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A spatial model for dormancy in random environment
Get PDFIn this paper, we introduce a spatial model for dormancy in random environment via a two-type branching random walk in continuous-time, where individuals can switch between dormant and active states through spontaneous switching independent of the random environment. However, the branching mechanism is governed by a random environment which dictates the branching rates. We consider three specific choices for random environments composed of particles: (1) a Bernoulli field of immobile particles, (2) one moving particle, and (3) a Poisson field of moving particles. In each case, the particles of the random environment can either be interpreted as emphcatalysts, accelerating the branching mechanism, or as emphtraps, aiming to kill the individuals. The different between active and dormant individuals is defined in such a way that dormant individuals are protected from being trapped, but do not participate in migration or branching. We quantify the influence of dormancy on the growth resp.,survival of the population by identifying the large-time asymptotics of the expected population size. The starting point for our mathematical considerations and proofs is the parabolic Anderson model via the Feynman-Kac formula. Especially, the quantitative investigation of the role of dormancy is done by extending the Parabolic Anderson model to a two-type random walk
Three-state -SOS models on binary Cayley trees
Get PDFWe consider a version of the solid-on-solid model on the Cayley tree of order two in which vertices carry spins of value 0,1 or 2 and the pairwise interaction of neighboring vertices is given by their spin difference to the power p>0. We exhibit all translation-invariant splitting Gibbs measures (TISGMs) of the model and demonstrate the existence of up to seven such measures, depending on the parameters. We further establish general conditions for extremality and non-extremality of TISGMs in the set of all Gibbs measures and use them to examine selected TISGMs for a small and a large p. Notably, our analysis reveals that extremality properties are similar for large p compared to the case p=1, a case that has been explored already in previous work. However, for the small p, certain measures that were consistently non-extremal for p=1 do exhibit transitions between extremality and non-extremality
Branched Itô formula and natural Itô--Stratonovich isomorphism
Get PDFBranched rough paths define integration theories that may fail to satisfy the usual integration by parts identity. The intrinsically-defined projection of the Connes-Kreimer Hopf algebra onto its primitive elements defined by Broadhurst and Kreimer, and further studied by Foissy, allows us to view it as a commutative B?-algebra and thus to write an explicit change- of-variable formula for solutions to rough differential equations. This formula, which is realised by means of an explicit morphism from the Grossman-Larson Hopf algebra to the Hopf algebra of differential operators, restricts to the well-known Itô formula for semimartingales. We establish an isomorphism with the shuffle algebra over primitives, extending Hoffman?s exponential for the quasi shuffle algebra, and in particular the usual Itô-Stratonovich correction formula for semimartingales. We place special emphasis on the one-dimensional case, in which certain rough path terms can be expressed as polynomials in the extended trace path, which for semimartingales restrict to the well-known Kailath-Segall polynomials. We end by describing an algebraic framework for generating examples of branched rough paths, and, motivated by the recent literature on stochastic processes, exhibit a few such examples above scalar 1/4-fractional Brownian motion, two of which are ?truly branched?, i.e. not quasi- geometric
Finite‐strain poro‐visco‐elasticity with degenerate mobility
No full textA quasistatic nonlinear model for poro-visco-elastic solids at finite strains is considered in the Lagrangian frame using the concept of second-order nonsimple materials and Kelvin--Voigt-type viscosity. The elastic stresses satisfy static frame-indifference, while the viscous stresses satisfy dynamic frame-indifference. The mechanical equation is coupled to a diffusion equation for a solvent or fluid content. The latter is pulled-back to the reference configuration. To treat the nonlinear dependence of the mobility tensor on the deformation gradient, the result by Healey and Krömer is used to show that the determinant of the deformation gradient is bounded away from zero. Moreover, the focus is on the physically relevant case of degenerate mobilities. The existence of weak solutions is shown using a staggered time-incremental scheme and suitable energy-dissipation inequalities
Interface dynamics in a degenerate Cahn--Hilliard model for viscoelastic phase separation
Get PDFThe formal sharp-interface asymptotics in a degenerate Cahn--Hilliard model for viscoelastic phase separation with cross-diffusive coupling to a bulk stress variable are shown to lead to non-local lower-order counterparts of the classical surface diffusion flow. The diffuse-interface model is a variant of the Zhou--Zhang--E model and has an Onsager gradient-flow structure with a rank-deficient mobility matrix reflecting the ODE character of stress relaxation. In the case of constant coupling, we find that the evolution of the zero level set of the order parameter approximates the so-called intermediate surface diffusion flow. For non-constant coupling functions monotonically connecting the two phases, our asymptotic analysis leads to a family of third order whose propagation operator behaves like the square root of the minus Laplace--Beltrami operator at leading order. In this case, the normal velocity of the moving sharp interface arises as the Lagrange multiplier in a constrained elliptic equation, which is at the core of our derivation. The constrained elliptic problem can be solved rigorously by a variational argument, and is shown to encode the gradient structure of the effective geometric evolution law. The asymptotics are presented for deep quench, an intermediate free boundary problem based on the double obstacle potential
Sharp-Interface Limits of Cahn--Hilliard Models and Mechanics with Moving Contact Lines
No full textWe consider the fluid-structure interaction of viscoelastic solids and Stokesian multiphase fluid flows with moving capillary interfaces and investigate the impact of moving contact lines. Thermodynamic consistency of Lagrangian diffuse and sharp-interface models is ensured even on the discrete level by providing a monolithic incremental time discretization and a finite element space discretization. We numerically analyze how phase-field models converge to sharp-interface limits when the interface thickness tends to zero, ε → 0 , and investigate scalings of the Cahn–Hilliard mobility m ( ε ) = m 0 ε α for α ― < α < α ― . In the presence of interfaces, certain sharp-interface limits are only valid for an interval α ― < α < α ― , i.e., there is an upper and lower bound on the range of valid scaling exponents α . We show that with moving contact lines scaling is more restrictive since α ≈ α ― causes significant errors due to excess diffusion. Similarly, we demonstrate that α ≈ α ― leads to nonconvergence to the sharp-interface limit. We propose 2 ≤ α ≤ 5 / 2 as a range of exponents that ensure optimal convergence of the phase field dynamics towards the sharp interface dynamics as 2 ≤ α ≤ 5 / 2
Monotone Discretizations for Elliptic Second Order Partial Differential Equations - Data of Reference Curves
No full textThis is the data to following monograph: Gabriel R. Barrenechea, Volker John, Petr Knobloch (eds), "Monotone Discretizations for Elliptic Second Order Partial Differential Equations", Springer Series in Computational Mathematics, Springer Nature. This monograph contains the first comprehensive presentation of monotone discretization for elliptic boundary value problems. All currently available relevant methods are studied in detail. Besides monotonicity or the satisfaction of discrete maximum principles, other properties of these methods, in particular their error analysis are discussed. Many concepts and techniques from the numerical analysis are explained. Numerical examples illustrate the behavior of many methods and numerical comparisons are presented. Here, the data of the reference curves used in the numerical simulations are provided
Decentralized Saddle Point Problems via Non-Euclidean Mirror Prox
Get PDFWe consider distributed convex-concave saddle point problems over arbitrary connected undirected networks and propose a decentralized distributed algorithm for their solution. The local functions distributed across the nodes are assumed to have global and local groups of variables. For the proposed algorithm we prove non-asymptotic convergence rate estimates with explicit dependence on the network characteristics. To supplement the convergence rate analysis, we propose lower bounds for strongly-convex-strongly-concave and convex-concave saddle-point problems over arbitrary connected undirected networks. We illustrate the considered problem setting by a particular application to distributed calculation of non-regularized Wasserstein barycenters
Stability of the higher-order splitting methods for the generalized nonlinear Schrödinger equation
No full textThe numerical solution of the generalized nonlinear Schrödinger equation by simple splitting methods can be disturbed by so-called spurious instabilities. We analyze these numerical instabilities for an arbitrary splitting method and apply our results to several well-known higher-order splittings. We find that the spurious instabilities can be suppressed to a large extent. However, they never disappear completely if one keeps the integration step above a certain limit and applies what is considered to be a more accurate higher-order method. The latter can be used to make calculations more accurate with the same numerically stable step, but not to make calculations faster with a much larger step
Strain distribution in zincblende and wurtzite GaAs nanowires bent by a one-sided (In, Al)As shell: Consequences for torsion, chirality, and piezoelectricity
No full textWe present a finite-strain model that is capable of describing the large deformations in bent nanowire heterostructures. The model incorporates a nonlinear strain formulation derived from the first Piola-Kirchhoff stress tensor, coupled with an energy functional that effectively captures the lattice-mismatch-induced strain field. We use the finite element method to solve the resulting partial differential equations and extract cross- sectional maps of the full strain tensor for both zincblende and wurtzite nanowires with lattice-mismatched core and one-sided stressor shell. In either case, we show that the bending is essentially exclusively determined by E-zz . However, the distinct difference in shear strain has important consequences with regard to both the mechanical deformation and the existence of transverse piezoelectric fields in the nanowires