Bootstrapping Topological Properties and Systemic Risk of Complex Networks Using the Fitness Model

In this paper we present a novel method to reconstruct global topological properties of a complex network starting from limited information. We assume to know for all the nodes a non-topological quantity that we interpret as fitness. In contrast, we assume to know the degree, i.e. the number of connections, only for a subset of the nodes in the network. We then use a fitness model, calibrated on the subset of nodes for which degrees are known, in order to generate ensembles of networks. Here, we focus on topological properties that are relevant for processes of contagion and distress propagation in networks, i.e. network density and k-core structure, and we study how well these properties can be estimated as a function of the size of the subset of nodes utilized for the calibration. Finally, we also study how well the resilience to distress propagation in the network can be estimated using our method. We perform a first test on ensembles of synthetic networks generated with the Exponential Random Graph model, which allows to apply common tools from statistical mechanics. We then perform a second test on empirical networks taken from economic and financial contexts. In both cases, we find that a subset as small as 10 % of nodes can be enough to estimate the properties of the network along with its resilience with an error of 5 %.     

Lower-Dimensional Invariant Tori for Perturbations of a Class of Non-convex Hamiltonian Functions

We consider a class of quasi-integrable Hamiltonian systems obtained by adding to a non-convex Hamiltonian function of an integrable system a perturbation depending only on the angle variables. We focus on a resonant maximal torus of the unperturbed system, foliated into a family of lower-dimensional tori of codimension 1, invariant under a quasi-periodic flow with rotation vector satisfying some mild Diophantine condition. We show that at least one lower-dimensional torus with that rotation vector always exists also for the perturbed system. The proof is based on multiscale analysis and resummation procedures of divergent series. A crucial role is played by suitable symmetries and cancellations, ultimately due to the Hamiltonian structure of the system.     

A nonautonomous chain rule in W 1,p and BV

In this paper we consider the chain rule formula for compositions ${x\mapsto F(x, u(x))}$ in the case when u has a Sobolev or BV regularity and F(x, z) is separately Sobolev, or BV, with respect to x and C ¹ with respect to z. Our results extend to this “nonautonomous” case the results known for compositions ${x\mapsto F(u(x))}$ .     

Justification of the coupled-mode approximation for a nonlinear elliptic problem with a periodic potential

Coupled-mode systems are used in physical literature to simplify the nonlinear Maxwell and Gross–Pitaevskii equations with a small periodic potential and to approximate localized solutions called gap solitons by analytical expressions involving hyperbolic functions. We justify the use of the 1D stationary coupled-mode system for a relevant elliptic problem by employing the method of Lyapunov–Schmidt reductions in Fourier space. In particular, existence of periodic/anti-periodic and decaying solutions is proved and the error terms are controlled in suitable norms. The use of multi-dimensional stationary coupled-mode systems is justified for analysis of bifurcations of periodic/anti-periodic solutions in a small multi-dimensional periodic potential.     

Justification of the Ginzburg–Landau approximation for an instability as it appears for Marangoni convection

The Ginzburg–Landau equation appears as a universal amplitude equation for spatially extended pattern forming systems close to the first instability. It can be derived via multiple scaling analysis for the Marangoni convection problem that is driven by temperature‐dependent surface tension and is the subject of our interest. In this paper, we prove estimates between this formal approximation and true solutions of a scalar pattern forming model problem showing the same spectral picture as the Marangoni convection problem in case of a thin fluid. The new difficulties come from neutral modes touching the imaginary axis for the wave number k = 0 and from identical group velocities at the critical wave number k = k_c and the wave number k = 0. The problem is solved by using the reflection symmetry of the system and by using the fact that the modes concentrate at integer multiples of the critical wave number k = k_c. The paper presents a method that is applicable whenever this kind of instability occurs. Copyright © 2012 John Wiley & Sons, Ltd.     

Comprehensive study of the vapour–liquid coexistence of the truncated and shifted Lennard–Jones fluid including planar and spherical interface properties

Vapour–liquid equilibria of the Lennard–Jones potential, truncated and shifted at 2.5σ, are studied using molecular dynamics simulations, an attractive option for studying inhomogeneous systems. Comprehensive simulation data are reported for three cases: no interface, a planar interface, and a spherical interface between the coexisting phases, covering a wide range of temperatures. Spherical droplets are also studied for a range of radii between 5 and 16σ. The size dependence of the surface tension, based on the Irving–Kirkwood pressure tensor, and other properties is quantified for spherical interfaces. All simulation results are correlated with a consistent set of empirical equations. A comparison with the results of other authors as well as with experimental data for noble gases and methane is also presented.     

Expressions for forces and torques in molecular simulations using rigid bodies

Expressions for intermolecular forces and torques, derived from pair potentials between rigid non-spherical units, are presented. The aim is to give compact and clear expressions, which are easily generalized, and which minimize the risk of error in writing molecular dynamics simulation programs. It is anticipated that these expressions will be useful in the simulation of liquid crystalline systems, and in coarse-grained modelling of macromolecules.     

The Raman Spectrum of The 3-Br Pyridine-Bromine Charge Transfer Complex

The formation of a charge transfer complex between pyridine and halogens or mixed halogens^(1,2) brings about perturbations in the infrared spectra of the pyridine molecule and many halogen complexes show two bands in the low frequency range, which can be attributed to the v _X-X and v _N…Y stretching vibrations. ^(3–5). However, by comparison, very little has been reported on the Raman spectra of such complexes. Klaboe⁽⁶⁾ studied the v _Br-Br Raman frequency of some bromine complexes and more recently the Raman spectra of pyridine complexed with Ni (CN)₄ ⁽⁷⁾ and TiCl₄ ⁽⁸⁾ have been reported. In this work, the Raman spectrum of the 3-Br pyridine-bromine charge transfer complex is investigated.