In this paper, we consider amaranth starch extracted from the seeds of Amaranthus hypochondriacus L. An amphiphilic character is conferred to the starch by a chemical modification, which involves an esterification by lauroyl chloride at three modification levels. The degree of substitution (DS) after the modification ranged from 0.06 to 1.16. X-ray photoelectron spectroscopy analysis confirmed the presence of fatty acyl chains on the surface of the esterified starches. The hydrophobicity of starches was confirmed by their adsorption isotherms, which showed a decrease in the moisture adsorption of lauroylated as DS increased. X-ray diffraction analysis revealed a higher crystallinity, which was observed in the two samples subjected to the highest levels of modification. A higher crystallinity is related to a higher gelatinization enthalpy. These results are in agreement with the thermal characterization obtained by differential scanning calorimetry (DSC). An inhibition of the retrogradation properties of lauroylated amaranth starches was also observed.

The paper is devoted to the spectral properties of one-dimensional Schr\"{o}dinger operators \begin{equation} S_{q}u\left(x\right)=\left(-\frac{d^{2}}{dx^{2}}+q\left(x\right)\right)u\left(x\right),\quad x\in\mathbb{R},\label{eq1} \end{equation} with potentials $q=q_{0}+q_{s}$, where $q_{0}\in L^{\infty}\left(\mathbb{R}\right)$ is a regular potential, and $q_{s}\in\mathcal{D}^{\prime}\left(\mathbb{R}\right)$ is a singular potential with support on a discrete infinite set $\mathcal{Y}\subset\mathbb{R}$. We consider the extension $\mathcal{H}$ of formal operator (\ref{eq1}) to an unbounded operator in $L^{2}\left(\mathbb{R}\right)$ defined by the Schr\"{o}dinger operator $S_{q_{0}}$ with regular potential $q_{0}$ and interaction conditions at the points of the set $\mathcal{Y}$. We study the closedness and self-adjointness of $\mathcal{H}$. If the set $\mathcal{Y}\simeq\mathbb{Z}$ has a periodic structure we give the description of the essential spectrum of operator $\mathcal{H}$ in terms of limit operators. For periodic potentials $q_{0}$ we consider the Floquet theory of $\mathcal{H}$, and apply the spectral parameter power series method for definition of band-gap structure of the spectrum. We also consider the case when the regular periodic part of the potential is perturbed by a slowly oscillating at infinity term. We show that this perturbation changes the structure of the spectra of periodic operators significantly. This works presents several numerical examples to demonstrate the effectiveness of our approach.

Avocado oil is a very valuable agro-industrial product which can be perishable in a short time if it is not stored in the right conditions. The encapsulation of the oils through the spray drying technique protects them from oxidation and facilitates their incorporation into different pharmaceutical products and food matrices; however, the selection of environmentally friendly emulsifiers is a great challenge. Four formulations of the following solid particles: Gum Arabic, HI-CAP^®100 starch, and phosphorylated waxy maize starch, were selected to prepare avocado oil Pickering emulsions. Two of the formulations have the same composition, but one of them was emulsified by rotor-stator homogenization. The rest of the emulsions were emulsified by combining rotor-stator plus ultrasound methods. The protective effect of mixed particle emulsifiers in avocado oil encapsulated by spray drying was based on the efficiency of encapsulation. The best results were achieved when avocado oil was emulsified with a mixture of phosphorylated starch/HI-CAP^®100, where it presented the highest encapsulation efficiency.