Stability radii of positive linear Volterra–Stieltjes equations

AbstractWe study stability radii of linear Volterra–Stieltjes equations under multi-perturbations and affine perturbations. A lower and upper bound for the complex stability radius with respect to multi-perturbations are given. Furthermore, in some special cases concerning the structure matrices, the complex stability radius can precisely be computed via the associated transfer functions. Then, the class of positive linear Volterra–Stieltjes equations is studied in detail. It is shown that for this class, complex, real and positive stability radius under multi-perturbations or multi-affine perturbations coincide and can be computed by simple formulae expressed in terms of the system matrices. As direct consequences of the obtained results, we get some results on robust stability of positive linear integro-differential equations and of positive linear functional differential equations. To the best of our knowledge, most of the results of this paper are new

Unified Dark Matter Scalar Field Models

In this work we analyze and review cosmological models in which the dynamics of a single scalar field accounts for a unified description of the Dark Matter and Dark Energy sectors, dubbed Unified Dark Matter (UDM) models. In this framework, we consider the general Lagrangian of k-essence, which allows to find solutions around which the scalar field describes the desired mixture of Dark Matter and Dark Energy. We also discuss static and spherically symmetric solutions of Einstein's equations for a scalar field with non-canonical kinetic term, in connection with galactic halo rotation curves.Comment: 57 pages, 6 figures, LaTeX file. Typos corrected; Added References; Revised according to reviewer's suggestions; Invited Review for the special issue "Focus Issue on Dark Matter" for Advances in Astronom

Geometric origin of mechanical properties of granular materials

Some remarkable generic properties, related to isostaticity and potential energy minimization, of equilibrium configurations of assemblies of rigid, frictionless grains are studied. Isostaticity -the uniqueness of the forces, once the list of contacts is known- is established in a quite general context, and the important distinction between isostatic problems under given external loads and isostatic (rigid) structures is presented. Complete rigidity is only guaranteed, on stability grounds, in the case of spherical cohesionless grains. Otherwise, the network of contacts might deform elastically in response to load increments, even though grains are rigid. This sets an uuper bound on the contact coordination number. The approximation of small displacements (ASD) allows to draw analogies with other model systems studied in statistical mechanics, such as minimum paths on a lattice. It also entails the uniqueness of the equilibrium state (the list of contacts itself is geometrically determined) for cohesionless grains, and thus the absence of plastic dissipation. Plasticity and hysteresis are due to the lack of such uniqueness and may stem, apart from intergranular friction, from small, but finite, rearrangements, in which the system jumps between two distinct potential energy minima, or from bounded tensile contact forces. The response to load increments is discussed. On the basis of past numerical studies, we argue that, if the ASD is valid, the macroscopic displacement field is the solution to an elliptic boundary value problem (akin to the Stokes problem).Comment: RevTex, 40 pages, 26 figures. Close to published paper. Misprints and minor errors correcte

A new gravitational wave generation algorithm for particle perturbations of the Kerr spacetime

We present a new approach to solve the 2+1 Teukolsky equation for gravitational perturbations of a Kerr black hole. Our approach relies on a new horizon penetrating, hyperboloidal foliation of Kerr spacetime and spatial compactification. In particular, we present a framework for waveform generation from point-particle perturbations. Extensive tests of a time domain implementation in the code {\it Teukode} are presented. The code can efficiently deliver waveforms at future null infinity. As a first application of the method, we compute the gravitational waveforms from inspiraling and coalescing black-hole binaries in the large-mass-ratio limit. The smaller mass black hole is modeled as a point particle whose dynamics is driven by an effective-one-body-resummed analytical radiation reaction force. We compare the analytical angular momentum loss to the gravitational wave angular momentum flux. We find that higher-order post-Newtonian corrections are needed to improve the consistency for rapidly spinning binaries. Close to merger, the subdominant multipolar amplitudes (notably the $m=0$ ones) are enhanced for retrograde orbits with respect to prograde ones. We argue that this effect mirrors nonnegligible deviations from circularity of the dynamics during the late-plunge and merger phase. We compute the gravitational wave energy flux flowing into the black hole during the inspiral using a time-domain formalism proposed by Poisson. Finally, a self-consistent, iterative method to compute the gravitational wave fluxes at leading-order in the mass of the particle is presented. For a specific case study with $\hat{a}$=0.9, a simulation that uses the consistent flux differs from one that uses the analytical flux by $\sim35$ gravitational wave cycles over a total of about $250$ cycles. In this case the horizon absorption accounts for about $+5$ gravitational wave cycles

Jammed frictionless discs: connecting local and global response

By calculating the linear response of packings of soft frictionless discs to quasistatic external perturbations, we investigate the critical scaling behavior of their elastic properties and non-affine deformations as a function of the distance to jamming. Averaged over an ensemble of similar packings, these systems are well described by elasticity, while in single packings we determine a diverging length scale $\ell^*$ up to which the response of the system is dominated by the local packing disorder. This length scale, which we observe directly, diverges as $1/\Delta z$, where $\Delta z$ is the difference between contact number and its isostatic value, and appears to scale identically to the length scale which had been introduced earlier in the interpretation of the spectrum of vibrational modes. It governs the crossover from isostatic behavior at the small scale to continuum behavior at the large scale; indeed we identify this length scale with the coarse graining length needed to obtain a smooth stress field. We characterize the non-affine displacements of the particles using the \emph{displacement angle distribution}, a local measure for the amount of relative sliding, and analyze the connection between local relative displacements and the elastic moduli.Comment: 19 pages, 15 figures, submitted to Phys. Rev.

A mean field description of jamming in non-cohesive frictionless particulate systems

A theory for kinetic arrest in isotropic systems of repulsive, radially-interacting particles is presented that predicts exponents for the scaling of various macroscopic quantities near the rigidity transition that are in agreement with simulations, including the non-trivial shear exponent. Both statics and dynamics are treated in a simplified, one-particle level description, and coupled via the assumption that kinetic arrest occurs on the boundary between mechanically stable and unstable regions of the static parameter diagram. This suggests the arrested states observed in simulations are at (or near) an elastic buckling transition. Some additional numerical evidence to confirm the scaling of microscopic quantities is also provided.Comment: 9 pages, 3 figs; additional clarification of different elastic moduli exponents, plus typo fix. To appear in PR