S-regular functions which preserve a complex slice

We study global properties of quaternionic slice regular functions (also called s-regular) defined on symmetric slice domains. In particular, thanks to new techniques and points of view, we can characterize the property of being one-slice preserving in terms of the projectivization of the vectorial part of the function. We also define a "Hermitian" product on slice regular functions which gives us the possibility to express the $*$-product of two s-regular functions in terms of the scalar product of suitable functions constructed starting from $f$ and $g$. Afterwards we are able to determine, under different assumptions, when the sum, the $*$-product and the $*$-conjugation of two slice regular functions preserve a complex slice. We also study when the $*$-power of a slice regular function has this property or when it preserves all complex slices. To obtain these results we prove two factorization theorems: in the first one, we are able to split a slice regular function into the product of two functions: one keeping track of the zeroes and the other which is never-vanishing; in the other one we give necessary and sufficient conditions for a slice regular function (which preserves all complex slices) to be the symmetrized of a suitable slice regular one.Comment: 23 pages, to appear in Annali di Matematica Pura e Applicat

∗*-exponential of slice-regular functions

According to [5] we define the $*$-exponential of a slice-regular function, which can be seen as a generalization of the complex exponential to quaternions. Explicit formulas for $\exp_*(f)$ are provided, also in terms of suitable sine and cosine functions. We completely classify under which conditions the $*$-exponential of a function is either slice-preserving or $\mathbb{C}_J$-preserving for some $J\in\mathbb{S}$ and show that $\exp_*(f)$ is never-vanishing. Sharp necessary and sufficient conditions are given in order that $\exp_*(f+g)=\exp_*(f)*\exp_*(g)$, finding an exceptional and unexpected case in which equality holds even if $f$ and $g$ do not commute. We also discuss the existence of a square root of a slice-preserving regular function, characterizing slice-preserving functions (defined on the circularization of simply connected domains) which admit square roots. Square roots of this kind of functions are used to provide a further formula for $\exp_{*}(f)$. A number of examples is given throughout the paper.Comment: 15 pages; to appear in Proceedings of the American Mathematical Societ

Applications of the Sylvester operator in the space of slice semi-regular functions

AbstractIn this paper we apply the results obtained in [3] to establish some outcomes of the study of the behaviour of a class of linear operators, which include the Sylvester ones, acting on slice semi-regular functions. We first present a detailed study of the kernel of the linear operator ℒf,g (when not trivial), showing that it has dimension 2 if exactly one between f and g is a zero divisor, and it has dimension 3 if both f and g are zero divisors. Afterwards, we deepen the analysis of the behaviour of the -product, giving a complete classification of the cases when the functions fv, gv and fvgv are linearly dependent and obtaining, as a by-product, a necessary and sufficient condition on the functions f and g in order their *-product is slice-preserving. At last, we give an Embry-type result which classifies the functions f and g such that for any function h commuting with f + g and f * g, we have that h commutes with f and g, too

∗*-Logarithm for Slice Regular Functions

In this paper, we study the (possible) solutions of the equation $\exp_{*}(f)=g$, where $g$ is a slice regular never vanishing function on a circular domain of the quaternions $\mathbb{H}$ and $\exp_{*}$ is the natural generalization of the usual exponential to the algebra of slice regular functions. Any function $f$ which satisfies $\exp_{*}(f)=g$ is called a $*$-logarithm of $g$. We provide necessary and sufficient conditions, expressed in terms of the zero set of the ``vector'' part $g_{v}$ of $g$, for the existence of a $*$-logarithm of $g$, under a natural topological condition on the domain $\Omega$. By the way, we prove an existence result if $g_{v}$ has no non-real isolated zeroes; we are also able to give a comprehensive approach to deal with more general cases. We are thus able to obtain an existence result when the non-real isolated zeroes of $g_{v}$ are finite, the domain is either the unit ball, or $\mathbb{H}$, or $\mathbb{D}$ and a further condition on the ``real part'' $g_{0}$ of $g$ is satisfied (see Theorem 6.19 for a precise statement). We also find some unexpected uniqueness results, again related to the zero set of $g_{v}$, in sharp contrast with the complex case. A number of examples are given throughout the paper in order to show the sharpness of the required conditions.Comment: 27 pages, 4 figure

A STEM Literacy Program for Students in Secondary-Tertiary Transition to Reduce the Gender Gap: a Focus on the Students' Perception

This study concerns the design and implementation of a STEM literacy program for 11th to 13th-grade high achieving students, mainly females. The program, funded by the Italian Ministry of Equal Opportunities, aims at reducing the gender gap in the STEM disciplines and at orienting students towards university studies. We carried out a qualitative analysis of the students’ perception in terms of (1) a-priori expectations about the STEM literacy program and (2) a-posteriori thoughts and reflections about the attended course. Our analysis shows that students aspiring to participate had strong motivations with respect to the program; moreover, most students who participated in the program displayed satisfaction and an increase of awareness about their learning. We put a specific focus on the mathematical sessions of the curriculum, involving students as designers of educational resources. Some differences between male and female students arose for what concerns the perception of the program and the awareness of the impact of the STEM literacy program on their own learning

INCB84344-201: Ponatinib and steroids in frontline therapy for unfit patients with Ph+ acute lymphoblastic leukemia

Tyrosine kinase inhibitors have improved survival for patients with Philadelphia chromosome-positive (Ph+) acute lymphoblastic leukemia (ALL). However, prognosis for old or unfit patients remains poor. In the INCB84344-201 (formerly GIMEMA LAL 1811) prospective, multicenter, phase 2 trial, we tested the efficacy and safety of ponatinib plus prednisone in newly diagnosed patients with Ph+ ALL 6560 years, or unfit for intensive chemotherapy and stem cell transplantation. Forty-four patients received oral ponatinib 45 mg/d for 48 weeks (core phase), with prednisone tapered to 60 mg/m2/d from days-14-29. Prophylactic intrathecal chemotherapy was administered monthly. Median age was 66.5 years (range, 26-85). The primary endpoint (complete hematologic response [CHR] at 24 weeks) was reached in 38/44 patients (86.4%); complete molecular response (CMR) in 18/44 patients (40.9%) at 24 weeks. 61.4% of patients completed the core phase. As of 24 April 2020, median event-free survival was 14.31 months (95% CI 9.30-22.31). Median overall survival and duration of CHR were not reached; median duration of CMR was 11.6 months. Most common treatment-emergent adverse events (TEAEs) were rash (36.4%), asthenia (22.7%), alanine transaminase increase (15.9%), erythema (15.9%), and \u3b3-glutamyltransferase increase (15.9%). Cardiac and vascular TEAEs occurred in 29.5% (grade 653, 18.2%) and 27.3% (grade 653, 15.9%), respectively. Dose reductions, interruptions, and discontinuations due to TEAEs occurred in 43.2%, 43.2%, and 27.3% of patients, respectively; 5 patients had fatal TEAEs. Ponatinib and prednisone showed efficacy in unfit patients with Ph+ ALL; however, a lower ponatinib dose may be more appropriate in this population. This trial was registered at www.clinicaltrials.gov as #NCT01641107

One parameter groups of volume preserving automorphisms of C2C^2

Si esaminano i gruppi ad un parametro nel gruppo degli automorfismi polinomiali di C$^{2}$ e nel gruppo degli shears, provando che sono coniugati a gruppi a un parametro nel gruppo degli automorfismi affini di C$^{2}$ o nel gruppo degli automorfismi elementari; da ciò si deducono risultati sul comportamento asintotico del gruppo ad un parametro, sui suoi punti periodici e sui suoi punti fissi.In this work we study the one-parameter groups in the group of all polynomial automorphisms of C$^{2}$ and in the group of all shears. We prove that any such one-parameter group is conjugated to a one-parameter group contained either in the group of all affine automorphisms of C$^{2}$ or in the group of elementary automorphisms. This implies some results on the asymptotic behaviour of the one-parameter group, on its periodic points and on its fixed points