acid Dimers

Carboxylic acids form exceptionally stable dimers and have been used to model proton and double proton transfer processes. The stabilization energies of the carboxylic acid dimers are very weakly dependent on the nature of the substitution. However, the electric field experienced by the OH group of a particular carboxylic acid is dependent more on the nature of the substitution on the dimer partner. In general, the electric field was higher when the partner was substituted with electron-donating group and lower with electron-withdrawing substituent on the partner. The Stark tuning rate (∆μ) of the O–H stretching vibrations calculated at the MP2/aug-cc-pVDZ level was found to be weakly dependent on the nature of substitution on the carboxylic acid. The average Stark tuning rate of O–H stretching vibrations of a particular carboxylic acid when paired with other acids was 5.7 cm–1 (MV cm–1)–1, while the corresponding average Stark tuning rate of the partner acids due to a particular carboxylic acid was 21.9 cm–1 (MV cm–1)–1. The difference in the Stark tuning rate is attributed to the primary and secondary effects of substitution on the carboxylic acid. The average Stark tuning rate for the anharmonic O–D frequency shifts is about 40-50% higher than the corresponding harmonic O–D frequency shifts calculated at B3LYP/aug-cc-pVDZ level, much greater than the typical scaling factors used, indicating the strong anharmonicity of O–H/O–D oscillators in carboxylic acid dimers. Finally, the linear correlation observed between pKa and the electric field was used to estimate the pKa of fluoroformic acid to be around 0.9.

A Meeting Point of Probability, Graphs, and Algorithms: The Lovász Local Lemma and Related Results—A Survey

A classic and fundamental result, known as the Lovász Local Lemma, is a gem in the probabilistic method of combinatorics. At a high level, its core message can be described by the claim that weakly dependent events behave similarly to independent ones. A fascinating feature of this result is that even though it is a purely probabilistic statement, it provides a valuable and versatile tool for proving completely deterministic theorems. The Lovász Local Lemma has found many applications; despite being originally published in 1973, it still attracts active novel research. In this survey paper, we review various forms of the Lemma, as well as some related results and applications.

Increase of the Informative Value of Magnetic Nondestructive Testing Using Vibration Induction Transducer on the Basis of Spectral Analysis

The ways of increasing the information content of magnetic flaw detection based on spectral analysis when registering magnetic fluxes of leakage from defects by a vibration induction probe (VIP) are considered. It is shown that the VIP signals, caused by the normal and tangential components of the strength of the magnetic scattering fluxes, are linearly independent, and the corresponding harmonic components of the spectra of these signals are weakly dependent on the noise component and carry information about the coordinates and parameters of the defect.

A UNIFORM BOUND ON THE OPERATOR NORM OF SUB-GAUSSIAN RANDOM MATRICES AND ITS APPLICATIONS

For an $N \times T$ random matrix $X(\beta )$ with weakly dependent uniformly sub-Gaussian entries $x_{it}(\beta )$ that may depend on a possibly infinite-dimensional parameter $\beta \in \mathbf {B}$ , we obtain a uniform bound on its operator norm of the form $\mathbb {E} \sup _{\beta \in \mathbf {B}} ||X(\beta )|| \leq CK \left (\sqrt {\max (N,T)} + \gamma _2(\mathbf {B},d_{\mathbf {B}})\right )$ , where C is an absolute constant, K controls the tail behavior of (the increments of) $x_{it}(\cdot )$ , and $\gamma _2(\mathbf {B},d_{\mathbf {B}})$ is Talagrand’s functional, a measure of multiscale complexity of the metric space $(\mathbf {B},d_{\mathbf {B}})$ . We illustrate how this result may be used for estimation that seeks to minimize the operator norm of moment conditions as well as for estimation of the maximal number of factors with functional data.