This paper is mainly a review of the multi–Hamiltonian nature of Toda and generalized Toda lattices corresponding to the classical simple Lie groups but it includes also some new results. The areas investigated include master symmetries, recursion operators, higher Poisson brackets, invariants and group symmetries for the systems. In addition to the positive hierarchy we also consider the negat...

The symmetry group method is important and widely used in the reduction and construction of explicit solutions of PDEs (partial differential equations). As shown in [1 – 4], the Lie symmetry group method can be used to find explicit solutions of PDEs using symmetry reduction and construction. This method is known as the classical method. By a classical symmetry group of a system of PDEs we mean...

We present a systematic first-principles study of the equilibrium bond lengths, harmonic frequencies, dissociation energies, ground state symmetries, and spin state splittings of 22 diatomic molecules comprised of a first-row 3d transition-metal and a main-group element (H, C, N, O, or F). Diatomic molecules are building blocks of the key molecular bonding motifs in biological and inorganic cat...

We review double field theory (DFT) with emphasis on the doubled spacetime and its generalized coordinate transformations, which unify diffeomorphisms and b-field gauge transformations. We illustrate how the composition of generalized coordinate transformations fails to associate. Moreover, in dimensional reduction, the O(d, d) T-duality transformations of fields can be obtained as generalized ...

Abstract Considering the classical definition of the Littlewood-Richardson rule and its 2-dimensional representation by means of rectangular tableaux, we exhibit 24 symmetries of this rule when considering dualization, conjugation and their composition. Extending the Littlewood-Richardson rule to sequences of nonnegative real numbers, six of these symmetries may be generalized. Our point is to ...

We investigate the intrinsic parity of black holes. It appears that discrete symmetries require the black hole Hilbert space to be larger than suggested by the usual quantum numbers M (mass), Q (charge) and J (angular momentum). Recent results on black hole production in trans-Planckian scattering lead to gravitational effects which do not decouple from low-energy physics. Dispersion relations ...

Different symmetry formalisms for difference equations on lattices are reviewed and applied to perform symmetry reduction for both linear and nonlinear partial difference equations. Both Lie point symmetries and generalized symmetries are considered and applied to the discrete heat equation and to the integrable discrete time Toda lattice. Résumé Deux formalismes différents pour étudier les sym...

A long-standing problem in the study of topological phases matter has been to understand types fractional insulator (FTI) possible 3+1 dimensions. Unlike ordinary insulators free fermions, FTI are characterized by \Theta ? -angles, long-range entanglement, and fractionalization. Starting from a...