Stability Estimates Research Articles

A note on the telegraph type differential equation with time involution

In the present paper, the initial value problem for the telegraph type involutory in t second order linear partial differential equation with damping term is investigated. The equivalent initial value problem for the fourth order partial differential equations to the initial value problem for this second order linear partial differential equations with involution and damping term is obtained. Applying the operator tools, the stability estimates for the solution and its first and second order derivatives of this problem are established.

A Fast Second-Order ADI Finite Difference Scheme for the Two-Dimensional Time-Fractional Cattaneo Equation with Spatially Variable Coefficients

This paper presents an efficient finite difference method for solving the time-fractional Cattaneo equation with spatially variable coefficients in two spatial dimensions. The main idea is that the original equation is first transformed into a lower system, and then the graded mesh-based fast L2-1σ formula and second-order spatial difference operator for the Caputo derivative and the spatial differential operator are applied, respectively, to derive the fully discrete finite difference scheme. By adding suitable perturbation terms, we construct an efficient fast second-order ADI finite difference scheme, which significantly improves computational efficiency for solving high-dimensional problems. The corresponding stability and error estimate are proved rigorously. Extensive numerical examples are shown to substantiate the accuracy and efficiency of the proposed numerical scheme.

Existence and stability of weak solutions of the Vlasov–Poisson system in localised Yudovich spaces

We consider the Vlasov–Poisson system both in the repulsive (electrostatic potential) and in the attractive (gravitational potential) cases. Our first main theorem yields the analog for the Vlasov–Poisson system of Yudovich’s celebrated well-posedness theorem for the Euler equations: we prove the uniqueness and the quantitative stability of Lagrangian solutions f=f(t,x,v) whose associated spatial density ρf=ρf(t,x) is potentially unbounded but belongs to suitable uniformly-localised Yudovich spaces. This requirement imposes a condition of slow growth on the function p↦‖ρf(t,⋅)‖Lp uniformly in time. Previous works by Loeper, Miot and Holding–Miot have addressed the cases of bounded spatial density, i.e. ‖ρf(t,⋅)‖Lp≲1 , and spatial density such that ‖ρf(t,⋅)‖Lp∼p1/α for α∈[1,+∞) . Our approach is Lagrangian and relies on an explicit estimate of the modulus of continuity of the electric field and on a second-order Osgood lemma. It also allows for iterated-logarithmic perturbations of the linear growth condition. In our second main theorem, we complement the aforementioned result by constructing solutions whose spatial density sharply satisfies such iterated-logarithmic growth. Our approach relies on real-variable techniques and extends the strategy developed for the Euler equations by the first and fourth-named authors. It also allows for the treatment of more general equations that share the same structure as the Vlasov–Poisson system. Notably, the uniqueness result and the stability estimates hold for both the classical and the relativistic Vlasov–Poisson systems.

Probability density function of roll amplitude for parametric rolling using moment equation

For the ship stability estimation and ship design, it is helpful to know the probability of the roll amplitude exceeding a certain threshold. Therefore, it is necessary to obtain the probability density function of the roll amplitude. In this study, first, we derive the moment values of roll amplitude by combining the moment equations and the linearity of expectation. With this, we propose a method to estimate the probability density function of the roll amplitude by using the obtained moment values. The results of our proposed method are compared to those obtained from Monte Carlo simulations. When the higher-order cumulant neglect closure method is used, the moment values resulting from the moment equations are close to the results of corresponding Monte Carlo simulations. In addition, our proposed method for deriving the probability density function of the roll amplitude is validated by comparison with Monte Carlo simulation results. In conclusion, we can state that the proposed methods for deriving the moments and the probability density function of the roll amplitude are available for practical use cases.

Optimal controls for forward-backward stochastic differential equations: Time-inconsistency and time-consistent solutions

This paper is concerned with an optimal control problem for a forward-backward stochastic differential equation (FBSDE, for short) with a recursive cost functional determined by a backward stochastic Volterra integral equation (BSVIE, for short). It is found that such an optimal control problem is time-inconsistent in general, even if the cost functional is reduced to a classical Bolza type one as in Peng [47], Lim–Zhou [38], and Yong [72]. Therefore, instead of finding a global optimal control (which is time-inconsistent), we will look for a time-consistent and locally optimal equilibrium strategy, which can be constructed via the solution of an associated equilibrium Hamilton–Jacobi–Bellman (HJB, for short) equation. A verification theorem for the local optimality of the equilibrium strategy is proved by means of the generalized Feynman–Kac formula for BSVIEs and some stability estimates of the representation parabolic partial differential equations (PDEs, for short). Under certain conditions, it is proved that the equilibrium HJB equation, which is a nonlocal PDE, admits a unique classical solution. As special cases and applications, the linear-quadratic problems, a mean-variance model, a social planner problem with heterogeneous Epstein–Zin utilities, and a Stackelberg game are briefly investigated. It turns out that our framework can cover not only the optimal control problems for FBSDEs studied in [47,38,72], and so on, but also the problems of the general discounting and some nonlinear appearance of conditional expectations for the terminal state, studied in Yong [73,75] and Björk–Khapko–Murgoci [6].

Linearized inverse problem for biharmonic operators at high frequencies

In this paper, we study the phenomenon of increasing stability in the inverse boundary value problems for the biharmonic equation. The inverse problem is to determine the potential from DtN map. It is a kind of nonlinear inverse problem. By considering a linearized form, we obtain an increasing Lipschitz‐like stability when is large. Furthermore, we extend the discussion to the linearized inverse biharmonic potential problem with attenuation, where an exponential dependence of the attenuation constant is traced in the stability estimate.

High‐frequency stability estimates for the linearized inverse boundary value problem for the biharmonic operator with attenuation on some bounded domains

In this article, high‐frequency stability estimates are explored for the determination of the zeroth‐order perturbation of the biharmonic operator with constant attenuation from the linearized partial Dirichlet‐to‐Neumann map when part of the boundary is inaccessible and flat. The results obtained suggest improvement of the stability with an appropriate choice of frequency.

Projective integration methods in the Runge–Kutta framework and the extension to adaptivity in time

Projective Integration methods are explicit time integration schemes for stiff ODEs with large spectral gaps. In this paper, we show that all existing Projective Integration methods can be written as Runge–Kutta methods with an extended Butcher tableau including many stages. We prove consistency and order conditions of the Projective Integration methods using the Runge–Kutta framework. Spatially adaptive Projective Integration methods are included via partitioned Runge–Kutta methods. New time adaptive Projective Integration schemes are derived via embedded Runge–Kutta methods and step size variation while their accuracy, stability, convergence, and error estimators are investigated analytically and numerically.

Propofol anesthesia destabilizes neural dynamics across cortex

Every day, hundreds of thousands of people undergo general anesthesia. One hypothesis is that anesthesia disrupts dynamic stability-the ability of the brain to balance excitability with the need to be stable and controllable. To test this hypothesis, we developed a method for quantifying changes in population-level dynamic stability in complex systems: delayed linear analysis for stability estimation (DeLASE). Propofol was used to transition animals between the awake state and anesthetized unconsciousness. DeLASE was applied to macaque cortex local field potentials (LFPs). We found that neural dynamics were more unstable in unconsciousness compared with the awake state. Cortical trajectories mirrored predictions from destabilized linear systems. We mimicked the effect of propofol in simulated neural networks by increasing inhibitory tone. This in turn destabilized the networks, as observed in the neural data. Our results suggest that anesthesia disrupts dynamical stability that is required for consciousness.

One-level and two-level operator splitting methods for the unsteady incompressible micropolar fluid equations with double diffusion convection

In this paper, a one-level operator splitting method (OOSM) and a two-level operator splitting method (TOSM) for the two-dimensional or three-dimensional (2D/3D) unsteady incompressible micropolar fluid equations with double diffusion convection (IMFDDC) are proposed and analyzed. Firstly, a OOSM is constructed, which consists of five steps. The projection strategy is adopted to get the values of linear velocity and pressure in the first two steps, then the three elliptic subproblems about the angular velocity, the temperature, and the concentration variables are solved in the last three steps. The original variables with the predefined boundary conditions need to be solved separately in each step. Next, in order to solve the problem efficiently, a TOSM is presented, which only solves the nonlinear equations in the coarse level subspaces and the Oseen type linearized equations in the fine level. The numerical analysis of stability and error estimates of the fully discrete methods show that TOSM can reach the same accuracy as the OOSM with H∼O(h2/3). Numerical examples are provided to confirm the effectiveness and reliability of the proposed methods.

Stability of Fixed Points of Partial Contractivities and Fractal Surfaces

In this paper, a large class of contractions is studied that contains Banach and Matkowski maps as particular cases. Sufficient conditions for the existence of fixed points are proposed in the framework of b-metric spaces. The convergence and stability of the Picard iterations are analyzed, giving error estimates for the fixed-point approximation. Afterwards, the iteration proposed by Kirk in 1971 is considered, studying its convergence, stability, and error estimates in the context of a quasi-normed space. The properties proved can be applied to other types of contractions, since the self-maps defined contain many others as particular cases. For instance, if the underlying set is a metric space, the contractions of type Kannan, Chatterjea, Zamfirescu, Ćirić, and Reich are included in the class of contractivities studied in this paper. These findings are applied to the construction of fractal surfaces on Banach algebras, and the definition of two-variable frames composed of fractal mappings with values in abstract Hilbert spaces.

Estimation of environment stability for fruit yield and capsaicin content by using two models in Capsicum chinense Jacq. (Ghost Pepper) with multi-year evaluation.

Capsicum chinense Jacq. (Ghost Pepper) is well-known for its high pungency and pleasant aroma. The recent years witnessed a significant decline in popularity of this important crop due to the use of inferior planting material and lack of elite lines. To maintain constant performance across a variety of settings, it is crucial to choose stable lines with high yield and capsaicin content, as these are the most promising traits of Ghost Pepper. In this study, 120 high-capsaicin genotypes were subjected to a 3-year (kharif 2017, 2018 and 2019) stability investigation utilizing two well-known stability methods: Eberhart-Russell (ER) and additive main effects and multiple interaction (AMMI). Three replications were used following Randomized Complete Block Design for 11 traits. The experiment soil was sandy loam with pH 4.9. Minimum and maximum temperature of 18.5 °C, 17.5 °C, 17.4 °C and 32.2 °C, 31.3 °C, 32.7 °C and rainfall of 1,781, 2,099, 1,972 mm respectively was recorded for the study period. The genotype-environment linear interaction (G×E Lin.) was highly significant for days to 50% flowering, capsaicin content, fruit length and girth, fruit yield per plant and number of fruits per plant at p < 0.005. G×E interaction for fruit yield and capsaicin content in AMMI-analysis of variance reported 67.07% and 71.51% contribution by IPCA-1 (interactive principal component axis) and 32.76% and 28.49% by IPCA-2, respectively. Eight genotypes were identified to be stable with high yield and capsaicin content. The identified stable lines can be opted for cultivation to reduce the impact of crop failure when grown in different macro-environments. Moreover, the pharmaceutical and spice sectors will also be benefitted from the lines with high capsaicin content. Further research assessing the lines' performance across various regions of India can provide a solid foundation for the crop's evaluation at national level.

A Reduced-Dimension Weighted Explicit Finite Difference Method Based on the Proper Orthogonal Decomposition Technique for the Space-Fractional Diffusion Equation

A kind of reduced-dimension method based on a weighted explicit finite difference scheme and the proper orthogonal decomposition (POD) technique for diffusion equations with Riemann–Liouville fractional derivatives in space are discussed. The constructed approximation method written in matrix form can not only ensure a sufficient accuracy order but also reduce the degrees of freedom, decrease storage requirements, and accelerate the computation rate. Uniqueness, stabilization, and error estimation are demonstrated by matrix analysis. The procedural steps of the POD algorithm, which reduces dimensionality, are outlined. Numerical simulations to assess the viability and effectiveness of the reduced-dimension weighted explicit finite difference method are given. A comparison between the reduced-dimension method and the classical weighted explicit finite difference scheme is presented, including the error in the L2 norm, the accuracy order, and the CPU time.

Determining coefficients for a fractional p-Laplace equation from exterior measurements

We consider an inverse problem of determining the coefficients of a fractional p -Laplace equation in the exterior domain. Assuming suitable local regularity of the coefficients in the exterior domain, we offer an explicit reconstruction formula in the region where the exterior measurements are performed. This formula is then used to establish a global uniqueness result for real-analytic coefficients. In addition, we also derive a stability estimate for the unique determination of the coefficients in the exterior measurement set.

Plasmon-enhanced electrocatalytic ethanol oxidation over Au@Pd nanostars

Hybrid noble-metal nanostructures, which integrate plasmonic characteristics and electrocatalytic performance, have garnered significant interest recently. In this study, Au@Pd nanostars were synthesized and employed as an advanced electrocatalyst for the ethanol oxidation reaction (EOR), which suffers from the kinetic limitations of the anodic half-reaction of direct ethanol fuel cells. Leveraging the superior surface plasmon resonance (SPR) property of Au nanostars, the synthesized Au@Pd nanostars exhibited superior plasmon-enhanced EOR activity of 4191.63 A g−1 under laser illumination, noting a 9.40-fold increase compared to dark conditions (445.99 A g−1). The influences of Pd loading and laser power on EOR activity were explored. The impact of Pd loading on EOR activity demonstrates a volcano-shaped correlation, where optimal Pd coverage enhances catalysis but excessive amounts dampen SPR performance, crucial for maximizing EOR activity. Within a specified range, EOR activity linearly increases with laser power due to its influence on SPR performance. Mechanistic investigation suggests the critical roles of photothermal effect and hot carriers’ separation in boosting EOR kinetics. The repeated experiments show little difference, ensuring the accuracy and reproducibility of the reported activity of Au@Pd nanostars for EOR. During the long-term stability estimation, the Au@Pd nanostars display the best durability over three electrocatalysts. Overall, our findings highlight the potential of plasmon-enhanced catalysis in advancing the performance of electrocatalysts for EOR and other conventional electrocatalytic reactions.

Inverse mean curvature flow and Ricci-pinched three-manifolds

Abstract Let ( M , g ) (M,g) be a noncompact, connected, complete Riemannian three-manifold with nonnegative Ricci curvature satisfying Ric ≥ ε ⁢ tr ⁡ ( Ric ) ⁢ g \mathrm{Ric}\geq\varepsilon\operatorname{tr}(\mathrm{Ric})g for some ε &gt; 0 \varepsilon&gt;0 . In this note, we give a new proof based on inverse mean curvature flow that ( M , g ) (M,g) is either flat or has non-Euclidean volume growth. In conjunction with the work of J. Lott [On 3-manifolds with pointwise pinched nonnegative Ricci curvature, Math. Ann. 388 (2024), 3, 2787–2806] and of M.-C. Lee and P. Topping [Three-manifolds with non-negatively pinched Ricci curvature, preprint (2022), https://arxiv.org/abs/2204.00504], this gives an alternative proof of a conjecture of R. Hamilton recently proven by A. Deruelle, F. Schulze, and M. Simon [Initial stability estimates for Ricci flow and three dimensional Ricci-pinched manifolds, preprint (2022), https://arxiv.org/abs/2203.15313] using Ricci flow.

(The limits of) eye-tracking with iPads.

Applications for eye-tracking-particularly in the clinic-are limited by a reliance on dedicated hardware. Here we compare eye-tracking implemented on an Apple iPad Pro 11" (third generation)-using the device's infrared head-tracking and front-facing camera-with a Tobii 4c infrared eye-tracker. We estimated gaze location using both systems while 28 observers performed a variety of tasks. For estimating fixation, gaze position estimates from the iPad were less accurate and precise than the Tobii (mean absolute error of 3.2° ± 2.0° compared with 0.75° ± 0.43°), but fixation stability estimates were correlated across devices (r = 0.44, p < 0.05). For tasks eliciting saccades >1.5°, estimated saccade counts (r = 0.4-0.73, all p < 0.05) were moderately correlated across devices. For tasks eliciting saccades >8° we observed moderate correlations in estimated saccade speed and amplitude (r = 0.4-0.53, all p < 0.05). We did, however, note considerable variation in the vertical component of estimated smooth pursuit speed from the iPad and a catastrophic failure of tracking on the iPad in 5% to 20% of observers (depending on the test). Our findings sound a note of caution to researchers seeking to use iPads for eye-tracking and emphasize the need to properly examine their eye-tracking data to remove artifacts and outliers.

Stable Recovery of Coefficients in an Inverse Fault Friction Problem

We consider the inverse fault friction problem of determining the friction coefficient in the Tresca friction model, which can be formulated as an inverse problem for differential inequalities. We show that the measurements of elastic waves during a rupture uniquely determine the friction coefficient at the rupture surface with explicit stability estimates.

Increasing stability for inverse source problem with limited-aperture far field data at multi-frequencies

We study the increasing stability of an inverse source problem for the Helmholtz equation from limited-aperture far field data at multiple wave numbers. The measurement data are given by the far field patterns u∞(xˆ,k) for all observation directions xˆ in some neighborhood of a fixed direction xˆ0 and for all wave numbers k belonging to a finite interval (0,K). In this paper, we discuss the increasing stability with respect to the width of the wavenumber interval K>1. In three dimensions we establish stability estimates of the L2-norm and H−1-norm of the source function from the far field data. The ill-posedness of the inverse source problem turns out to be of Hölder type while increasing the wavenumber band K. We also discuss an analytic continuation argument of the far-field data with respect to the wavenumbers at a fixed direction.

A fuzzy q-closest alignment model

The paper examines the issue of well-posedness of the Cucker-Smale model with communication restricted to the q-closest neighbors, known also as the Cucker-Dong model. With agents oscillating on the boundary of different clusters, the system becomes difficult to precisely define, which leads to further problems with kinetic limits as the number of agents tends to infinity. We introduce the fuzzy q-closest system, which circumvents the issues with well-posedness. For such a system we prove a stability estimate for measure-valued solutions and perform the kinetic mean-field limit.