Fabio Musso

The evolution of the meningeal vascular system in the human genus: From brain shape to thermoregulation

The imprints of the middle meningeal vessels make it possible to analyze vascularization in fossil specimens. The association between changes in the cortical anatomy and vascular organization raises questions about the actual physiological meaning of these features, most of all when dealing with the origin of the modern human brain. Metabolism and thermoregulation may be relevant factors in influencing morphological adaptations between brain and vessels. This study is aimed at investigating the relationships between endocranial morphology and endocranial vessels in modern humans and to analyze the pattern of heat dissipation through the endocranial surface in fossil specimens.

(Super)integrability from coalgebra symmetry: Formalism and applications

The coalgebra approach to the construction of classical integrable systems from Poisson coalgebras is reviewed, and the essential role played by symplectic realizations in this framework is emphasized. Many examples of Hamiltonians with either undeformed or q-deformed coalgebra symmetry are given, and their Liouville superintegrability is discussed. Among them, (quasi-maximally) superintegrable systems on N-dimensional curved spaces of nonconstant curvature are analysed in detail. Further generalizations of the coalgebra approach that make use of comodule and loop algebras are presented. The generalization of such a coalgebra symmetry framework to quantum mechanical systems is straightforward.

Algebraic Extensions of Gaudin Models

Abstract We perform a Inönü–Wigner contraction on Gaudin models, showing how the integrability property is preserved by this algebraic procedure. Starting from Gaudin models we obtain new integrable chains, that we call Lagrange chains, associated to the same linear r-matrix structure. We give a general construction involving rational, trigonometric and elliptic solutions of the classical Yang-Baxter equation. Two particular examples are explicitly considered: the rational Lagrange chain and the trigonometric one. In both cases local variables of the models are the generators of the direct sum of N interacting tops.

On quantum deformations of (anti-)de Sitter algebras in (2+1) dimensions

Quantum deformations of (anti-)de Sitter (A)dS algebras in (2+1) dimensions are revisited, and several features of these quantum structures are reviewed. In particular, the classification problem of (2+1) (A)dS Lie bialgebras is presented and the associated noncommutative quantum (A)dS spaces are also analysed. Moreover, the flat limit (or vanishing cosmological constant) of all these structures leading to (2+1) quantum Poincare algebras and groups is simultaneously given by considering the cosmological constant as an explicit Lie algebra parameter in the (A)dS algebras. By making use of this classification, a three-parameter generalization of the K-deformation for the (2+1) (A)dS algebras and quantum spacetimes is given. Finally, the same problem is studied in (3+1) dimensions, where a two-parameter generalization of the κ-(A)dS deformation that preserves the space isotropy is found.

Quantifying Patterns of Endocranial Heat Distribution: Brain Geometry and Thermoregulation

The mechanisms involved in brain thermoregulation are still poorly known, and many disagreements still exist concerning the selective cooling capacity of the brain volume. This issue has also been discussed in human evolution and paleoneurology, speculating on possible changes associated with hominid encephalization. Although the vascular system is supposed to be the main component responsible for thermoregulation, brain geometry also plays an important role in the pattern of heat distribution.

A Stochastic Version of the Eigen Model

We exhibit a stochastic discrete time model that produces the Eigen model (Naturwissenschaften 58:465–523, 1971) in the deterministic and continuous time limits. The model is based on the competition among individuals differing in terms of fecundity but with the same viability. We explicitly write down the Markov matrix of the discrete time stochastic model in the two species case and compute the master sequence concentration numerically for various values of the total population. We also obtain the master equation of the model and perform a Van Kampen expansion. The results obtained in the two species case are compared with those coming from the Eigen model. Finally, we comment on the range of applicability of the various approaches described, when the number of species is larger than two.

Quantum algebras as quantizations of dual Poisson–Lie groups

A systematic computational approach for the explicit construction of any quantum Hopf algebra (Uz(g), Δz) starting from the Lie bialgebra (g, δ) that gives the first-order deformation of the coproduct map Δz is presented. The procedure is based on the well-known ‘quantum duality principle’, namely the fact that any quantum algebra can be viewed as the quantization of the unique Poisson–Lie structure (G*, Λg) on the dual group G*, which is obtained by exponentiating the Lie algebra g* defined by the dual map δ*. From this perspective, the coproduct for Uz(g) is just the pull-back of the group law for G*, and the Poisson analogues of the quantum commutation rules for Uz(g) are given by the unique Poisson–Lie structure Λg on G* whose linearization is the Poisson analogue of the initial Lie algebra g. This approach is shown to be a very useful technical tool in order to solve the Lie bialgebra quantization problem explicitly since, once a Lie bialgebra (g, δ) is given, the full dual Poisson–Lie group (G*, Λ) can be obtained either by applying standard Poisson–Lie group techniques or by implementing the algorithm presented here with the aid of symbolic manipulation programs. As a consequence, the quantization of (G*, Λ) will give rise to the full Uz(g) quantum algebra, provided that ordering problems are appropriately fixed through the choice of certain local coordinates on G* whose coproduct fulfils a precise ‘quantum symmetry’ property. The applicability of this approach is explicitly demonstrated by reviewing the construction of several instances of quantum deformations of physically relevant Lie algebras such as , the (2+1) anti-de Sitter algebra so(2, 2) and the Poincare algebra in (3+1) dimensions.

The spin 1/2 Calogero-Gaudin System and its q−Deformation

The spin-½ Calogero-Gaudin system and its q-deformation are exactly solved: a complete set of commuting observables is diagonalized, and the corresponding eigenvectors and eigenvalues are explicitly calculated. The method of solution is purely algebraic and relies on the co-algebra symmetry of the model.

Exact solution of the quantum Calogero–Gaudin system and of its q deformation

A complete set of commuting observables for the Calogero–Gaudin system is diagonalized, and the explicit form of the corresponding eigenvalues and eigenfunctions is derived. We use a purely algebraic procedure exploiting the coalgebra invariance of the model; with the proper technical modifications this procedure can be applied to the q-deformed version of the model, which is then also exactly solved.

Integrable deformations of Rössler and Lorenz systems from Poisson–Lie groups

Abstract A method to construct integrable deformations of Hamiltonian systems of ODEs endowed with Lie–Poisson symmetries is proposed by considering Poisson–Lie groups as deformations of Lie–Poisson (co)algebras. Moreover, the underlying Lie–Poisson symmetry of the initial system of ODEs is used to construct integrable coupled systems, whose integrable deformations can be obtained through the construction of the appropriate Poisson–Lie groups that deform the initial symmetry. The approach is applied in order to construct integrable deformations of both uncoupled and coupled versions of certain integrable types of Rossler and Lorenz systems. It is worth stressing that such deformations are of non-polynomial type since they are obtained through an exponentiation process that gives rise to the Poisson–Lie group from its infinitesimal Lie bialgebra structure. The full deformation procedure is essentially algorithmic and can be computerized to a large extent.