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Consider the recursive relation generating a new positive integer $n_{\ell +1}$ from the positive integer $n_{\ell }$ according to the following simple rules: if the integer $n_{\ell }$ is odd, $n_{\ell +1}=3n_{\ell }+1$; if the integer $n_{\ell }$ is even, $n_{\ell +1}=n_{\ell }/2$. The so-called Collatz conjecture states that, starting from any positive integer $N$, the recursion characterized by the continued application of these rules ends up in the cycle $4,$ $2,1$. This conjecture is generally believed to be true (on the basis of extensive numerical checks), but it is as yet unproven. In this paper -- based on the assumption that the Collatz conjecture is indeed true -- we present a quite simple extension of it, which entails the possibility to divide all natural numbers into $3$ disjoint classes, to each of which we conjecture -- on the basis of (not very extensive) numerical checks -- $1/3$ of all natural numbers belong; or, somewhat equivalently, to $2$ disjoint classes, to which we conjecture that respectively $1/3$ and $2/3$ of all natural numbers belong.

In this short communication we introduce a rather simple autonomous system of 2 nonlinearly-coupled first-order Ordinary Differential Equations (ODEs), whose initial-values problem is explicitly solvable by algebraic operations. Its ODEs feature 2 right-hand sides which are the ratios of 2 homogeneous polynomials of first degree divided by the same homogeneous polynomial of second degree. The model features only 4 arbitrary parameters. We also report its isochronous variant featuring 4 nonlinearly-coupled first-order ODEs in 4 dependent variables, featuring 9 arbitrary parameters., 3 pages

The main purpose of my PhD was the combination of the principles of transition metal catalysis with photoredox catalysis. We focused our attention on the development of novel dual catalytic protocols for the functionalization of carbonyl compounds through the generation of transient nucleophilic organometallic species. Specifically, we focused on the development of new methodologies combining photoredox catalysis with titanium and nickel in low oxidation state. Firstly, a Barbier-type allylation of aromatic and aliphatic aldehydes –catalytic in titanium– in the presence of a blue photon-absorbing dye was developed. Parallelly, we were pleased to observe that the developed methodology could also be extended to the propargylation of aldehydes under analogous conditions. After an extensive re–optimization of all the reaction parameters, we developed an enantioselective and diastereoselective pinacol coupling of aromatic aldehydes promoted by non-toxic, cheap and easy to synthetize titanium complexes. The key feature, that allows the complete (dia)stereocontrol played by titanium, is the employment of a red-absorbing organic dye. The tailored (photo)redox properties of the red-absorbing organic dye [nPr–DMQA+][BF4–] promote the selective reduction of Ti(IV) to Ti(III). Moreover, even if the major contribution in dual photoredox and nickel catalysis is devoted to the realization of cross-coupling-type reactions, we wanted to evaluate different possible scenarios. Our focus was on the possibility of exploiting intermediates arising from the oxidative addition of nickel complexes as transient nucleophilic species. The first topic considered regarded the possibility to perform allylation of aldehydes by dual photoredox and nickel catalysis. In the first instance, a non–stereocontrolled version of the reaction was presented. Finally, after a long series of drastic modification of the reaction conditions, a highly enantioselective variant of the protocol was also reported. All the reported methodologies are supported by careful photophysical analysis and, in some cases, computational modelling.

A system of 4 nonlinearly-coupled Ordinary Differential Equations has been recently introduced to investigate the evolution of human respiratory virus epidemics. In this paper we point out that some explicit solutions of that system can be obtained by algebraic operations, provided the parameters of the model satisfy certain constraints., Comment: 10 pages

The generic monic polynomial of degree N features N a priori arbitrary coefficients $c_m$ and N zeros $z_n$. In this paper we limit consideration to $N = 8$ and $N = 9$. We show that if the $N$ -- a priori arbitrary -- coefficients $c_m$ of these polynomials are appropriately defined -- as it were, a posteriori -- in terms of 6 arbitrary parameters, then the $N$ roots of these polynomials can be explicitly computed in terms of radicals of these 6 parameters. We also report the constraints on the N coefficients $c_m$ implied by the fact that they are so defined in terms of 6 arbitrary parameters; as well as the explicit determination of these 6 parameters in terms of the N coefficients $c_m$., 5 pages

In this short paper we identify special systems of (an arbitrary number) N of first-order Difference Equations with nonlinear homogeneous polynomials of arbitrary degree M in their right-hand sides, which feature very simple explicit solutions. A novelty of these findings is to consider special systems characterized by constraints involving both their parameters and their initial data, 3 Pages. arXiv admin note: substantial text overlap with arXiv:2106.06634

The asymmetric May-Leonard model is a prototypical system of 3 nonlinearly coupled first-order Ordinary Differential Equations with second-degree polynomial right-hand sides. In this short paper we identify a class of special solutions of this system which do not seem to have been previously advertised in spite of their rather elementary character., Comment: 3 pages

The generic monic polynomial of sixth degree features 6 a priori arbitrary coefficients. We show that if these 6 coefficients are appropriately defined in two different ways|in terms of 5 arbitrary parameters, then the 6 roots of the corresponding polynomial can be explicitly computed in terms of radicals of these parameters. We also report the 2 constraints on the 6 coefficients of the polynomial implied by the fact that they are so defined in terms of 5 arbitrary parameters; as well as the explicit determination of these 5 parameters in terms of the 6 coefficients of the sextic polynomial., Comment: 4 pages

In this paper we report a few examples of algebraically solvable dynamical systems characterized by 2 coupled Ordinary Differential Equations which read as follows: x_n = P(n) (x1, x2) , n = 1, 2 , with P(n) (x1, x2) specific polynomials of relatively low degree in the 2 dependent variables x1 = x1 (t) and x2 = x2 (t) . These findings are obtained via a new twist of a recent technique to identify dynamical systems solvable by algebraic operations, themselves explicitly identified as corresponding to the time evolutions of the zeros of polynomials the coefficients of which evolve according to algebraically solvable (systems of) evolution equations., 11 pages, Sabmitted 27 August 2018, to be published as a chapter in a collective book to celebrate the 65th birthdate of Emma Previato (in press)

Recently new solvable systems of nonlinear evolution equations -- including ODEs, PDEs and systems with discrete time -- have been introduced. These findings are based on certain convenient formulas expressing the $k$-th time-derivative of a root of a time-dependent monic polynomial in terms of the $k$-th time-derivative of the coefficients of the same polynomial and of the roots of the same polynomial as well as their time-derivatives of order less than $k$. These findings were restricted to the case of generic polynomials without any multiple root. In this paper some of these findings -- those for $k=1$ and $k=2$ -- are extended to polynomials featuring one double root; and a few representative examples are reported of new solvable systems of nonlinear evolution equations.

Novel classes of dynamical systems are introduced, including many-body problems characterized by nonlinear equations of motion of Newtonian type ("acceleration equals forces") which determine the motion of points in the complex plane. These models are solvable, namely their configuration at any time can be obtained from the initial data by algebraic operations, amounting to the determination of the zeros of a known time-dependent polynomial in the independent variable z. Some of these models are multiply periodic, isochronous or asymptotically isochronous; others display scattering phenomena., Comment: arXiv admin note: text overlap with arXiv:1510.05017

We define the generalized hypergeometric polynomial of degree N in terms of the generalized hypergeometric function that depends on p parameters a_1, ..., a_p and q parameters b_1, ..., b_q. The parameters are "generic", possibly complex, numbers. In this paper we obtain a set of N nonlinear algebraic equations satisfied by the N zeros z_n of this polynomial. We moreover manufacture an NxN matrix L in terms of the 1+p+q parameters N, a_j, b_k characterizing this polynomial, and of its N zeros z_n. We show that the matrix L features N eigenvalues that depend only on the q parameters b_k, implying that this matrix is isospectral for the variations of the p parameters a_j. These eigenvalues are integer (or rational) numbers if the q parameters b_k are themselves integer (or rational) numbers: a nontrivial diophantine property., arXiv admin note: text overlap with arXiv:1504.01748

We introduce a monic polynomial p_N(z) of degree N whose coefficients are the zeros of the N-th degree Hermite polynomial. Note that there are N! such different polynomials p_N(z), depending on the ordering assignment of the N zeros of the Hermite polynomial of order N. We construct two NxN matrices M_1 and M_2 defined in terms of the N zeros of the polynomial p_N(z). We prove that the eigenvalues of M_1 and M_2 are the first N integers respectively the first N squared-integers, a remarkable isospectral and Diophantine property. The technique whereby these findings are demonstrated can be extended to other named polynomials.

This thesis is focused on the development of new synthetic processes for the production of already known Active Pharmaceutical Ingredients (APIs). The research work was performed in the laboratories of Dipharma Francis srl, a company which operates in the generic drug market. The launch of the generic version of a drug in the market often leads to lowering of product prices for both the branded product and the generic ones. For this reason, the process adopted to produce an API has to be innovative, efficient, safe and, of course, cheaper than the existing ones, in order to be competitive in the market. During my Ph.D. I worked on the synthesis of some APIs, in particular here I report the feasibility and development studies of an alternative process to produce silodosin. Silodosin is an API used as a treatment for the symptoms of Benign Prostatic Hyperplasia (BPH). In order to establish the synthetic strategy and to outline our freedom to operate, an accurate survey of the whole patent literature about silodosin has been done. During the feasibility study several synthetic approaches have been tried in order to functionalise indoline at positions 5 and 7. A copper(I) catalysed C-arylation reaction and a regioselective electrophilic aromatic substitution revealed to be the best choices to introduce respectively substituents in position 5 and 7 of indoline. The synthesis continued with a diastereoselective reductive amination which after crystallisation yielded optically pure amine intermediate that is the key intermediate for the synthesis of silodosin and ended converting amine intermediate into Silodosin using already reported procedures. Our new process to prepare silodosin starting from commercially available and cheap indoline consists of 11 steps. The whole synthetic route has been performed in gram scale using only 4 purifications of key intermediates. Silodosin has been finally obtained in a 10% overall yield, with a purity greater than 99% measured by HPLC and an optical purity greater than 99% measured by HPLC on chiral stationary phase.

In this paper we provide properties -- which are, to the best of our knowledge, new -- of the zeros of the polynomials belonging to the q-Askey scheme. These findings include Diophantine relations satisfied by these zeros when the parameters characterizing these polynomials are appropriately restricted., This version differs from version v1 by one sentence at the end of section 4.1. arXiv admin note: text overlap with arXiv:1407.3379

Recently – via a simple trick, amounting essentially to a change of independent, and possibly as well of dependent, variables – the possibility has been noted to modify a quite general evolution system so that the modified system possess a lot of completely periodic, indeed isochronous, solutions. Generally these isochronous solutions emerge out of an open domain of initial data having full dimensionality in the space of initial data. And many of the isochronous systems obtained in this manner seem rather interesting. In this paper these developments are reviewed, mainly in the context of dynamical systems (systems of ODEs – in particular, systems interpretable as many-body problems), and some specific examples are discussed in detail, including an analysis of the transition (to motions with higher periods, or aperiodic, or perhaps chaotic) occurring when the initial data get outside of the region producing isochronous motions. The applicability of this approach in the context of nonlinear evolution PDEs is also outlined.

International audience; The Hamiltonian system characterized by the Newtonian equations of motion x¨n=∑Nm=1,m≠nx˙nx˙mw(xn−xm), w(x)=2 ζ(x)+λx, where ζ(x)≡ζ(x|w,w′) is the Weierstrass zeta function and λ is an arbitrary constant, is solvable provided there holds the single constraint (on the initial data) ∑Nn=1x˙n(t)=∑Nn=1x˙n(0)=0.