Background: Ghrelin is the orexigenic hormone secreted mainly by the stomach. Its involvement in neoplastic development has been studied in gastrointestinal adenocarcinomas. Our paper aims to evaluate the influence of the ghrelin axis in gastrointestinal stromal tumors (GISTs). Materials and Methods: The study design included two groups of patients, 46 with gastric GISTs and 30 with obesity. Archived tissue samples were evaluated for the presence of gastritis and H. pylori. Immunohistochemical expression of ghrelin and its receptor (GHS-R) was assessed. Results: All GISTs showed absent immunohistochemical expression for ghrelin, while GHS-R displayed a particular pattern, with notable differences in intensity (p = 0.0256) and percentage of stained cells (p < 0.00001) in the periphery vs. core of tumors. Positive ghrelin expression was lower in the gastric mucosa of the first group compared to the second group (p < 0.001). Conclusion: The ghrelin axis can influence GISTs carcinogenesis through activation of GHS-R. A previously described direct autocrine/paracrine mechanism is not supported by our findings.

We introduce the concept of almost $\mathcal {P}$ -numbers where $\mathcal {P}$ is a class of groups. We survey the existing results in the literature for almost cyclic numbers, and give characterisations for almost abelian and almost nilpotent numbers proving these two concepts are equivalent.

In this paper, we introduce a new function computing the harmonic mean of element orders of a finite group. We present a series of properties for this function, and then we study groups for which the value of the function is an integer.

In this paper, we extend the parallelism between perfect numbers and Leinster groups, by introducing the new concepts of almost and quasi-Leinster groups which parallel almost and quasi-perfect numbers. These are small deviations from perfect numbers; very few results and/or examples are known about them. We investigate nilpotent almost/quasi-/Leinster groups and find some examples and conditions for the existence of such groups for classes of nonnilpotent groups: ZM (Zassenhaus metacyclic) groups, affine groups, dihedral groups and dicyclic groups.

We are surveying recent results that describe second order differential operators having only algebraic solutions in the sense of Galois theory. We call such operators algebraic. For hypergeometric operators, this problem was studied by Schwarz and Klein who also gave results that describe all second order linear differential operators with a full set of algebraic solutions. Starting from their work, we see algebraic operators as pull-backs of algebraic hypergeometric operators via Belyi functions. We are surveying some of the main results describing second order operators with a full set of algebraic solutions, especially those obtained by using the properties of the pull-back functions. Using the Grothendieck correspondence, these properties transfer to properties for their corresponding dessins d’enfants.

Our work adds to the picture of second order differential operators with a full set of algebraic solutions, which we will call algebraic. We see algebraic Heun operators as pull-backs of algebraic hypergeometric operators via Belyi functions. We focus on the case when the hypergeometric one has a tetrahedral monodromy group. We find arithmetic conditions for the pull-back functions to exist. For each distribution of the singular points in the ramified fibers, we identify the minimal values of the exponent differences and we explicitly construct the dessin d’enfant corresponding to the pull-back function in the minimal cases. Then by allowing some parameters to vary, we find infinite families of such graphs, hence of Heun operators with tetrahedral monodromy.