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Photoredox reactions involving allyl and vinyl nucleophiles in the presence of nickel enable precise control over diastereoselectivity and enantioselectivity, without the need for noble metals or costly and highly reactive stoichiometric reductants. More information can be found in the review article by P. G. Cozzi and co‐workers.

Dual photoredox catalysis has revolutionized the field of cross‐coupling reactions, enabling the discovery of numerous highly efficient reactions. This breakthrough is attributed to the exceptional combination of nickel catalysis with photoredox catalysis. Nickel exhibits both oxidative addition and reductive elimination processes, and a wide range of oxidation states (ranging from I to IV) accessible within a single catalytic cycle. Furthermore, nickel complexes are capable of catalyzing various processes through radical mechanisms. The latest feature has proven to be incredibly potent in facilitating the formation of new C−C and C−X bonds (X=H, O, S, N). The powerful combination of photoredox and nickel catalysis reveals an expansive domain of unexplored possibilities. It offers unparalleled opportunities for improving reactions and exploring innovative pathways. Under photoredox conditions, it is possible to form nucleophilic reactive organometallic intermediates, useful in reaction with electrophiles. We have devoted a research program towards the rediscovery and use of organometallic reagents, introduced by Corey, Hegedus, and Semmelack many years ago. The results unveiled the extraordinary capabilities of photoredox catalysis, enabling the creation and efficient utilization of potent nucleophilic organometallic reagents under mild conditions, free from the need for strong bases or stoichiometric metal reductants. Nickel photoredox reactions with allyl and vinyl nucleophiles allowed diastereo and enantio control, in absence of metals or expensive and pyrophoric stoichiometric reductants.

A practical and effective photoredox propargylation of aldehydes promoted by 10 mol % of [Cp2TiCl2] is presented. No stoichiometric metals or scavengers are used for the process. A catalytic amount of the cheap and simply prepared organic dye 3DPAFIPN is used as the reductant for titanium. The reaction displayed a broad scope, and no traces of allenyl isomers were detected for simple propargyl bromide, whereas mixtures of propargyl and allenyl isomers were observed for substituted propargyl bromides.

In this paper we identify certain peculiar systems of 2 discrete-timeevolution equations,which are algebraically solvable. Here lis the “discrete-time” independentvariable taking integervalues (l= 0, 1, 2, . . .), xn≡ xn(l) are 2 dependent variables, and are the corresponding 2 updatedvariables. In a previous paper the 2 functions F(n)(x1, x2), n= 1, 2, were defined as follows: F(n)(x1, x2) = P2(xn, xn+1), n= 1, 2 mod[2], with P2(x1, x2) a specific second-degree homogeneous polynomialin the 2 (indistinguishable!) dependent variables x1(l) and x2(l). In the present paper we further clarify some aspects of that model and we present its extension to the case when a specific homogeneous function of arbitrary(integer) degree k(hence a polynomial of degree kwhen k> 0) in the 2 dependent variables x1(l) and x2(l).

In earlier work, a Hamiltonian describing the classicalmotion of a particle moving in two dimensions under the combined influence of a perpendicular magnetic field and of a damping force proportional to the particle velocity, was indicated. Here we derive the quantumpropagator for the Hamiltonian in different representations, one corresponding to momentum space, the other to position, and the third to a natural choice of “velocity” variables. We call attention to the following noteworthy fact: the Hamiltonian contains three parameters which do not in any way influence the motion of the position of the particle. However, at the quantum level, the propagator, even in the position representation, depends in an intricate way on these classically irrelevant parameters. This creates considerable doubt as to the validity of such a quantization procedure, as the physical results predicted differ for various Hamiltonians, all of which describe the dissipative dynamics equally well.

The evolution equations mentioned in the title of this paper read as follows:where ℓis the “discrete-time” independentvariable taking integervalues (ℓ= 0,1,2, … ), xn ≡ xn(ℓ) are the 2 dependent variables, , and the 2 functions P(n)(x1, x2), n= 1,2, are 2 polynomialsin the 2 dependent variables x1(ℓ) and x2(ℓ). The results reported in this paper have been obtained by an appropriate modification of a recently introduced technique to obtain analogous results in continuous-time t—in which case xn≡ xn(t) and the above recursion relations are replaced by first-order ODEs. Their potential interest is due to the relevance of this kind of evolution equations in various applicative contexts.

The evolution equations mentioned in the title of this paper read as follows:

In this paper we identify certain peculiar systems of 2 discrete-timeevolution equations,

In earlier work, a Hamiltonian describing the classicalmotion of a particle moving in two dimensions under the combined influence of a perpendicular magnetic field and of a damping force proportional to the particle velocity, was indicated. Here we derive the quantumpropagator for the Hamiltonian in different representations, one corresponding to momentum space, the other to position, and the third to a natural choice of “velocity” variables. We call attention to the following noteworthy fact: the Hamiltonian contains three parameters which do not in any way influence the motion of the position of the particle. However, at the quantum level, the propagator, even in the position representation, depends in an intricate way on these classically irrelevant parameters. This creates considerable doubt as to the validity of such a quantization procedure, as the physical results predicted differ for various Hamiltonians, all of which describe the dissipative dynamics equally well.

As is well-known, any ordinary differential equation in one dimension can be cast as the Euler—Lagrange equation of an appropriate Lagrangian. Additionally, if the initial equation is autonomous, the Lagrangian can always be chosen to be time-independent. In two dimensions, however, the situation is more complex, and there exist systems of ODEs which cannot be described by any Lagrangian. In this paper we display Hamiltonians which describe the behaviour of a charged particle moving in a plane under the combined influence of a constant electric field (in the plane) and a constant magnetic field (orthogonal to the plane) as well as a friction force proportional to the velocity (“cyclotron with friction”).

As is well-known, any ordinary differential equation in one dimension can be cast as the Euler–Lagrange equation of an appropriate Lagrangian. Additionally, if the initial equation is autonomous, the Lagrangian can always be chosen to be time-independent. In two dimensions, however, the situation is more complex, and there exist systems of ODEs which cannot be described by any Lagrangian. In this paper we display Hamiltonians which describe the behaviour of a charged particle moving in a plane under the combined influence of a constant electric field (in the plane) and a constant magnetic field (orthogonal to the plane) as well as a friction force proportional to the velocity (“cyclotron with friction”).

We use previous results concerning the time evolution of the zeros xn(t) of time-dependent polynomials pN(z;t) or entire functions F(z;t) of the complex variable z, in order to identify lots of nonlinearly-coupled, finite or infinite, systems of Ordinary Differential Equations the solutions of which feature remarkable Diophantine properties.

We use previous results concerning the time evolution of the zeros xn(t) of time-dependent polynomials pN(z;t) or entire functions F(z;t) of the complex variable z, in order to identify lots of nonlinearly-coupled, finite or infinite, systems of Ordinary Differential Equations the solutions of which feature remarkable Diophantine properties.

A novel, remarkably neat, differential algorithm is introduced, which is suitable to evaluate allthe zeros of a genericpolynomial of arbitrarydegree N.

A simple application of a neat formula relating the time evolution of the N zerosof a (monic) time-dependent polynomial of degree Nin the complexvariable wto the time evolution of its N coefficientsallows to identify integrable Hamiltonian N-body problems in the plane featuring N arbitrary functions, the equations of motions of which are of Newtoniantype: accelerations equal forces nonlinearly dependent on the coordinates of the Nparticle. The motions generally take place in the complex z-plane, or, equivalently, in the Cartesian xy-plane with z= x+ iy. It is also easy to identify qualitative features of special subclasses of these models, for instance cases in which allthe motions starting from an arbitrary realset of initial data are confinedand multiply periodic. It is also indicated how to generate from these models hierarchies of analogous models with analogous properties.

A technique to identify new C-integrable and S-integrable systems of nonlinear partial differential equations is reported, with two representative examples displayed and tersely discussed.

A technique to identify new C-integrable and S-integrable systems of nonlinear partial differential equations is reported, with two representative examples displayed and tersely discussed.

A simple application of a neat formula relating the time evolution of the Nzeros of a (monic) time-dependent polynomial of degree Nin the complexvariable wto the time evolution of its N coefficientsallows to identify integrable Hamiltonian N-body problems in the plane featuring N arbitrary functions, the equations of motions of which are of Newtoniantype: accelerations equal forces nonlinearly dependent on the coordinates of the Nparticle. The motions generally take place in the complex z-plane, or, equivalently, in the Cartesian xy-plane with z = x+iy. It is also easy to identify qualitative features of special subclasses of these models, for instance cases in which allthe motions starting from an arbitrary realset of initial data are confined and multiply periodic. It is also indicated how to generate from these models hierarchies of analogous models with analogous properties.

A novel, remarkably neat, differential algorithm is introduced, which is suitable to evaluate allthe zeros of a genericpolynomial of arbitrarydegree N.

New solvabledynamical systems are identified and the properties of their solutions are tersely discussed.

Let pN(z; t) be a (monic) time-dependent polynomial of arbitrary degree Nin z,and let zn≡ zn(t) be its Nzeros: . In this paper we report a convenient expression of the k-th time-derivative of the zero zn(t). This formula plays a key role in the identification of classes of solvabledynamical systems describing the motion of point-particles moving in the complex z-plane while nonlinearly interacting among themselves; one such example, featuring many arbitraryparameters, is reported, including its variation describing the motion of many particles moving in the realCartesian xy-plane and interacting among themselves via rotation-invariantNewtonian equations of motion (”accelerations equal forces”).

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 complexplane. These models are solvable, namely their configuration at any time can be obtained from the initial data by algebraicoperations, amounting to the determination of the zerosof a known time-dependent polynomial in the independent variable z. Some of these models are multiply periodic, isochronousor asymptotically isochronous; others display scatteringphenomena.

AbstractA N-body problem “of goldfish type” is introduced, the Newtonian (“acceleration equal force”) equations of motion of which describe the motion of Npointlike unit-mass particles moving in the complex z-plane. The model—for arbitrary N—is solvable, namely its configuration (positions and velocities of the N“particles”) at any later time tcan be obtained from its configuration at the initial time by algebraic operations. It features specific nonlinear velocity-dependent many-body forces depending on N2arbitrary (complex) coupling constants. Sufficient conditions on these constants are identified which cause the model to be isochronous—so that all its solutions are then periodic with a fixed period independent of the initial data. A variant with twice as many arbitrary coupling constants, or even more, is also identified.

A new solvable many-body problem of goldfish type is introduced and the behavior of its solutions is tersely discussed.

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 complexplane. These models are solvable, namely their configuration at any time can be obtained from the initial data by algebraicoperations, amounting to the determination of the zerosof a known time-dependent polynomial in the independent variable z. Some of these models are multiply periodic, isochronousor asymptotically isochronous; others display scatteringphenomena.

New solvabledynamical systems are identified and the properties of their solutions are tersely discussed.

Let pN(z;t) be a (monic) time-dependent polynomial of arbitrary degree Nin z,and let zn≡ zn(t) be its Nzeros: pN(z;t) = . In this paper we report a convenient expression of the k-th time-derivative (t) of the zero zn(t). This formula plays a key role in the identification of classes of solvabledynamical systems describing the motion of point-particles moving in the complex z-plane while nonlinearly interacting among themselves; one such example, featuring many arbitraryparameters, is reported, including its variation describing the motion of many particles moving in the realCartesian xy-plane and interacting among themselves via rotation-invariantNewtonian equations of motion (“accelerations equal forces”).

A N-body problem “of goldfish type” is introduced, the Newtonian (“acceleration equal force”) equations of motion of which describe the motion of Npointlike unit-mass particles moving in the complex z-plane. The model—for arbitrary N—is solvable, namely its configuration (positions and velocities of the N“particles”) at any latertime tcan be obtained from its configuration at the initialtime by algebraicoperations. It features specific nonlinear velocity-dependent many-body forces depending on N2arbitrary (complex) coupling constants. Sufficient conditions on these constants are identified which cause the model to be isochronous—so that allits solutions are then periodic with a fixed period independent of the initial data. A variant with twiceas many arbitrary coupling constants, or even more, is also identified.

A new solvable many-body problem of goldfish type is introduced and the behavior of its solutions is tersely discussed.

Certain nonlinearly-coupled systems of Ndiscrete-time evolution equations are identified, which can be solved by algebraic operations; and some remarkable Diophantine findings are thereby obtained. These results might be useful to test the accuracy of numerical routines yielding the Nroots of polynomials of arbitrary degree N.

Certain nonlinearly-coupled systems of Ndiscrete-time evolution equations are identified, which can be solved by algebraic operations; and some remarkable Diophantine findings are thereby obtained. These results might be useful to test the accuracy of numerical routines yielding the Nroots of polynomials of arbitrary degree N.

The development of modern technologies casts a new light on the natural products as an invaluable source of lead compounds that could guide drug discovery. Cancer stem cells are a subpopulation of cancer cells with a high clonogenic capacity and the ability to reform the parental tumours upon transplantation. They have been proposed to drive tumorigenesis and metastases. In this review, we present the ability of forty-nine different natural products to influence the biology of cancer stem cells.

A system of algebraic equations satisfied by the zeros of the sum of three polynomials are reported.

The class of solvable N-body problems of “goldfish” type has been recently extended by including (the additional presence of) three-bodyforces. In this paper we show that the equilibria of some of these systems are simply related to the Nroots xnof the polynomial equation , where is the Jacobi polynomial of order N, the parameters αand βare related to parameters of the N-body problem (which can be arbitrarilyassigned) and wis an arbitrarynumber. By investigating the behavior of these solvable N-body systems in the infinitesimal neighborhood of these equilibria, the eigenvalues associated to certain N× Nmatrices explicitly given in terms of the Nnumbers xn(and of additional free parameters of the N-body problem) are explicitly identified. In some cases—corresponding to isochronous N-body problems—these findings have a Diophantineconnotation, inasmuch as these eigenvalues are then rationalnumbers.

The class of solvable N-body problems of “goldfish” type has been recently extended by including (the additional presence of) three-bodyforces. In this paper we show that the equilibria of some of these systems are simply related to the Nroots xnof the polynomial equation (x)= w, where (x) is the Jacobi polynomial of order N, the parameters αand βare related to parameters of the N-body problem (which can be arbitrarilyassigned) and wis an arbitrarynumber. By investigating the behavior of these solvable N-body systems in the infinitesimal neighborhood of these equilibria, the eigenvalues associated to certain N × Nmatrices explicitly given in terms of the Nnumbers xn(and of additional free parameters of the N-body problem) are explicitly identified. In some cases—corresponding to isochronous N-body problems—these findings have a Diophantineconnotation, inasmuch as these eigenvalues are then rationalnumbers.

A system of algebraic equations satisfied by the zeros of the sum of three polynomials are reported.

Some properties—including relations having a Diophantine character—of the zeros of the sum of two polynomials are reported.

Certain techniques to obtain properties of the zeros of polynomials satisfying second-order ODEs are reviewed. The application of these techniques to the classical polynomials yields formulas which were already known; new are instead the formulas for the zeros of the (recently identified, and rather explicitly known) polynomials satisfying a (recently identified) second-order ODE which features many free parameters and onlypolynomial solutions. Some of these formulas have a Diophantine connotation. Techniques to manufacture infinite sequences of second-order ODEs featuring onlypolynomial solutions are also reported.

Certain techniques to obtain properties of the zeros of polynomials satisfying second-order ODEs are reviewed. The application of these techniques to the classical polynomials yields formulas which were already known; new are instead the formulas for the zeros of the (recently identified, and rather explicitly known) polynomials satisfying a (recently identified) second-order ODE which features many free parameters and onlypolynomial solutions. Some of these formulas have a Diophantine connotation. Techniques to manufacture infinite sequences of second-order ODEs featuring onlypolynomial solutions are also reported.

AbstractRecently a solvable N-body problem featuring several free parameters has been investigated, and conditions on these parameters have been identified which guarantee that this system is isochronous(allits solutions are periodicwith a fixedperiod) and that it possesses equilibria. The Ncoordinates characterizing the equilibrium configurations are in some cases explicitly known, in others coincide with the Nzeros of certain para-Jacobi polynomials or are arbitrary numbers. In the present paper the behavior of this N-body system in the immediate vicinity of its equilibria is studied, and Diophantinerelations satisfied by the Ncoordinates are thereby identified.

Recently a solvable N-body problem featuring several free parameters has been investigated, and conditions on these parameters have been identified which guarantee that this system is isochronous(allits solutions are periodicwith a fixedperiod) and that it possesses equilibria. The Ncoordinates characterizing the equilibrium configurations are in some cases explicitly known, in others coincide with the Nzeros of certain para-Jacobi polynomials or are arbitrary numbers. In the present paper the behavior of this N-body system in the immediate vicinity of its equilibria is studied, and Diophantinerelations satisfied by the Ncoordinates are thereby identified.

The stability of a photocatalyst under irradiation is important in photoredox applications. In this work, we investigated the stability of a thermally activated delayed fluorescence (TADF) photocatalyst {3DPAFIPN [2,4,6-tris(diphenylamino)-5-fluoroisophthalonitrile]}, recently employed in photoredox-mediated processes, discovering that in the absence of quenchers the chromophore is unstable and is efficiently converted by irradiation with visible light into another species based on the carbazole-1,3-dicarbonitrile moiety. The new species obtained is itself a TADF emitter and finds useful applications in photoredox transformations. At the excited state, it is a strong reductant and was efficiently applied to cobalt-mediated allylation of aldehydes, whereas other TADFs (4CzIPN and 3DPAFIPN) failed to promote efficient photocatalytic cycles.

Mechanisms are elucidated underlying the existence of dynamical systems whose genericsolutions approach asymptotically(at large time) isochronousevolutions: alltheir dependent variables tend asymptoticallyto functions periodicwith the samefixed period. We focus on two such mechanisms, emphasizing their generality and illustrating each of them via a representative example. The first example belongs to a recently discovered class of integrableindeed solvablemany-body problems. The second example consists of a broad class of (generally nonintegrable) models obtained by deforming appropriately the well-known (integrableand isochronous) many-body problem with inverse-cube two-body forces and a one-body linear (“harmonic oscillator”) force.

AbstractMechanisms are elucidated underlying the existence of dynamical systems whose genericsolutions approach asymptotically(at large time) isochronousevolutions: alltheir dependent variables tend asymptoticallyto functions periodicwith the samefixed period. We focus on two such mechanisms, emphasizing their generality and illustrating each of them via a representative example. The first example belongs to a recently discovered class of integrableindeed solvablemany-body problems. The second example consists of a broad class of (generally nonintegrable) models obtained by deforming appropriately the well-known (integrableand isochronous) many-body problem with inverse-cube two-body forces and a one-body linear (“harmonic oscillator”) force.

An isochronousdynamical system is characterized by the existence of an opendomain of initial data such that allmotions evolving from it are completely periodicwith a fixedperiod (independent of the initial data). Taking advantage of a recently introduced trick, two new Hamiltonianclasses of such systems are identified.

AbstractAn isochronousdynamical system is characterized by the existence of an opendomain of initial data such that allmotions evolving from it are completely periodicwith a fixedperiod (independent of the initial data). Taking advantage of a recently introduced trick, two new Hamiltonianclasses of such systems are identified.

AbstractThe (Hamiltonian, rotation- and translation-invariant) “goldfish” N-body problem in the plane is characterized by the Newtonian equations of motionwritten here in their complex version, entailing the identification of the real“physical” plane with the complexplane. In this paper we exhibit in completely explicit form the solution of the initial-value problem for this N-body model in the special case in which the two-body interaction only acts among “nearest neighbors” (namely, only among particles whose labels differ by one unit: an,m= 0 unless |n − m|= 1) and the corresponding coupling constants all equal minus one half, an,n+1= an+1,n= −1/2, n= 1, 2, ..., N −1. This result implies that, if ωis a real nonvanishing constant, say, without loss of generality, ω> 0, then allthe solutions of this N-body model are completely periodic indeed isochronouswith period T= 2π / ω.An analogous conclusion holds as well for the model in which also the first and last particle interact with the same coupling constant, namely a1,N= aN,1= −1/2 (rather than vanishing).

AbstractIn two previous papers the quantization was discussed of three one-degree-of-freedom Hamiltonians featuring a constant c,the value of which does not influence at all the corresponding classical dynamics (which is characterized by isochronoussolutions, allof them periodic with period 2π: “nonlinear harmonic oscillators”), but turned instead out to influence, possibly quite nontrivially, the results in the quantized case. The quantization of two analogous Hamiltonian systems is discussed in this paper. The outcome is analogous, if perhaps even more clear cut, than in the previous three cases; it also confirms that different quantized spectra may be obtained if quantization is performed beforeor aftera (nonlinear) canonical transformation.