Continuous symmetry of C60 fullerene and its derivatives

Conventionally, the Ih symmetry of fullerene C60 is accepted which is supported by numerous calculations. However, this conclusion results from the consideration of the molecule electron system, of its odd electrons in particular, in a close-shell approximation without taking the electron spin into account. Passing to the open-shell approximation has lead to both the energy and the symmetry lowering up to Ci. Seemingly contradicting to a high-symmetry pattern of experimental recording, particularly concerning the molecule electronic spectra, the finding is considered in the current paper from the continuous symmetry viewpoint. Exploiting both continuous symmetry measure and continuous symmetry content, was shown that formal Ci symmetry of the molecule is by 99.99% Ih. A similar continuous symmetry analysis of the fullerene monoderivatives gives a reasonable explanation of a large variety of their optical spectra patterns within the framework of the same C1 formal symmetry exhibiting a strong stability of the C60 skeleton.Comment: 11 pages. 5 figures. 6 table

Average Symmetry and Complexity of Binary Sequences

The concept of complexity as average symmetry is here formalised by introducing a general expression dependent on the relevant symmetry and a related discrete set of transformations. This complexity has hybrid features of both statistical complexities and of those related to algorithmic complexity. Like the former, random objects are not the most complex while they still are more complex than the more symmetric ones (as in the latter). By applying this definition to the particular case of rotations of binary sequences, we are able to find a precise expression for it. In particular, we then analyse the behaviour of this measure in different well-known automatic sequences, where we find interesting new properties. A generalisation of the measure to statistical ensembles is also presented and applied to the case of i.i.d. random sequences and to the equilibrium configurations of the one-dimensional Ising model. In both cases, we find that the complexity is continuous and differentiable as a function of the relevant parameters and agrees with the intuitive requirements we were looking for.Comment: 9 pages, 5 figure

Discovering the manifold facets of a square integrable representation: from coherent states to open systems

Group representations play a central role in theoretical physics. In particular, in quantum mechanics unitary --- or, in general, projective unitary --- representations implement the action of an abstract symmetry group on physical states and observables. More specifically, a major role is played by the so-called square integrable representations. Indeed, the properties of these representations are fundamental in the definition of certain families of generalized coherent states, in the phase-space formulation of quantum mechanics and the associated star product formalism, in the definition of an interesting notion of function of quantum positive type, and in some recent applications to the theory of open quantum systems and to quantum information.Comment: 13 page

Isotropy of Angular Frequencies and Weak Chimeras With Broken Symmetry

The notion of a weak chimeras provides a tractable definition for chimera states in networks of finitely many phase oscillators. Here we generalize the definition of a weak chimera to a more general class of equivariant dynamical systems by characterizing solutions in terms of the isotropy of their angular frequency vector - for coupled phase oscillators the angular frequency vector is given by the average of the vector field along a trajectory. Symmetries of solutions automatically imply angular frequency synchronization. We show that the presence of such symmetries is not necessary by giving a result for the existence of weak chimeras without instantaneous or setwise symmetries for coupled phase oscillators. Moreover, we construct a coupling function that gives rise to chaotic weak chimeras without symmetry in weakly coupled populations of phase oscillators with generalized coupling

Operators versus functions: from quantum dynamical semigroups to tomographic semigroups

Quantum mechanics can be formulated in terms of phase-space functions, according to Wigner's approach. A generalization of this approach consists in replacing the density operators of the standard formulation with suitable functions, the so-called generalized Wigner functions or (group-covariant) tomograms, obtained by means of group-theoretical methods. A typical problem arising in this context is to express the evolution of a quantum system in terms of tomograms. In the case of a (suitable) open quantum system, the dynamics can be described by means of a quantum dynamical semigroup 'in disguise', namely, by a semigroup of operators acting on tomograms rather than on density operators. We focus on a special class of quantum dynamical semigroups, the twirling semigroups, that have interesting applications, e.g., in quantum information science. The 'disguised counterparts' of the twirling semigroups, i.e., the corresponding semigroups acting on tomograms, form a class of semigroups of operators that we call tomographic semigroups. We show that the twirling semigroups and the tomographic semigroups can be encompassed in a unique theoretical framework, a class of semigroups of operators including also the probability semigroups of classical probability theory, so achieving a deeper insight into both the mathematical and the physical aspects of the problem.Comment: 12 page