We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Social Network Analysis

QUANTUM gravitational effects are usually ignored in calculations of the formation and evolution of black holes. The justification for this is that the radius of curvature of space-time outside the event horizon is very large compared to the Planck length (Għ/c3)1/2 ≈ 10−33 cm, the length scale on which quantum fluctuations of the metric are expected to be of order unity. This means that the energy density of particles created by the gravitational field is small compared to the space-time curvature. Even though quantum effects may be small locally, they may still, however, add up to produce a significant effect over the lifetime of the Universe ≈ 1017 s which is very long compared to the Planck time ≈ 10−43 s. The purpose of this letter is to show that this indeed may be the case: it seems that any black hole will create and emit particles such as neutrinos or photons at just the rate that one would expect if the black hole was a body with a temperature of (κ/2π) (ħ/2k) ≈ 10−6 (M/M)K where κ is the surface gravity of the black hole1. As a black hole emits this thermal radiation one would expect it to lose mass. This in turn would increase the surface gravity and so increase the rate of emission. The black hole would therefore have a finite life of the order of 1071 (M/M)−3 s. For a black hole of solar mass this is much longer than the age of the Universe. There might, however, be much smaller black holes which were formed by fluctuations in the early Universe2. Any such black hole of mass less than 1015 g would have evaporated by now. Near the end of its life the rate of emission would be very high and about 1030 erg would be released in the last 0.1 s. This is a fairly small explosion by astronomical standards but it is equivalent to about 1 million 1 Mton hydrogen bombs. It is often said that nothing can escape from a black hole. But in 1974, Stephen Hawking realized that, owing to quantum effects, black holes should emit particles with a thermal distribution of energies — as if the black hole had a temperature inversely proportional to its mass. In addition to putting black-hole thermodynamics on a firmer footing, this discovery led Hawking to postulate 'black hole explosions', as primordial black holes end their lives in an accelerating release of energy.

Black Hole Explosions

We give a general introduction to quantum phase transitions in strongly-correlated electron systems. These transitions which occur at zero temperature when a non-thermal parameter $g$ like pressure, chemical composition or magnetic field is tuned to a critical value are characterized by a dynamic exponent $z$ related to the energy and length scales $\Delta$ and $\xi$. Simple arguments based on an expansion to first order in the effective interaction allow to define an upper-critical dimension $D_{C}=4$ (where $D=d+z$ and $d$ is the spatial dimension) below which mean-field description is no longer valid. We emphasize the role of pertubative renormalization group (RG) approaches and self-consistent renormalized spin fluctuation (SCR-SF) theories to understand the quantum-classical crossover in the vicinity of the quantum critical point with generalization to the Kondo effect in heavy-fermion systems. Finally we quote some recent inelastic neutron scattering experiments performed on heavy-fermions which lead to unusual scaling law in $\omega /T$ for the dynamical spin susceptibility revealing critical local modes beyond the itinerant magnetism scheme and mention new attempts to describe this local quantum critical point.

https://arxiv.org/pdf/cond-mat/0102119.pdf

Quantum Phase Transitions

Advanced Mathematical Methods for Scientists and Engineers

It is shown that all four superintegrable quantum systems on the Euclidean plane possess the same underlying hidden algebra sl(3). The gauge-rotated Hamiltonians, as well as their integrals of motion, once rewritten in appropriate coordinates, preserve a flag of polynomials. This flag corresponds to highest-weight finite-dimensional representations of the sl(3)-algebra, realized by first-order differential operators.

Exact solvability of superintegrable systems

The notion of group entropy is proposed. It enables the unification and generaliztion of many different definitions of entropy known in the literature, such as those of Boltzmann-Gibbs, Tsallis, Abe, and Kaniadakis. Other entropic functionals are introduced, related to nontrivial correlation laws characterizing universality classes of systems out of equilibrium when the dynamics is weakly chaotic. The associated thermostatistics are discussed. The mathematical structure underlying our construction is that of formal group theory, which provides the general structure of the correlations among particles and dictates the associated entropic functionals. As an example of application, the role of group entropies in information theory is illustrated and generalizations of the Kullback-Leibler divergence are proposed. A new connection between statistical mechanics and zeta functions is established. In particular, Tsallis entropy is related to the classical Riemann zeta function.

http://www.cbpf.br/GrupPesq/StatisticalPhys/pdftheo/Tempesta2011.pdf

Group entropies, correlation laws, and zeta functions.

We introduce a 2N-parametric family of maximally superintegrable systems in N dimensions, obtained as a reduction of an anisotropic harmonic oscillator in a 2N-dimensional configuration space. These systems possess closed bounded orbits and integrals of motion which are polynomial in the momenta. They generalize known examples of superintegrable models in the Euclidean plane.

/pdf/reduction-of-superintegrable-systems-the-anisotropic-59tmsv8vau.pdf

Reduction of superintegrable systems: the anisotropic harmonic oscillator.

Superintegrable deformations of the Smorodinsky-Winternitz Hamiltonian by A. Ballesteros, F. J. Herranz, F. Musso, and O. Ragnisco Isochronous motions galore: Nonlinearly coupled oscillators with lots of isochronous solutions by F. Calogero and J.-P. Francoise Nambu dynamics, deformation quantization, and superintegrability by T. L. Curtright and C. K. Zachos Maximally superintegrable systems of Winternitz type by C. Gonera Cubic integrals of motion and quantum superintegrability by S. Gravel Superintegrability, Lax matrices and separation of variables by J. Harnad and O. Yemolayeva Maximally superintegrable Smorodinsky-Winternitz systems on the $N$-dimensional sphere and hyperbolic spaces by F. J. Herranz, A. Ballesteros, M. Santander, and T. Sanz-Gil Invariant Wirtinger projective connection and Tau-functions on spaces of branched coverings by A. Kokotov and D. Korotkin Dyon-oscillator duality. Hidden symmetry of the Yang-Coulomb monopole by L. G. Mardoyan Supersymmetric Calogero-Moser-Sutherland models: Superintegrability structure and eigenfunctions by P. Desrosiers, L. Lapointe, and P. Mathieu Complete sets of invariants for classical systems by W. Miller, Jr. Higher-order symmetry operators for Schrodinger equation by A. G. Nikitin Symmetries and Lagrangian time-discretizations of Euler equations by A. V. Penskoi Two exactly-solvable problems in one-dimensional quantum mechanics on circle by L. G. Mardoyan, G. S. Pogosyan, and A. N. Sissakian Higher-order superintegrability of a rational oscillator with inversely quadratic nonlinearities: Euclidean and non-Euclidean cases by M. F. Ranada and M. Santander A survey of quasi-exactly solvable systems and spin Calogero-Sutherland models by F. Finkel, D. Gomez-Ullate, A. Gonzalez-Lopez, M. A. Rodriguez, and R. Zhdanov On the classification of third-order integrals of motion in two-dimensional quantum mechanics by M. Sheftel Towards a classification of cubic integrals of motion by R. G. McLenaghan, R. G. Smirnov, and D. The Integrable systems whose spectral curves are the graph of a function by K. Takasaki On superintegrable systems in $E_2$: Algebraic properties and symmetry preserving discretization by P. Tempesta Perturbations of integrable systems and Dyson-Mehta integrals by A. V. Turbiner Separability and the Birkhoff-Gustavson normalization of the perturbed harmonic oscillators with homogeneous polynomial potentials by Y. Uwano Integrability and superintegrability without separability by J. Berube and P. Winternitz Applications of CRACK in the classification of integrable systems by T. Wolf The prolate spheroidal phenomenon as a consequence of bispectrality by G. A. Grunbaum and M. Yakimov On a trigonometric analogue of Atiyah-Hitchin bracket by O. Yermolayeva Separation of variables in time-dependent Schrodinger equations by A. Zhalij and R. Zhdanov New types of solvability in PT symmetric quantum theory by M. Znojil.

Superintegrability in Classical and Quantum Systems

The relation is established between some concepts of quantum mechanics and those of soliton theory. In particular, superintegrable systems in two-dimensional quantum mechanics are shown to be invariant under generalized Lie symmetries and to allow recursion operators.

Piergiulio Tempesta

Papers

Exact solvability of superintegrable systems

Group entropies, correlation laws, and zeta functions.

Reduction of superintegrable systems: the anisotropic harmonic oscillator.

Superintegrability in Classical and Quantum Systems

Superintegrable systems in quantum mechanics and classical Lie theory