Rumor Processes in Random Environment on \mathbb{N} and on Galton–Watson Trees

The aim of this paper is to study rumor processes in random environment. In a rumor process a signal starts from the stations of a fixed vertex (the root) and travels on a graph from vertex to vertex. We consider two rumor processes. In the firework process each station, when reached by the signal, transmits it up to a random distance. In the reverse firework process, on the other hand, stations do not send any signal but they “listen” for it up to a random distance. The first random environment that we consider is the deterministic 1-dimensional tree $\mathbb{N}$ with a random number of stations on each vertex; in this case the root is the origin of $\mathbb{N}$ . We give conditions for the survival/extinction on almost every realization of the sequence of stations. Later on, we study the processes on Galton–Watson trees with random number of stations on each vertex. We show that if the probability of survival is positive, then there is survival on almost every realization of the infinite tree such that there is at least one station at the root. We characterize the survival of the process in some cases and we give sufficient conditions for survival/extinction.     

A Fleming–Viot Particle Representation¶of the Dirichlet Laplacian

We consider a model with a large number N of particles which move according to independent Brownian motions. A particle which leaves a domain D is killed; at the same time, a different particle splits into two particles. For large N, the particle distribution density converges to the normalized heat equation solution in D with Dirichlet boundary conditions. The stationary distributions converge as N→∞ to the first eigenfunction of the Laplacian in D with the same boundary conditions. Received: 11 November 1999 / Accepted: 19 May 2000     

Regularity and Existence of Global Solutions to the Ericksen–Leslie System in {\mathbb{R}^2}

In this paper, we first establish the regularity theorem for suitable weak solutions to the Ericksen–Leslie system in \({\mathbb{R}^2}\) . Building on such a regularity, we then establish the existence of a global weak solution to the Ericksen–Leslie system in \({\mathbb{R}^2}\) for any initial data in the energy space, under the physical constraints on the Leslie coefficients ensuring the dissipation of energy of the system, which is smooth away from at most finitely many times. This extends earlier works by Lin et al. (Arch Ration Mech Anal 197:297–336, 2010) on a simplified nematic liquid crystal flow to the general Ericksen–Leslie system.     

The Local Gromov–Witten Theory of $${\mathbb{C}\mathbb{P}^1}$$ and Integrable Hierarchies

In this paper we begin the study of the relationship between the local Gromov–Witten theory of Calabi–Yau rank two bundles over the projective line and the theory of integrable hierarchies. We first of all construct explicitly, in a large number of cases, the Hamiltonian dispersionless hierarchies that govern the full-descendent genus zero theory. Our main tool is the application of Dubrovin’s formalism, based on associativity equations, to the known results on the genus zero theory from local mirror symmetry and localization. The hierarchies we find are apparently new, with the exception of the resolved conifold in the equivariantly Calabi–Yau case. For this example the relevant dispersionless system turns out to be related to the long-wave limit of the Ablowitz–Ladik lattice. This identification provides us with a complete procedure to reconstruct the dispersive hierarchy which should conjecturally be related to the higher genus theory of the resolved conifold. We give a complete proof of this conjecture for genus g ≤ 1; our methods are based on establishing, analogously to the case of KdV, a “quasi-triviality” property for the Ablowitz–Ladik hierarchy at the leading order of the dispersive expansion. We furthermore provide compelling evidence in favour of the resolved conifold/Ablowitz–Ladik correspondence at higher genus by testing it successfully in the primary sector for g = 2.     

Uniform in N global well-posedness of the time-dependent Hartree–Fock–Bogoliubov equations in $$\mathbb {R}^{1+1}$$

We prove the global well-posedness of the time-dependent Hartree–Fock–Bogoliubov (TDHFB) equations in \(\mathbb {R}^{1+1}\) with two-body interaction potential of the form \(N^{-1}v_N(x) = N^{\beta -1} v(N^\beta x)\) where \(v\ge 0\) is a sufficiently regular radial function, i.e., \(v \in L^1(\mathbb {R})\cap C^\infty (\mathbb {R})\). In particular, using methods of dispersive PDEs similar to the ones used in Grillakis and Machedon (Commun Partial Differ Equ 42:24–67, 2017), we are able to show for any scaling parameter \(\beta >0\) the TDHFB equations are globally well-posed in some Strichartz-type spaces independent of N, cf. (Bach et al. in The time-dependent Hartree–Fock–Bogoliubov equations for Bosons, 2016. arXiv:1602.05171).     

Renormalisation of Φ4-Theory on Non-Commutative $$\mathbb{R}^{4}$$ to All Orders

We present the main ideas and techniques of the proof that the duality-covariant four-dimensional non-commutative ⁴-model is renormalisable to all orders. This includes the reformulation as a dynamical matrix model, the solution of the free theory by orthogonal polynomials as well as the renormalisation byflow equations involving power-counting theorems for ribbon graphs drawn on Riemann surfaces     

Long-Time Dynamics of Korteweg–de Vries Solitons Driven by Random Perturbations

This paper deals with the transmission of a soliton in a random medium described by a randomly perturbed Korteweg–de Vries equation. Different kinds of perturbations are addressed, depending on their specific time or position dependences, with or without damping. We derive effective evolution equations for the soliton parameter by applying a perturbation theory of the inverse scattering transform and limit theorems of stochastic calculus. Original results are derived that are very different compared to a randomly perturbed Nonlinear Schrödinger equation. First the emission of a soliton gas is proved to be a very general feature. Second some perturbations are shown to involve a speeding-up of the soliton, instead of the decay that is usually observed in random media.     

$${\mathbb{Z}_N}$$ -Curves Possessing No Thomae Formulae of Bershadsky–Radul Type

A \({\mathbb{Z}_N}\) -curve is one of the form \({y^{N}=(x-\lambda_{1})^{m_{1}}\cdots(x-\lambda_{s})^{m_{s}}}\) . When N = 2 these curves are called hyperelliptic and for them Thomae proved his classical formulae linking the theta functions corresponding to their period matrices to the branching values λ₁, . . . , λ _s . In his work on Fermionic fields on \({\mathbb{Z}_N}\) -curves with arbitrary N, Bershadsky and Radul discovered the existence of generalized Thomae’s formulae for these curves which they wrote down explicitly in the case in which all rotation numbers m _i equal 1. This work was continued by several authors and new Thomae’s type formulae for \({\mathbb{Z}_N}\) -curves with other rotation numbers m _i were found. In this article we prove that for some choices of the rotation numbers the corresponding \({\mathbb{Z}_N}\) -curves do not admit such generalized Thomae’s formulae.     

Time-Dependent Vacuum Energy Induced by {\mathcal{D}}–Particle Recoil

We consider cosmology in the framework of a material reference system of particles, including the effects of quantum recoil induced by closed-string probe particles. We find a time-dependent contribution to the cosmological vacuum energy, which relaxes to zero as 1/t ² for large times t. If this energy density is dominant, the Universe expands with a scale factor R(t)t ². We show that this possibility is compatible with recent observational constraints from high–redshift supernovae, and may also respect other phenomenological bounds on time variation in the vacuum energy imposed by early cosmology.     

Trapping State in a System of Single Three-level Trapped Ion Driven by External Fields

We investigate the interaction of a single three-level trapped ion with two laser beams. By applying a unitary transformation and a small rotating transformation, an exact solution to this quantum system is obtained without performing the Lamb-Dicke approximation, and the trapping state is observed.     

Beam-target double-spin asymmetry A{LT} in charged pion production from deep inelastic scattering on a transversely polarized {3}He target at 1.4<Q{2}<2.7 GeV{2}

We report the first measurement of the double-spin asymmetry A{LT} for charged pion electroproduction in semi-inclusive deep-inelastic electron scattering on a transversely polarized {3}He target. The kinematics focused on the valence quark region, 0.16相似文献   

Low optical loss nano-structured \mathrm{TiO }_{2} planar waveguides by sol–gel route for photonic crystal applications

Nano-structured $\mathrm{TiO }_{2}$ planar waveguides were prepared by sol–gel route: titanium tetraisopropoxide was dissolved in isopropanol, and then hydrolyzed by adding a water/isopropanol mixture with a controlled hydrolysis ratio. The resulting sol was deposited by “dip-coating” on a glass substrate with a controlled withdrawal speed. The obtained films were annealed for 2 h at 350 and $500\,^{\circ }\mathrm C $ , respectively. The structural and morphological properties of the synthesized films were analyzed by X-ray diffraction, scanning electron microscopy, and atomic force microscopy. Optical properties such as refractive index, thickness, number of propagating modes, and attenuation coefficient were measured at 632.8 nm by m-lines spectroscopy as a function of the elaboration parameters. The films exhibit diffraction pattern consistent with an anatase phase and the $\mathrm{TiO }_{2}$ planar waveguides are multimodes and demonstrate propagation losses as low as 0.3 dB/cm.     

Particle size effects on magnetic properties of yttrium iron garnets prepared by a sol–gel method

We present detailed measurements of field—and temperature—dependence of magnetization in nanocrystalline YIG (Y₃Fe₅O₁₂) particles. The fine powders were prepared using sol–gel method. Samples with particle sizes ranging from 45 to 450 nm were obtained. We observe that the saturation magnetization decreases as the particle size is reduced due to enhancement of the surface spin effects. Below a critical diameter D_s≅190 nm, the particles become single domains and the coercive forces reaches a maximum at diameters close to the critical value. As the particle size decreases the coercivity diminishes and at D_p≃35 nm diameters the upper limit of superparamagnetic behavior is reached. A quantitative comparison of temperature and particle size dependence of coercivity shows a satisfactory agreement that is expected for an assembly of randomly oriented particles.