Several papers, mainly written by J. de la Call and co-authors, contain modifications of classical sequences of positive linear operators to obtain new sequences converging to limits which are not necessarily the identity operator. Such results were obtained using probabilistic methods. Recently, results of this type have been obtained with analytic methods. Semi-exponential operators have also been introduced, extending the theory of exponential operators. We combine these two approaches, applying the semi-exponential operators in a new context and enlarging the list of operators representable as limits of other operators.

Noncoding RNAs, particularly microRNAs (miRNAs) and small interfering RNAs (siRNAs), have emerged as key players in the pathogenesis and therapeutic strategies for inflammatory bowel disease (IBD). MiRNAs, small endogenous RNA molecules that silence target mRNAs to regulate gene expression, are closely linked to immune responses and inflammatory pathways in IBD. Notably, miR-21, miR-146a, and miR-155 are consistently upregulated in IBD, influencing immune cell modulation, cytokine production, and the intestinal epithelial barrier. These miRNAs serve as biomarkers for disease progression and severity, as well as therapeutic targets for controlling inflammation. This comprehensive review highlights the intricate interplay between the gut microbiota, fecal microbiota transplantation (FMT), and miRNA regulation. It concludes that microbiota and FMT influence miRNA activity, presenting a promising avenue for personalized IBD treatment.

Liver cirrhosis is marked by sodium and water retention, portal hypertension and sharply reduced survival after decompensation. Sodium–glucose cotransporter-2 inhibitors (SGLT2i) induce insulin-independent glycosuria and natriuresis and have proven cardio-renal benefits, prompting interest in their role as adjuncts for ascites. This review synthesizes current evidence on efficacy, safety and mechanistic plausibility of SGLT2i in cirrhosis. Observational cohorts and case series suggest that adding SGLT2i to standard diuretics increases natriuresis, lowers ascites burden and paracentesis requirements, improves weight and aminotransferases and may reduce hepatic decompensation and hepatocellular carcinoma risk. Safety remains paramount: hypotension, acute kidney injury and hepatorenal syndrome-related acute kidney injury, genitourinary infections, electrolyte disturbances and rare euglycemic ketoacidosis necessitate careful patient selection, slow titration and close monitoring, especially in decompensated disease and when combined with loop diuretics or mineralocorticoid receptor antagonists. Overall, the balance of data supports cautious optimism: SGLT2i represent a promising adjunct within protocolized care pathways for selected patients, while definitive trials powered for hepatic outcomes are still required to clarify indications, timing, dosing and long-term impact.

This paper is concerned with the orthogonal polynomials. Upper and lower bounds of Legendre polynomials are obtained. Furthermore, entropies associated with discrete probability distributions is a topic considered in this paper. Bounds of the entropies which improve some previously known results are obtained in terms of inequalities. In order to illustrate the results obtained in this paper and to compare them with other results from the literature some graphs are provided.