Remarks on the Cauchy functional equation and variations of it

This paper examines various aspects related to the Cauchy functional equation $f(x+y)=f(x)+f(y)$, a fundamental equation in the theory of functional equations. In particular, it considers its solvability and its stability relative to subsets of multi-dimensional Euclidean spaces and tori. Several new types of regularity conditions are introduced, such as a one in which a complex exponent of the unknown function is locally measurable. An initial value approach to analyzing this equation is considered too and it yields a few by-products, such as the existence of a non-constant real function having an uncountable set of periods which are linearly independent over the rationals. The analysis is extended to related equations such as the Jensen equation, the multiplicative Cauchy equation, and the Pexider equation. The paper also includes a rather comprehensive survey of the history of the Cauchy equation.Comment: To appear in Aequationes Mathematicae (important remark: the acknowledgments section in the official paper exists, but it appears before the appendix and not before the references as in the arXiv version); correction of a minor inaccuracy in Lemma 3.2 and the initial value proof of Theorem 2.1; a few small improvements in various sections; added thank

MicroRNA Dysregulation in Colon Cancer Microenvironment Interactions: The Importance of Small Things in Metastases

The influence of the microenvironment through the various steps of cancer progression is signed by different cytokines and growth factors, that could directly affect cell proliferation and survival, either in cancer and stromal cells. In colon cancer progression, the cooperation between hypoxia, IL-6 and VEGF-A165 could regulate the DNA repair capacity of the cell, whose impairment is the first step of colon cancer development. This cooperation redirects the activity of proteins involved in the metabolic shift and cell death, affecting the cell fate. The pathways triggered by micro environmental factors could modulate cancer-related gene transcription, affecting also small non coding mRNA, microRNAs. MicroRNAs have emerged as key post-transcriptional regulators of gene expression, directly involved in human cancers. The present review will focus first on the intertwined connection between cancer microenvironment and aberrant expression of microRNAs which contribute to carcinogenesis. In particular, the epigenetic mechanisms triggered by tissue microenvironment will be discussed, in view of the recent identification of miRNAs able to directly or indirectly modulate the epigenetic machinery (epi-miRNAs) and that are involved in the epithelial to mesenchimal transition and metastases development

Social organization and endocrine profiles of Australoheros facetus, an exotic freshwater fish in southern Portugal

Australoheros facetus is a neotropical cichlid and an exotic species in the Guadiana and Odelouca basins (Southern Portugal). In this research, we aimed to characterize the main behavioural patterns and circulating hormones, 17 beta-estradiol (E-2) for females, and testosterone, 11-ketotestosterone (11KT) and cortisol for both sexes, during the formation of social groups, a crucial step in the life-history of A. facetus. A pair-breeding strategy with territorial behaviour and aggressive interactions was found, with a positive correlation between dominance and size for both sexes. There were no significant differences between non-territorial and territorial individuals in the baseline levels of hormones, but 11KT was higher in males when they became territorial, as E-2 for territorial females, while there was no clear pattern for testosterone. In contrast, cortisol was higher in non-territorial males and correlated negatively with social dominance. These results suggest that size is the main driver for social dominance and that formation of stable hierarchies result in higher circulating 11KT and lowers stress in territorial males. Related to A. facetus' behaviour, aggressiveness and biparental care may be an advantage during the invasion process of A. facetus, facilitating colonization of new habitats.National Council for Technological and Scientific Development of Brazil (CNPq), Program Science without Borders [245971/20122]FCT-Foundation for Science and Technology [UID/Multi/04326/2013, SFRH/BPD/67008/2009]info:eu-repo/semantics/publishedVersio