A scaling law for a model of epitaxially strained elastic films with dislocations

In this work we derive a lower bounds for the Hausdorff and fractal dimensions of the global attractor of the Sabra shell model of turbulence in different regimes of parameters. We show that for a particular choice of the forcing and for sufficiently small viscosity term $\nu$, the Sabra shell model has a global attractor of large Hausdorff and fractal dimensions proportional to $\log_\lambda \nu^{-1}$ for all values of the governing parameter $\epsilon$, except for $\epsilon=1$. The obtained lower bounds are sharp, matching the upper bounds for the dimension of the global attractor obtained in our previous work. Moreover, we show different scenarios of the transition to chaos for different parameters regime and for specific forcing. In the ``three-dimensional'' regime of parameters this scenario changes when the parameter $\epsilon$ becomes sufficiently close to 0 or to 1. We also show that in the ``two-dimensional'' regime of parameters for a certain non-zero forcing term the long-time dynamics of the model becomes trivial for any value of the viscosity.

In this paper we present two types of local error estimators for the primal finite‐element‐method (FEM) by duality arguments. They are first derived from the (explicit) residual error estimation method (REM) and then—as a new contribution—from the (implicit) posterior equilibrium method (PEM) using improved boundary tractions, gained by local post‐processing with local Neumann problems, with applications in elastic problems. For the displacements a local error estimator with an upper bound is derived and also a local estimator for stresses. Furthermore—for better numerical efficiency—the residua are projected energy‐invariant onto reference elements, where the local Neumann problems have to be solved. Comparative examples between REM‐ and PEM‐type local estimators show superior effectivity indices for the latter one. Copyright © 2001 John Wiley & Sons, Ltd.

This paper is motivated by the complex blister patterns sometimes seen in thin elastic films on thick, compliant substrates. These patterns are often induced by an elastic misfit that compresses the film. Blistering permits the film to expand locally, reducing the elastic energy of the system. It is therefore natural to ask: what is the minimum elastic energy achievable by blistering on a fixed area fraction of the substrate? This is a variational problem involving both the elastic deformation of the film and substrate and the geometry of the blistered region. It involves three small parameters: the nondimensionalized thickness of the film, the compliance ratio of the film/substrate pair, and the mismatch strain. In formulating the problem, we use a small‐slope (Föppl–von Kármán) approximation for the elastic energy of the film, and a local approximation for the elastic energy of the substrate.For a one‐dimensional version of the problem, we obtain “matching” upper and lower bounds on the minimum energy, in the sense that both bounds have the same scaling behavior with respect to the small parameters. The upper bound is straightforward and familiar: it is achieved by periodic blistering on a specific length scale. The lower bound is more subtle, since it must be proved without any assumption on the geometry of the blistered region.For a two‐dimensional version of the problem, our results are less complete. Our upper and lower bounds only “match” in their scaling with respect to the nondimensionalized thickness, not in the dependence on the compliance ratio and the mismatch strain. The lower bound is an easy consequence of our one‐dimensional analysis. The upper bound considers a two‐dimensional lattice of blisters and uses ideas from the literature on the folding or “crumpling” of a confined elastic sheet. Our main two‐dimensional result is that in a certain parameter regime, the elastic energy of this lattice is significantly lower than that of a few large blisters. © 2015 Wiley Periodicals, Inc.

We develop an analytical description of static ferrodark solitons, the $\mathbb{Z}_2$ topological defects in the magnetic order, in the easy-plane phase of ferromagnetic spin-1 Bose-Einstein condensates. We find that the type-I ferrodark soliton has a single width while the type-II ferrodark soliton exhibits two characteristic length scales. The proposed ansatzes show excellent agreement with numerical results. Spin-singlet amplitudes, nematic tensor densities and nematic currents of ferrodark solitons are also discussed. The $\mathbb{Z}_2$ topological defects in the mass superfluid order, dark-dark-dark vector solitons, are obtained exactly in the parameter regime where exact ferrodark solitons exist. The dark-dark-dark vector soliton has higher excitation energy than ferrodark solitons.

Topological defects at the (001) surface of MgO: energetics and reactivity

How can we understand heteroepitaxial growth modes with a computer simulation? (A perspective on the article, “heteroepitaxial growth modes with dislocations in a two-dimensional elastic lattice model”)

We study a fully back-reacted non-abelian vortex solution in an extension of the holographic superconductor setup. The thermodynamic properties of the vortex are computed. We show that, in some regime of parameters, the non-abelian vortex solution has a lower free energy than a competing abelian vortex solution. The solution is dual to a finite-temperature perturbed conformal field theory with a topological defect, on which operators related to the Goldstone modes of a spontaneously broken symmetry are localized. We compute numerically the retarded Green function of these operators and we find, in the classical approximation in the bulk, a gapless ℂℙ1 excitation on the vortex world line.

We investigate island formation during heteroepitaxial growth using an atomistic model that incorporates deposition, activated diffusion, and stress relaxation. For high misfit the system naturally evolves into a state characterized by a narrow island size distribution. The simulations indicate the existence of a strain assisted kinetic mechanism responsible for the self-assembling process, involving enhanced detachment of atoms from the edge of large islands and biased adatom diffusion.

We obtain the exact nontopological soliton lattice solutions of the associated Lamé equation in different parameter regimes and compute the corresponding energy for each of these solutions. We show that in specific limits these solutions give rise to nontopological (pulse-like) single solitons, as well as to different types of topological (kink-like) single soliton solutions of the associated Lamé equation. Following Manton, we also compute, as an illustration, the asymptotic interaction energy between these soliton solutions in one particular case. Finally, in specific limits, we deduce the soliton lattices, as well as the topological single soliton solutions of the Lamé equation, and also the sine-Gordon soliton solution.

We review some mathematical results on the Ginzburg–Landau model with and without magnetic field. The Ginzburg–Landau energy is the standard model for superconductivity, able to predict the existence of vortices (which are quantized, topological defects) in certain regimes of the applied magnetic field. We focus particularly on deriving limiting (or reduced) energies for the Ginzburg–Landau energy functional, depending on the various parameter regimes, in the spirit of -convergence. These passages to the limit allow to perform a sort of dimension-reduction and to deduce a rather complete characterization of the behavior of vortices for energy-minimizers, in agreement with the physics results. We also describe the behavior of energy critical points, the stability of the solutions, the motion of vortices for solutions of the gradient-flow of the Ginzburg–Landau energy, and show how they are also governed by those of the limiting energies. Mathematics Subject Classification (2000). Primary 00A05; Secondary 00B10.

An important component of the cyber-defense mechanism is the adequate staffing levels of its cybersecurity analyst workforce and their optimal assignment to sensors for investigating the dynamic alert traffic. The ever-increasing cybersecurity threats faced by today’s digital systems require a strong cyber-defense mechanism that is both reactive in its response to mitigate the known risk and proactive in being prepared for handling the unknown risks. In order to be proactive for handling the unknown risks, the above workforce must be scheduled dynamically so the system is adaptive to meet the day-to-day stochastic demands on its workforce (both size and expertise mix). The stochastic demands on the workforce stem from the varying alert generation and their significance rate, which causes an uncertainty for the cybersecurity analyst scheduler that is attempting to schedule analysts for work and allocate sensors to analysts. Sensor data are analyzed by automatic processing systems, and alerts are generated. A portion of these alerts is categorized to be significant , which requires thorough examination by a cybersecurity analyst. Risk, in this article, is defined as the percentage of significant alerts that are not thoroughly analyzed by analysts. In order to minimize risk, it is imperative that the cyber-defense system accurately estimates the future significant alert generation rate and dynamically schedules its workforce to meet the stochastic workload demand to analyze them. The article presents a reinforcement learning-based stochastic dynamic programming optimization model that incorporates the above estimates of future alert rates and responds by dynamically scheduling cybersecurity analysts to minimize risk (i.e., maximize significant alert coverage by analysts) and maintain the risk under a pre-determined upper bound. The article tests the dynamic optimization model and compares the results to an integer programming model that optimizes the static staffing needs based on a daily-average alert generation rate with no estimation of future alert rates (static workforce model). Results indicate that over a finite planning horizon, the learning-based optimization model, through a dynamic (on-call) workforce in addition to the static workforce, (a) is capable of balancing risk between days and reducing overall risk better than the static model, (b) is scalable and capable of identifying the quantity and the right mix of analyst expertise in an organization, and (c) is able to determine their dynamic (on-call) schedule and their sensor-to-analyst allocation in order to maintain risk below a given upper bound. Several meta-principles are presented, which are derived from the optimization model, and they further serve as guiding principles for hiring and scheduling cybersecurity analysts. Days-off scheduling was performed to determine analyst weekly work schedules that met the cybersecurity system’s workforce constraints and requirements.

We consider a singularly-perturbed nonconvex energy functional which arises in the study of microstructures in shape memory alloys. The scaling law for the minimal energy predicts a transition from a parameter regime in which uniform structures are favored, to a regime in which the formation of fine patterns is expected. We focus on the transition regime and derive the reduced model in the sense of $\Gamma$-convergence. The limit functional turns out to be similar to the Mumford-Shah functional with additional constraints on the jump set of admissible functions. One key ingredient in the proof is an approximation result for $SBV^p$ functions whose jump sets have a prescribed orientation.

This paper provides a solution for sensor bias estimation based on local tracks at a single time without a priori association for a decentralized multiple sensor tracking system. Each local tracker generates its own local state estimates ignoring the bias. The fusion center then performs track-to-track fusion occasionally after estimating the sensor biases based on the common targets tracked by different sensors. The likelihood function of the bias in a multisensor-multitarget scenario is derived. Using this likelihood, it is shown that the difference of the local estimates is the sufficient statistic for estimating the biases. A least squares solution of the bias estimates and corresponding Cramer-Rao Lower Bound (CRLB) are presented assuming uncorrelatedness as well as accounting for the crosscorrelation between the local estimation errors. Two approaches to estimate the sensor biases in the absence of known track-to-track association, namely, the Maximum Likelihood estimator combined with Probabilistic Data Association (ML-PDA) and an estimator based on soft data association, are proposed. These methods are compared with the baseline solution with known (perfect) track-to-track association by Monte Carlo simulations. The experimental results indicate that the bias estimator based on the soft data association provides nearly optimal performance and has less computational load than the one using ML-PDA.

This paper investigates the problem of robust fusion state estimation for multi-sensor systems under the influence of time-correlated fading channels, incorporating a dynamic event-triggered mechanism (DETM). The randomly occurring parameter uncertainties are characterized by a stochastic variable following a Bernoulli distribution, while sensor measurements are transmitted to the corresponding estimators through time-correlated fading channels and dynamic event-triggered mechanisms. The DETM dynamically adjusts the triggering threshold via regulation and memory factors, enhancing adaptability in data transmission while effectively reducing redundant communication overhead. Furthermore, an augmented state model is constructed by integrating system states, channel coefficients, and the event-triggering mechanism, thereby comprehensively capturing the impact of dynamic environments on state estimation. Based on this model, a local state estimation algorithm is designed to ensure the convergence of the upper bound of the local estimation error covariance, which is further minimized at each time step through adaptive adjustment of local estimator gains. Subsequently, the local estimates obtained from multiple estimators are fused using the covariance intersection fusion strategy, improving the overall estimation accuracy. Simulation experiments demonstrate that the proposed recursive fusion state estimation framework significantly reduces communication overhead and enhances estimation performance in the presence of both time-correlated fading channels and randomly occurring parameter uncertainties, while maintaining an acceptable computational cost. Compared to the traditional Kalman filtering method, the proposed recursive fusion state estimation algorithm improves estimation accuracy by 58% while increasing computational time by only 32.4%. Additionally, the DETM effectively reduces communication frequency by 36.7%

Microcosm biofilm has been applied to induce carious lesions in dentin. However, no study has been done to compare the impact of the type of model for providing nutrients to microcosm biofilm formation on dentin.ObjectiveThis study compared the performance of two kinds of models (static and semi-dynamic) on the biofilm formation and the development of dentin carious lesions.Material and MethodsIn both models, biofilm was produced using inoculum from pooled human saliva mixed with McBain saliva for the first 8 h (5% CO2 and 37°C). Afterwards, for the static model, the samples were placed in 24-wells microplate containing McBain saliva with 0.2% sucrose, which was replaced at 24 h. In the semi-dynamic model, the samples were submitted to artificial mouth system with continuous flow of McBain saliva with 0.2% sucrose (0.15 ml/min, 37°C) for 10 h a day (for the other 14 h, no flow was applied, similarly to the static model). After 5 days, biofilm viability was measured by fluorescence and dentin demineralization by transverse microradiography.ResultsBiofilm viability was significantly lower for the static compared with semi-dynamic model, while dentin demineralization was significantly higher for the first one (p<0.05). The static model was able to produce a higher number of typical subsurface lesions compared with the semi-dynamic model (p<0.05).ConclusionsThe type of model (static and semi-dynamic) applied in the microcosm biofilm may have influence on it's viability and the severity/profile of dentin carious lesions.