The Birkhoff theorem for unitary matrices of arbitrary dimensions

It was shown recently that Birkhoff's theorem for doubly stochastic matrices can be extended to unitary matrices with equal line sums whenever the dimension of the matrices is prime. We prove a generalization of the Birkhoff theorem for unitary matrices with equal line sums for arbitrary dimension.Comment: This manuscript presents a proof for the general unitary birkhoff theorem, conjectured in arXiv:1509.0862

Edge modes in self-gravitating disc-planet interactions

We study the stability of gaps opened by a giant planet in a self-gravitating protoplanetary disc. We find a linear instability associated with both the self-gravity of the disc and local vortensity maxima which coincide with gap edges. For our models, these edge modes develop and extend to twice the orbital radius of a Saturn mass planet in discs with disc-to-star mass ratio >0.06, corresponding to a Toomre Q < 1.5 at the outer disc boundary. Unlike the local vortex-forming instabilities associated with gap edges in weakly or non-self-gravitating low viscosity discs, the edge modes are global and exist only in sufficiently massive discs, but for the typical viscosity values adopted for protoplanetary discs. Analytic modelling and linear calculations show edge modes may be interpreted as a localised disturbance associated with a gap edge inducing activity in the extended disc, through the launching of density waves excited at Lindblad resonances. Nonlinear hydrodynamic simulations are performed to investigate the evolution of edge modes in disc-planet systems. The form and growth rates of unstable modes are consistent with linear theory. Their dependence on viscosity and gravitational softening is also explored. We also performed a first study of the effect of edge modes on planetary migration. We found that if edge modes develop, then the average disc-on-planet torque becomes more positive with increasing disc mass. In simulations where the planet was allowed to migrate, although a fast type III migration could be seen that was similar to that seen in non-self-gravitating discs, we found that it was possible for the planet to interact gravitationally with the spiral arms associated with an edge mode and that this could result in the planet being scattered outwards. Thus orbital migration is likely to be complex and non monotonic in massive discs of the type we consider.Comment: 26 pages, 21 figures. Accepted by MNRAS. Abstract displayed is shortene

Eta absorption by mesons

Using the $[SU(3)_{\mathrm{L}} \times SU(3)_{\mathrm{R}}]_{\mathrm{global}}\times [SU(3)_V]_{\mathrm{local}}$ chiral Lagrangian with hidden local symmetry, we evaluate the cross sections for the absorption of eta meson ($\eta$) by pion ($\pi$), rho ($\rho$), omega ($\omega$), kaon ($K$), and kaon star ($K^*$) in the tree-level approximation. With empirical masses and coupling constants as well as reasonable values for the cutoff parameter in the form factors at interaction vertices, we find that most cross sections are less than 1 mb, except the reactions $\rho\eta\to K\bar K^*(\bar KK^*)$, $\omega\eta\to K\bar K^*(\bar KK^*)$, $K^*\eta\to\rho K$, and $K^*\eta\to\omega K$, which are a few mb, and the reactions $\pi\eta\to K\bar K$ and $K\eta\to\pi K$, which are more than 10 mb. Including these reactions in a kinetic model based on a schematic hydrodynamic description of relativistic heavy ion collisions, we find that the abundance of eta mesons likely reaches chemical equilibrium with other hadrons in nuclear collisions at the Relativistic Heavy Ion Collider.Comment: 29 pages, 10 figures, version to appear in Nucl. Phys.

Type III migration in a low viscosity disc

We study the type III migration of a Saturn mass planet in low viscosity discs. The planet is found to experience cyclic episodes of rapid decay in orbital radius, each amounting to a few Hill radii. We find this to be due to the scattering of large- scale vortices present in the disc. The origin and role of vortices in the context of type III migration is explored. It is shown through numerical simulations and semi- analytical modelling that spiral shocks induced by a sufficiently massive planet will extend close to the planet orbital radius. The production of vortensity across shock tips results in thin high vortensity rings with a characteristic width of the local scale height. For planets with masses equal to and above that of Saturn, the rings are co-orbital features extending the entire azimuth. Linear stability analysis show there exists unstable modes that are localised about local vortensity minima which coincide with gap edges. Simulations show that vortices are non-linear a outcome. We used hydrodynamic simulations to examine vortex-planet interactions. Their effect is present in discs with kinematic viscosity less than about an order of magnitude smaller than the typically adopted value of \nu = 10^{-5}\Omega_pr_p(0)^2, where r_p(0) and \Omega_p are the initial orbital radius and angular velocity of the planet respectively. We find that the magnitude of viscosity affects the nature of type III migration but not the extent of the orbital decay. The role of vortices as a function of initial disc mass is also explored and it is found that the amount of orbital decay during one episode of vortex-planet interaction is independent of initial disc mass. We incorporate the concept of the co-orbital mass deficit in the analysis of our results and link it to the presence of vortices at gap edges.Comment: 20 pages, 20 figures, accepted for publication in MNRA

Analysis Of Time-dependent Bearing Capacity Of A Driven Pile In Clayey Soils By Total Stress Method

This paper proposes an analytical approach to evaluate the time-dependent bearing capacity of a driven pile in clayey soils by taking the pile installation and subsequent reconsolidation effects into consideration. The process of pile installation is modeled by undrained expansion of a spherical cavity at the pile tip and a cylindrical cavity around the pile shaft. The cavity expansion solution, which is based on a K0-consolidated anisotropic modified Cam-clay model (K0-AMCC), is used to capture the pile installation effects. After pile installation, the dissipation of the excess pore water and the increase of the effective stress in the surrounding soil are evaluated by the radial consolidation theory. Based on the effective stress, the strength of the remolded soil is quantified by the modified Cam-clay (MCC) model and the spatially mobilized plane (SMP) criterion. With the three-dimensional strength of the surrounding soil, the time-dependent bearing capacity of the driven pile is evaluated by the total stress (α) method. To verify the proposed analytical approach, three groups of centrifuge model tests were performed, and the proposed approach was applied to predict the time-dependent bearing capacity of the tested piles. It is shown that reasonable predictions can be made by the method proposed in this paper

Unified Solution To Drained Expansion Of A Spherical Cavity In Clay And Sand

This paper presents a novel unified solution to drained expansion of a spherical cavity in both clay and sand. The large-strain theory and a critical state model with a unified hardening parameter are used to describe the elastoplastic behavior of the soils after yielding. The elastoplastic constitutive tensor of the critical state model is developed to be a system of first-order differential equations for the drained expansion of a spherical cavity. The problem is formulated as an initial value problem in terms of the Lagrangian scheme by introducing an auxiliary variable and is solved numerically. With the present solution, curves for the expansion pressures, the distributions of stress components, and the stress paths are plotted to illustrate the different expansion responses in clay and sand. The proposed solution not only incorporates the dilatancy and peak strength of dense sand, but it can also reduce to the solution for clay and loose sand when ignoring the dilatancy and peak strength. Therefore, the present solution can be applied to interpret the cone penetration test and the pile installation, as well as to evaluate the pile end bearing capacity in various kinds of soils

A Feasible Approach To Predicting Time-dependent Bearing Performance Of Jacked Piles From CPTu Measurements

In this paper, a simple but feasible approach is proposed to predict the time-dependent load carrying behaviors of jacked piles from CPTu measurements. The corrected cone resistance, which considers the unequal area of the cone rod and the cone, is used to determine the soil parameters used in the proposed approach. The pile installation effects on the changes in the stress state of the surrounding soil are assessed by an analytical solution to undrained expansion of a cylindrical cavity in K0-consolidated anisotropic clayey soil. Considering the similarity and scale effects between the piezocone and the pile, the CPTu measurements are properly incorporated in the shaft and end resistance factors as well as in the load-transfer curves to predict the time-dependent load carrying behaviors of the pile. Centrifuge model tests are conducted and the measured load carrying behaviors of the model piles are compared with the predictions to validate the proposed approach. The proposed approach not only greatly saves the time of conducting time-consuming pile load tests, but also effectively avoids solving the complex partial differential equations involved in the consolidation analysis, and hence is feasible enough to determine the time-dependent load carrying behaviors of jacked piles in clay

Near-Horizon Virasoro Symmetry and the Entropy of de Sitter Space in Any Dimension

De Sitter spacetime is known to have a cosmological horizon that enjoys thermodynamic-like properties similar to those of a black hole horizon. In this note we show that a universal argument can be given for the entropy of de Sitter spacetime in arbitrary dimensions, by generalizing a recent near horizon symmetry plus conformal field theory argument of Carlip for black hole entropy. The implications of this argument are also discussed.Comment: 13 pages, no figure. Add one reference and correct a minor typo in pp.6, no change was made in tex