Almost sure exponential stability of numerical solutions for stochastic delay differential equations

Using techniques based on the continuous and discrete semimartingale convergence theorems, this paper investigates if numerical methods may reproduce the almost sure exponential stability of the exact solutions to stochastic delay differential equations (SDDEs). The important feature of this technique is that it enables us to study the almost sure exponential stability of numerical solutions of SDDEs directly. This is significantly different from most traditional methods by which the almost sure exponential stability is derived from the moment stability by the Chebyshev inequality and the Borel–Cantelli lemma

Approximate solutions of stochastic differential delay equations with Markovian switching

Our main aim is to develop the existence theory for the solutions to stochastic differential delay equations with Markovian switching (SDDEwMSs) and to establish the convergence theory for the Euler-Maruyama approximate solutions under the local Lipschitz condition. As an application, our results are used to discuss a stochastic delay population system with Markovian switching

Isostaticity, auxetic response, surface modes, and conformal invariance in twisted kagome lattices

Model lattices consisting of balls connected by central-force springs provide much of our understanding of mechanical response and phonon structure of real materials. Their stability depends critically on their coordination number $z$. $d$-dimensional lattices with $z=2d$ are at the threshold of mechanical stability and are isostatic. Lattices with $z<2d$ exhibit zero-frequency "floppy" modes that provide avenues for lattice collapse. The physics of systems as diverse as architectural structures, network glasses, randomly packed spheres, and biopolymer networks is strongly influenced by a nearby isostatic lattice. We explore elasticity and phonons of a special class of two-dimensional isostatic lattices constructed by distorting the kagome lattice. We show that the phonon structure of these lattices, characterized by vanishing bulk moduli and thus negative Poisson ratios and auxetic elasticity, depends sensitively on boundary conditions and on the nature of the kagome distortions. We construct lattices that under free boundary conditions exhibit surface floppy modes only or a combination of both surface and bulk floppy modes; and we show that bulk floppy modes present under free boundary conditions are also present under periodic boundary conditions but that surface modes are not. In the the long-wavelength limit, the elastic theory of all these lattices is a conformally invariant field theory with holographic properties, and the surface waves are Rayleigh waves. We discuss our results in relation to recent work on jammed systems. Our results highlight the importance of network architecture in determining floppy-mode structure.Comment: 12 pages, 7 figure

Performance of Photosensors in the PandaX-I Experiment

We report the long term performance of the photosensors, 143 one-inch R8520-406 and 37 three-inch R11410-MOD photomultipliers from Hamamatsu, in the first phase of the PandaX dual-phase xenon dark matter experiment. This is the first time that a significant number of R11410 photomultiplier tubes were operated in liquid xenon for an extended period, providing important guidance to the future large xenon-based dark matter experiments.Comment: v3 as accepted by JINST with modifications based on reviewers' comment

On an elliptic Kirchhoff-type problem depending on two parameters

In this paper, we consider the Dirichlet problem associated to an elliptic Kirchhoff-type equation depending on two parameters. Under rather general and natural assumptions, we prove that, for certain values of the parameters, the problem has at least three solutions

On Necessary and Sufficient Conditions for Preserving Convergence Rates to Equilibrium in Deterministically and Stochastically Perturbed Differential Equations with Regularly Varying Nonlinearity

This paper develops necessary and sufficient conditions for the preservation of asymptotic convergence rates of deterministically and stochastically perturbed ordinary differential equations with regularly varying nonlinearity close to their equilibrium. Sharp conditions are also established which preserve the asymptotic behaviour of the derivative of the underlying unperturbed equation. Finally, necessary and sufficient conditions are established which enable finite difference approximations to the derivative in the stochastic equation to preserve the asymptotic behaviour of the derivative of the unperturbed equation, even though the solution of the stochastic equation is nowhere differentiable, almost surely

Elasticity of highly cross-linked random networks

Starting from a microscopic model of randomly cross-linked particles with quenched disorder, we calculate the Laudau-Wilson free energy S for arbitrary cross-link densities. Considering pure shear deformations, S takes the form of the elastic energy of an isotropic amorphous solid state, from which the shear modulus can be identified. It is found to be an universal quantity, not depending on any microscopic length-scales of the model.Comment: 6 pages, 5 figure

Modal and non-modal stabilities of flow around a stack of plates

Modal and non-modal stabilities of flow around a stack of flat plates are investigated by means of asymptotic stability and transient growth analyses respectively. It is observed that over the parameters considered, both the base flow and the stabilities vary as a function of ReW2/(W−1)2, i.e. the product of the Reynolds number and the square of the expansion ratio of the stack. The most unstable modes are found to be located downstream of the recirculation bubble while the global optimal initial perturbations (resulting in maximum energy growth over the entire domain) and the weighted optimal initial perturbations (resulting in maximum energy growth in the close downstream region of the stack) concentrate around the stack end owing to the Orr mechanism. In direct numerical simulations (DNS) of the base flow initially perturbed by the modes, it is noticed that the weighted optimal initial perturbation induces periodic vortex shedding downstream of the stack much faster than the most unstable mode. This observation suggests that the widely reported vortex shedding in flow around a stack of plates, e.g. in thermoacoustic devices, is associated with perturbations around the stack end