From Fibonacci Sequence to the Golden Ratio

We consider the well-known characterization of the Golden ratio as limit of the ratio of consecutive terms of the Fibonacci sequence, and we give an explanation of this property in the framework of the Difference Equations Theory. We show that the Golden ratio coincides with this limit not because it is the root with maximum modulus and multiplicity of the characteristic polynomial, but, from a more general point of view, because it is the root with maximum modulus and multiplicity of a restricted set of roots, which in this special case coincides with the two roots of the characteristic polynomial. This new perspective is the heart of the characterization of the limit of ratio of consecutive terms of all linear homogeneous recurrences with constant coefficients, without any assumption on the roots of the characteristic polynomial, which may be, in particular, also complex and not real

On the definition of periodic decimal representations: An alternative point of view

Until the seventeenth century, rational numbers were represented as fractions. It was starting from that century, thanks to Simon Stevin, that for all practical purposes the use of decimal notation became widespread. Although decimal numbers are widely used, their organic development is lacking, while a vast literature propagates misconceptions about them, among both students and teachers. In particular, the case of the period 9 is especially interesting. In this paper, changing the point of view, we propose substituting the usual definitions of the periodic decimal representation of a rational number - the one obtained through long division introduced in the secondary school and the one obtained through the series concept for undergraduate level - with one which, instead of infinite progressions, uses a known property of fractions deducible from Euler\u27s Totient Theorem. The long division becomes a convenient algorithm for obtaining the decimal expansion of a rational number. The new definition overcomes the difficulties noted in the literature. An elementary proof of Euler\u27s Totient Theorem will be given depending only on Euclidean Division Theorem

The Napoleon’s Theorem: Synthesis and Related Investigations

We recall a synthetic-geometric demonstration of Napoleon Theorem,which makes use of the Fermat point. Some insights on the topic related toconvex Quadrilaterals will be made. Il Teorema di Napoleone: Sintesi e ApprofondimentiSi richiama una dimostrazione di tipo sintetico-geometrico del Teoremadi Napoleone, che fa uso del Punto di Fermat, e si presentano alcuni appro-fondimenti sul tema relativi ai Quadrilateri Convessi.Parole Chiave: Il Teorema di Napoleone; quadrilateri convessi, costruzioniiterate di poligoni

On an assumption of geometric foundation of numbers

In line with the latest positions of Gottlob Frege, this article puts forward the hypoth- esis that the cognitive bases of mathematics are geometric in nature. Starting from the geometry axioms of the Elements of Euclid, we introduce a geometric theory of propor- tions along the lines of the one introduced by Grassmann in Ausdehnungslehre in 1844. Assuming as axioms, the cognitive contents of the theorems of Pappus and Desargues, through their configurations, in an Euclidean plane a natural field structure can be iden- tified that reveals the purely geometric nature of complex numbers. Reasoning based on figures is becoming a growing interdisciplinary field in logic, philosophy and cognitive sciences, and is also of considerable interest in the field of education, moreover, recently, it has been emphasized that the mutual assistance that geometry and complex numbers give is poorly pointed out in teaching and that a unitary vision of geometrical aspects and calculation can be clarifying

The BciAi4SLA Project: Towards a User-Centered BCI

The brain–computer interfaces (BCI) are interfaces that put the user in communication with an electronic device based on signals originating from the brain. In this paper, we describe a proof of concept that took place within the context of BciAi4Sla, a multidisciplinary project involving computer scientists, physiologists, biomedical engineers, neurologists, and psychologists with the aim of designing and developing a BCI system following a user-centered approach, involving domain experts and users since initial prototyping steps in a design–test–redesign development cycle. The project intends to develop a software platform able to restore a communication channel in patients who have compromised their communication possibilities due to illness or accidents. The most common case is the patients with amyotrophic lateral sclerosis (ALS). In this paper, we describe the background and the main development steps of the project, also reporting some initial and promising user evaluation results, including real-time performance classification and a proof-of-concept prototype

REGISTRI: Regorafenib in first-line of KIT/PDGFRA wild type metastatic GIST: a collaborative Spanish (GEIS), Italian (ISG) and French Sarcoma Group (FSG) phase II trial

Wild type GIST; Biomarker; Clinical trialGIST de tipo salvaje; Biomarcador; Ensayo clínicoGIST de tipus salvatge, Biomarcador; Assaig clínicBackground Approximately 15% of adult GIST patients harbor tumors that are wild-type for KIT and PDGFRα genes (KP-wtGIST). These tumors usually have SDH deficiencies, exhibit a more indolent behavior and are resistant to imatinib. Underlying oncogenic mechanisms in KP-wtGIST include overexpression of HIF1α high IGFR signaling through the MAPK pathway or BRAF activating mutation, among others. As regorafenib inhibits these signaling pathways, it was hypothesized that it could be more active as upfront therapy in advanced KP-wtGIST. Methods Adult patients with advanced KP-wtGIST after central confirmation by NGS, naïve of systemic treatment for advanced disease, were included in this international phase II trial. Eligible patients received regorafenib 160 mg per day for 21 days every 28 days. The primary endpoint was disease control rate (DCR), according to RECIST 1.1 at 12 weeks by central radiological assessment. Results From May 2016 to October 2020, 30 patients were identified as KP-wtGIST by Sanger sequencing and 16 were confirmed by central molecular screening with NGS. Finally, 15 were enrolled and received regorafenib. The study was prematurely closed due to the low accrual worsened by COVID outbreak. The DCR at 12 weeks was 86.7% by central assessment. A subset of 60% experienced some tumor shrinkage, with partial responses and stabilization observed in 13% and 87% respectively, by central assessment. SDH-deficient GIST showed better clinical outcome than other KP-wtGIST. Conclusions Regorafenib activity in KP-wtGIST compares favorably with other tyrosine kinase inhibitors, especially in the SDH-deficient GIST subset and it should be taken into consideration as upfront therapy of advanced KP-wtGIST.GEIS, Bayer and SELNET. SELNET has received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement Nº. 825806

On generalised FC-groups in which normality is a transitive relation

We extend to soluble FC∗-groups, the class of generalised FC-groups introduced in [F. de Giovanni, A. Russo, G. Vincenzi, `Groups with restricted conjugacy classes', Serdica Math. J. 28(3) (2002), 241-254], the characterisation of finite soluble T-groups obtained recently in [G. Kaplan, `On T-groups, supersolvable groups and maximal subgroups', Arch. Math. 96 (2011), 19-25]

Are Two Riboses Better Than One? The Case of the Recognition and Activation of Adenosine Receptors

Traditionally, molecular recognition between the orthosteric site of adenosine receptors and their endogenous ligand occurs with a 1 : 1 stoichiometry. Inspired by previous mechanistic insights derived from supervised molecular dynamics (SuMD) simulations, which suggested an alternative 2 : 1 binding stoichiometry, we synthesized BRA1, a bis-ribosyl adenosine derivative, tested its ability to bind to and activate members of the adenosine receptor family, and rationalized its activity through molecular modeling