The rehydration properties of Ca-, Mg-, Na-, and K-saturated homoionic beidellites after heating at various temperatures were compared with those of montmorillonites. The behavior of interlayer Na+ during dehydration and rehydration was also investigated by means of one-dimensional Fourier analysis. The K- and Mg-saturated specimens exhibited fast and slow rehydration rates, respectively, during exposure to air of 50% RH after heating at 800°C. These homoionic specimens showed strong rehydration properties on saturation with deionized water after heating <900°C for 1 hr. On the basis of Fourier analysis, the interlayer cations appeared to have migrated into the hexagonal holes of SiO4 network on thermal dehydration, and the migrated cations returned to the interlayer space on rehydration. This behavior of the interlayer cations appears to have been strongly dependent on value of the octahedral negative charge and on the sizes of interlayer cations. The small octahedral negative charge of beidellite produced a weaker attractive electrostatic force between the octahedral sheets and the migrated interlayer cations. Therefore, the migrated interlayer cations in beidellite were easily extracted from the hexagonal holes, and rehydration was rapid. The small cation migrated easily into hexagonal holes and was fixed to the holes. On the contrary, large cations were probably difficult to fix and were easily extracted from the hexagonal holes. Consequently, the rehydration rate of K-saturated beidellite was fast, and that of Mg-saturated beidellite was slow.

The Mg-vermiculite from Santa Olalla has been treated with aliphatic amides—formamide (FM), acetamide (AM) and propionamide (PM)—in aqueous solution. These treatments produce the transformation towards NH4-vermiculite and interstratified NH4-vermiculite-Mg-vermiculite phases. The NH4-vermiculite, Mg-vermiculite and interstratified (mixed-layer) phases have been identified from basal X-ray diffraction (XRD) interval peaks between 10.3 Å and 14.4 Å, and confirmed by direct Fourier transform method, as well as by atomic absorption spectrometry (AAS), infrared (IR) spectroscopy and thermal analysis.

According to their NH4-vermiculite/Mg-vermiculite probability coefficients ratio (PA/PB), and PAA, these interstratified phases can be divided into 3 categories: 1) If the PA/PB ratio is ≥ 7/3 and PAA ≥ 0.7, there are interstratified phases with a strong tendency toward segregation (case of FM, AM and PM). 2) If the PA/PB ratio is between 5/5 and 6/4, with PAA in the range 0.45–0.6, there are nearly regular alternating and random interstratified phases (case of AM and PM). 3) If the PA/PB ratio is ≤ 5/5 and PAA ≤ 0.45, there are interstratified phases with a strong tendency toward alternation (case of PM).

Experimental evidence reported in the present work indicates that the mechanism of interaction of Mg-vermiculite with FM, AM and PM in an aqueous medium takes place by ion exchange of NH4 between the layers. The hydrolysis of these aliphatic amides leads to the liberation of NH4+ into the medium. It has been found that the NH4+ sorption depends on the physico-chemical characteristics of the particular aliphatic amide, and the transformation of Mg-vermiculite to interstratified and/or NH4-vermiculite phases depends on the amide concentration. These treatments allow one to control the formation of interstratified and NH4-vermiculite phases.

In the classical framework, a random walk on a group is a Markov chain with independent and identically distributed increments. In some sense, random walks are time and space homogeneous. This paper is devoted to a class of inhomogeneous random walks on $\mathbb{Z}^d$ termed ‘Markov additive processes’ (also known as Markov random walks, random walks with internal degrees of freedom, or semi-Markov processes). In this model, the increments of the walk are still independent but their distributions are dictated by a Markov chain, termed the internal Markov chain. While this model is largely studied in the literature, most of the results involve internal Markov chains whose operator is quasi-compact. This paper extends two results for more general internal operators: a local limit theorem and a sufficient criterion for their transience. These results are thereafter applied to a new family of models of drifted random walks on the lattice $\mathbb{Z}^d$ .

Celebrated theorems of Roth and of Matoušek and Spencer together show that the discrepancy of arithmetic progressions in the first $n$ positive integers is $\Theta (n^{1/4})$. We study the analogous problem in the $\mathbb {Z}_n$ setting. We asymptotically determine the logarithm of the discrepancy of arithmetic progressions in $\mathbb {Z}_n$ for all positive integer $n$. We further determine up to a constant factor the discrepancy of arithmetic progressions in $\mathbb {Z}_n$ for many $n$. For example, if $n=p^k$ is a prime power, then the discrepancy of arithmetic progressions in $\mathbb {Z}_n$ is $\Theta (n^{1/3+r_k/(6k)})$, where $r_k \in \{0,1,2\}$ is the remainder when $k$ is divided by $3$. This solves a problem of Hebbinghaus and Srivastav.

Determining spatial resolution from images is crucial when optimizing focus, determining smallest resolvable object, and assessing size measurement uncertainties. However, no standard algorithm exists to measure resolution from electron microscopy (EM) images, though several have been proposed, where most require user decisions. We present the Spatial Image Resolution Assessment by Fourier analysis (SIRAF) algorithm that uses fast Fourier transform analysis to estimate resolution directly from a single image without user inputs. The method is derived from the underlying assumption that objects display intensity transitions, resembling a step function blurred by a Gaussian point spread function. This hypothesis is tested and verified on simulated EM images with known resolution. To identify potential pitfalls, the algorithm is also tested on simulated images with a variety of settings, and on real SEM images acquired at different magnification and defocus settings. Finally, the versatility of the method is investigated by assessing resolution in images from several microscopy techniques. It is concluded that the algorithm can assess resolution from a large selection of image types, thereby providing a measure of this fundamental image parameter. It may also improve autofocus methods and guide the optimization of magnification settings when balancing spatial resolution and field of view.

Fourier analysis can provide policymakers useful information for analysing the pandemic behaviours. This paper proposes a Fourier analysis approach for examining the cycle length and the power spectrum of the pandemic by converting the number of deaths due to coronavirus disease 2019 in the US to the frequency domain. Policymakers can control the pandemic by using observed cycle length whether they should strengthen their policy or not. The proposed Fourier method is useful for analysing waves in other medical applications.

In this work, a new method to determine and correct the linear drift for any crystalline orientation in a single-column-resolved high-resolution scanning transmission electron microscopy (HR-STEM) image, which is based on angle measurements in the Fourier space, is presented. This proposal supposes a generalization and the improvement of a previous work that needs the presence of two symmetrical planes in the crystalline orientation to be applicable. Now, a mathematical derivation of the drift effect on two families of asymmetric planes in the reciprocal space is inferred. However, though it was not possible to find an analytical solution for all conditions, a simple formula was derived to calculate the drift effect that is exact for three specific rotation angles. Taking this into account, an iterative algorithm based on successive rotation/drift correction steps is devised to remove drift distortions in HR-STEM images. The procedure has been evaluated using a simulated micrograph of a monoclinic material in an orientation where all the reciprocal lattice vectors are different. The algorithm only needs four iterations to resolve a 15° drift angle in the image.