Recursion operators, higher-order symmetries and superintegrability in quantum mechanics

A connection between the theory of superintegrable quantum-mechanical systems, which admit a maximal number of integrals of motion, and the standard Lie group theory is established. It is shown that the flows generated by first- and second-order Lie symmetries of the bidimensional Schrödinger equation can be classified and interpreted as quantum-mechanical operators which commute with integrable or superintegrable Hamiltonians. In this way, all known superintegrable potentials in the plane are naturally obtained and slightly more general integrals of motion are found.     

On Appell sequences of polynomials of Bernoulli and Euler type

A construction of new sequences of generalized Bernoulli polynomials of first and second kind is proposed. These sequences share with the classical Bernoulli polynomials many algebraic and number theoretical properties. A class of Euler-type polynomials is also presented.     

Multiple-scale analysis of dynamical systems on the lattice

We propose a new approach to the multiple-scale analysis of difference equations, in the context of the finite operator calculus. We derive the transformation formulae that map any given dynamical system, continuous or discrete, into a rescaled discrete system, by generalizing a classical result due to Jordan. Under suitable analytical hypotheses on the function space we consider, the rescaled equations are of finite order. Our results are applied to the study of multiple-scale reductions of dynamical systems, and in particular to the case of a discrete nonlinear harmonic oscillator.     

Lie Symmetries and Weak Transversality

We discuss the notion of weak transversality and clarify its role in the context of Lie group theory with several examples. In particular, we present some new results concerning models of fluid dynamics.     

Convulsant alkaloids from Dioscorea dumetorum

Investigation of the West African medicinal yam $\text{Dioscorea dumetorum}$ Pax has resulted in the isolation and stereostructural elucidation of the previously reported dihydrodioscorine as well as the new alkaloid dumetorine.     

Integrable maps from Galois differential algebras,Borel transforms and number sequences

A new class of integrable maps, obtained as lattice versions of polynomial dynamical systems is introduced. These systems are obtained by means of a discretization procedure that preserves several analytic and algebraic properties of a given differential equation, in particular symmetries and integrability (see Tempesta, 2010 [40]). Our approach is based on the properties of a suitable Galois differential algebra, that we shall call a Rota algebra. A formulation of the procedure in terms of category theory is proposed. In order to render the lattice dynamics confined, a Borel regularization is also adopted. As a byproduct of the theory, a connection between number sequences and integrability is discussed.     

Longipenol,a novel tetracyclic diterpene from,the termite soldier longipeditermes longipes

Longipenol, a tetracyclic diterpenoid from soldiers of the nasute termite $\text{longipeditermes longipes}$, has been shown by detailed carbon and proton NMR studies (COSY, NOSY) to have the novel spiro-fused structure (1).     

Beyond the Shannon–Khinchin formulation: The composability axiom and the universal-group entropy

The notion of entropy is ubiquitous both in natural and social sciences. In the last two decades, a considerable effort has been devoted to the study of new entropic forms, which generalize the standard Boltzmann–Gibbs (BG) entropy and could be applicable in thermodynamics, quantum mechanics and information theory. In Khinchin (1957), by extending previous ideas of Shannon (1948) and Shannon and Weaver (1949), Khinchin proposed a characterization of the BG entropy, based on four requirements, nowadays known as the Shannon–Khinchin (SK) axioms.     

Diels-alder reactions of 3-(2-nitrovinyl)indoles: Formation of carbazoles and bridged carbazoles

The Diels-Alder reactions of 1-substituted-3-(2-nitrovinyl)indoles 3 with quinones and acetylenes give aromatized 1:1 adducts (- nitrous acid) ( 1 ) or (- nitrous acid, -2 hydrogens) 2,5 . Likewise, dimerization (-2 nitrous acids) of 3 gives aromatized 2-(3-indolyl)carbazoles 4 . In contrast, 3 reacts with maleimides 6 to give 1:2 adducts (- nitrous acid or -2 hydrogens) 10 and 11 , respectively, along with smaller amounts of 1:1 adducts (- nitrous acid, -2 hydrogens; or -4 hydrogens) 12 and 13 , respectively. A mechanism for formation of the nitro products 11 and 13 is discussed. A 1:2 adduct (-2 hydrogens) 19 was also obtained from a Diels-Alder reaction between maleimide and the vinylindole produced in situ by condensing 1-methylindole with acetone. The stereochemisty of this 1:2 adduct has been determined by X-ray crystallography.     

Structure of the bromination product of 2-ethyl-1,4-benzothiazin-3(4H)-one

2-Ethyl-1,4-benzothiazin-3(4H)-one and bromine react smoothly to give Z-2-(1-bromoethylidine)-2(H)-1,4-benzothiazin-3(4H)-one, which results from a complex bromination-oxidation sequence. The structure of this product was determined by an X-ray study.