Time Rescaling Research Articles

Mathematical modeling and analysis of biological control strategy of aphid population

<abstract><p>To study the biological control strategy of aphids, in this paper we propose host-parasitoid-predator models for the interactions among aphids, parasitic wasps and aphidophagous Coccinellids incorporating impulsive releases of Coccinellids, and then study the long-term control and limited time optimal control of aphids by adjusting release amount and release timing of Coccinellids. For the long-term control, the existence and stability of the aphid-eradication periodic solution are investigated and threshold conditions about the release amount and release period to ensure the ultimate extinction of the aphid population are obtained. For the limited-time control, three different optimal impulsive control problems are studied. A time rescaling technique and an optimization algorithm based on gradient are applied, and the optimal release amounts and timings of natural enemies are gained. Our simulations indicate that in the limited-time control, the optimal selection of release timing should be given higher priority compared with the release amount.</p></abstract>

Toward the Removal of Model Dependency in Soil Moisture Climate Data Records by Using an $L$-Band Scaling Reference

Building climate data records of soil moisture (SM) requires computing long time series by merging retrievals from sensors on-board different satellites, which implies to perform a bias correction or rescaling on the original time series. Due to their long time span and high temporal frequency, model data could be used as a common reference for the rescaling. However, avoiding model dependence in observational climate data records is needed for some applications. In this article, the possibility of using as reference remote sensing data from one of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L$</tex-math></inline-formula> -band sensors specifically designed to measure SM is discussed. Advanced Microwave Scanning Radiometer 2 SM time series were rescaled by matching their cumulative distribution functions (CDFs) to those of Soil Moisture and Ocean Salinity (SMOS), Soil Moisture Active Passive (SMAP), and Global Land Data Assimilation System (GLDAS) NOAH model time series. The CDF computation was investigated as a function of the time series length, finding significant differences from four to nine years. Replacing temporal by spatial variance does not allow us to compute better CDFs from short time series. The rescaled time series show a high correlation ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$R&gt;0.8$</tex-math></inline-formula> ) to the original ones and a low bias with respect to the reference ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$&lt; $</tex-math></inline-formula> 0.03 m <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$^{3}\cdot$</tex-math></inline-formula> m <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$^{-3}$</tex-math></inline-formula> ). The time series rescaled using several SMOS or SMAP datasets were also evaluated against <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in situ</i> measurements and show performances similar to or slightly better than those rescaled using the model GLDAS. The impact of random errors and gaps of the observational data into the rescaling was evaluated. These results show that it is actually possible to use <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L$</tex-math></inline-formula> -band data as reference to rescale time series from other sensors to build long time series of SM.

Experimental study of a millimeter-sized Ga-In drop ablated by a nanosecond laser pulse

The motion of millimeter gallium-indium (Ga-In) drops subject to intense Neodymium-doped Yttrium Aluminum Garnet (Nd: YAG) laser blasts in the air is investigated experimentally. The drop first experiences plasma emissions and then undergoes interfacial instabilities. The effective ablation pulse energy is quantified by the laser-induced shockwave propagation. The laser-blast-induced concave expansion and spanwise depression history is measured, and the data collapse on straight lines with proper rescaling of pulse energy and time. The propulsion speed of the drop is described by a semi-empirical model that considers the laser energy and fluence at the threshold of ablation. The data show that this propulsion speed scaling remains valid to the millimeter drop ablated by the pulsed laser with beam spot much smaller than the drop, although the original scaling was derived and verified for the indium-tin (In-Sn) droplet of tens of micrometers impacted by a laser pulse with the focal point larger than the droplet.

Atomic transport properties of liquid iron at conditions of planetary cores.

Atomic transport properties of liquid iron are important for understanding the core dynamics and magnetic field generation of terrestrial planets. Depending on the sizes of planets and their thermal histories, planetary cores may be subject to quite different pressures (P) and temperatures (T). However, previous studies on the topic mainly focus on the P-T range associated with the Earth's outer core; a systematic study covering conditions from small planets to massive exoplanets is lacking. Here, we calculate the self-diffusion coefficient D and viscosity η of liquid iron via ab initio molecular dynamics from 7.0 to 25 g/cm3 and 1800 to 25 000 K. We find that D and η are intimately related and can be fitted together using a generalized free volume model. The resulting expressions are simpler than those from previous studies where D and η were treated separately. Moreover, the new expressions are in accordance with the quasi-universal atomic excess entropy (Sex) scaling law for strongly coupled liquids, with normalized diffusivity D⋆ = 0.621 exp(0.842Sex) and viscosity η⋆ = 0.171 exp(-0.843Sex). We determine D and η along two thermal profiles of great geophysical importance: the iron melting curve and the isentropic line anchored at the ambient melting point. The variations of D and η along these thermal profiles can be explained by the atomic excess entropy scaling law, demonstrating the dynamic invariance of the system under uniform time and space rescaling. Accordingly, scale invariance may serve as an underlying mechanism to unify planetary dynamos of different sizes.

Continuity of pullback attractors for evolution processes associated with semilinear damped wave equations with time-dependent coefficients

In this paper we consider the semilinear damped wave problem of the form { ( α ( t ) u t ) t − β ( t ) Δ u + γ ( t ) u t + δ ( t ) u = β ( t ) f ( u ) , x ∈ Ω , t > τ , u ( x , t ) = 0 , x ∈ ∂ Ω , t ⩾ τ , u ( x , τ ) = u τ ( x ) , u t ( x , τ ) = v τ ( x ) , x ∈ Ω , where Ω is a bounded smooth domain in R N , N ⩾ 3 , τ ∈ R , f is a real valued function of a real variable with some suitable conditions of growth, regularity and dissipativity, and α , β , γ and δ are continuous real valued functions of a real variable with some suitable conditions of growth, regularity and signs. Using rescaling of time we prove existence, regularity, gradient-like structure, upper and lower semicontinuity of the pullback attractors for the evolution processes associated with this boundary initial value problem in a suitable phase space.

Quadratic systems with an invariant algebraic curve of degree 3 and a Darboux invariantq

Let \(QS\) be the class of non-degenerate planar quadratic differential ystems and \(QS_3\) its subclass formed by the systems possessing an invariant cubic \(f(x,y)=0\). In this article, using the action of the group of real affine transformations and time rescaling on \(QS\), we obtain all the possible normal forms for the quadratic systems in \(QS_3\). Working with these normal forms we complete the characterization of the phase portraits in \(QS_3\) having a Darboux invariant of the form \(f(x, y)e^{st}\), with \(s \in \mathbb R\).&#x0D; For more information see https://ejde.math.txstate.edu/Volumes/2021/69/abstr.html

Surfactant and dilatational viscosity effects on the deformation of liquid droplets in an electric field

HypothesisA conducting droplet suspended in an insulating continuous phase, e.g. an aqueous electrolyte in an oil, is deformed by an applied electric field to nonspherical equilibrium shapes, and can even break-up under strong fields. Many technologies use electro-deformation to manipulate fluid dispersions, with surfactants present on the droplet interfaces forming stabilizing monolayers. While surfactants lower the interface tension which facilitates electro-deformation, the monolayer elasticity resists deformation. High molecular weight surfactants, with large dilatational viscosities, can potentially retard the deformation dynamics. NumericsA boundary integral method simulates the dynamic interfacial deformation of a perfectly conducting droplet in a dielectric in a uniform field. The interface contains an insoluble monolayer which is a Newtonian fluid with constant dilatational viscosity obeying a Langmuir state equation. A range of initial surfactant surface concentrations are studied, with elasticity proportional to concentration. FindingsEquilibrium drop deformations, unaffected by surface viscosity, are strongly resisted by elasticity at high surface concentrations, and field strengths necessary for break-up increase with elasticity. Dilatational viscosity scales with the ratio, κ∗, the surface viscosity (divided by the droplet radius) to the bulk viscosity, and can extend the deformation time. Extended times are described by a time rescaling proportional to κ∗.

Parabolic Kirchhoff equations with non-homogeneous flux boundary conditions: well-posedness, regularity and asymptotic behavior

We investigate well-posedness, regularity and asymptotic behavior of parabolic Kirchhoff equations on bounded domains of , N ⩾ 2, with non-homogeneous flux boundary conditions of Neumann or Robin type. The data in the problem satisfy (f, g, u(0)) ∈ L 2(Ω) × L 2(∂Ω) × H 1(Ω). Approximated solutions are constructed using time rescaling and a complete set in H 1(Ω) relating the equation and the boundary condition. Uniform global estimates are derived and used to prove existence, uniqueness, continuous dependence on data, a priori estimates and higher regularity for the parabolic problem. Existence and uniqueness of stationary solutions are shown, as well as a description about their role on the asymptotic behavior regarding to the evolutionary equation. Furthermore, a sufficient condition for the existence of isolated local energy minimizers is provided. They are shown to be asymptotically stable stationary solutions for the parabolic equation.

Multiwave pandemic dynamics explained: how to tame the next wave of infectious diseases

Pandemics, like the 1918 Spanish Influenza and COVID-19, spread through regions of the World in subsequent waves. Here we propose a consistent picture of the wave pattern based on the epidemic Renormalisation Group (eRG) framework, which is guided by the global symmetries of the system under time rescaling. We show that the rate of spreading of the disease can be interpreted as a time-dilation symmetry, while the final stage of an epidemic episode corresponds to reaching a time scale-invariant state. We find that the endemic period between two waves is a sign of instability in the system, associated to near-breaking of the time scale-invariance. This phenomenon can be described in terms of an eRG model featuring complex fixed points. Our results demonstrate that the key to control the arrival of the next wave of a pandemic is in the strolling period in between waves, i.e. when the number of infections grows linearly. Thus, limiting the virus diffusion in this period is the most effective way to prevent or delay the arrival of the next wave. In this work we establish a new guiding principle for the formulation of mid-term governmental strategies to curb pandemics and avoid recurrent waves of infections, deleterious in terms of human life loss and economic damage.

Gravitational energy of the spherically symmetric static and charged black holes encircled by the dark energy through approximate Lie symmetries

We investigate the gravitational energy of the Reissner-Nordström black hole encircled by dark energy using the approach of approximate Lie symmetries of second-order. We confine our calculations to following values of the equation of state parameter ω: (i) (quintessence regime of dark energy), (ii) ω Λ = −1 (the pure cosmological constant regime), (iii) (frustrated cosmic strings network). In order to examine the gravitational energy for the Reissner-Nordström black hole spacetime encircled by the dark energy, we construct a system of second-order perturbed geodesic equations, which is done primarily by considering the mass M and charge Q of the black hole in terms of some small perturbation parameter ϵ. It is observed that we do not obtain any non-trivial approximate symmetry generator for any value of the state parameter ω, we get only trivial second-order approximate symmetry generators. As a result of these trivial approximate symmetries we get a rescaling of the proper time s which gives the rescaling of energy in the Reissner-Nordström spacetime encircled by the dark energy. It is found that for and ω Λ = −1, the rescaling factor for energy is dependent on r and for , the rescaling factor is independent of the radial distance r. Furthermore, we investigate that the effect of cosmological constant and quintessence dark energy decreases the energy of the underlying spacetime while the equation of state parameter , for which the Universe is static, increases the energy of the Reissner-Nordström black hole spacetime encircled by the dark energy.

Fast convex optimization via inertial dynamics combining viscous and Hessian-driven damping with time rescaling

&lt;p style='text-indent:20px;'&gt;In a Hilbert setting, we develop fast methods for convex unconstrained optimization. We rely on the asymptotic behavior of an inertial system combining geometric damping with temporal scaling. The convex function to minimize enters the dynamic via its gradient. The dynamic includes three coefficients varying with time, one is a viscous damping coefficient, the second is attached to the Hessian-driven damping, the third is a time scaling coefficient. We study the convergence rate of the values under general conditions involving the damping and the time scale coefficients. The obtained results are based on a new Lyapunov analysis and they encompass known results on the subject. We pay particular attention to the case of an asymptotically vanishing viscous damping, which is directly related to the accelerated gradient method of Nesterov. The Hessian-driven damping significantly reduces the oscillatory aspects. We obtain an exponential rate of convergence of values without assuming the strong convexity of the objective function. The temporal discretization of these dynamics opens the gate to a large class of inertial optimization algorithms.&lt;/p&gt;

TRAVELING WAVES AND THEIR EVOLUTION FOR THE ZK(N, 2N, -N) EQUATION

<abstract> In this paper, using the approach of dynamical systems we investigate the traveling waves for the ZK(n, 2n, -n) equation including the types and evolution of traveling waves. The traveling wave problem is converted into the analysis of phase portraits of the corresponding traveling wave system, which is 5-parametric and has a singular line in its phase space. The orbits passing through this singular line in phase portraits are determined by a time rescaling. After converting the orbits in these phase portraits into traveling waves, we state all types of traveling waves and give at least one exact traveling wave solution for each type of bounded traveling waves in our main results. Finally, we discuss the evolution of these traveling waves among themselves when parameters vary by the bifurcations happening in the phase portraits of the traveling wave system. </abstract>

Study of the sterile insect release technique for a two-sex mosquito population model.

In this paper, to study the large-scale time control and limited-time control of mosquito population in a field, a two-sex mosquito population model with stage structure and impulsive releases of sterile males is proposed. For the large-scale time control, a wild mosquito-free periodic solution is given and conditions under which it is globally stable are obtained by the use of the monotone system theory. Besides, based on the stability analysis, threshold conditions under which the wild mosquito population is eliminated or not are obtained. Then we study three different optimal release strategies for the limited-time control, which takes into account both of the population control level of wild mosquitoes and the economic input. To solve technical problems in optimal impulsive control, a time rescaling technique is applied and the gradients of cost function with respect to all control parameters are obtained. In addition, by the aid of numerical simulation, we get the optimal release amounts and release timings for each release strategy. Our study indicates that the optimal release timing control is superior to the optimal release amount control. However, simultaneous optimal selection of release amount and release timing leads to the best control performance.

Impulsive release strategies of sterile mosquitos for optimal control of wild population

To investigate the release strategies of sterile mosquitoes for the wild population control, we propose mathematical models for the interaction between two-mosquito populations incorporating impulsive releases of sterile ones. The long-term control model is first studied, and the existence and stability of the wild mosquito-extinction periodic solution are exploited. Thresholds of the release amount and release period which can guarantee the elimination of the wild mosquitos are obtained. Then for the limited-time control model, three different optimal strategies in impulsive control are investigated. By applying a time rescaling technique and an optimization algorithm based on gradient, the optimal impulsive release timings and amounts of sterile mosquitoes are obtained. Our results show that the optimal selection of release timing is more important than the optimal selection of release amount, while mixed optimal control has the best comprehensive effect.

Improved anisotropic hydrodynamics ansatz

We introduce an improved form for the anisotropic hydrodynamics distribution function which explicitly takes into account the free-streaming and equilibrating contributions separately. We demonstrate that with this improvement one can better reproduce exact results available in the literature for the evolution of moments of the distribution function, in particular, for moments which contain no powers of the longitudinal momentum in their definition (m=0 moments). Using the resulting dynamical equations, we extract the non-equilibrium attractor associated with our improved aHydro ansatz and demonstrate that the improvement also allows one to better reproduce the exact dynamical attractor obtained using kinetic theory in the relaxation time approximation, particularly at early rescaled times and for m=0 moments.

Aging in a mean field elastoplastic model of amorphous solids

We construct a mean-field elastoplastic description of the dynamics of amorphous solids under arbitrary time-dependent perturbations, building on the work of Lin and Wyart [Phys. Rev. X 6, 011005 (2016)] for steady shear. Local stresses are driven by power-law distributed mechanical noise from yield events throughout the material, in contrast to the well-studied Hébraud–Lequeux model where the noise is Gaussian. We first use a mapping to a mean first passage time problem to study the phase diagram in the absence of shear, which shows a transition between an arrested and a fluid state. We then introduce a boundary layer scaling technique for low yield rate regimes, which we first apply to study the scaling of the steady state yield rate on approaching the arrest transition. These scalings are further developed to study the aging behavior in the glassy regime for different values of the exponent μ characterizing the mechanical noise spectrum. We find that the yield rate decays as a power-law for 1 &amp;lt; μ &amp;lt; 2, a stretched exponential for μ = 1, and an exponential for μ &amp;lt; 1, reflecting the relative importance of far-field and near-field events as the range of the stress propagator is varied. A comparison of the mean-field predictions with aging simulations of a lattice elastoplastic model shows excellent quantitative agreement, up to a simple rescaling of time.

Numerical Algorithms For State-Linear Optimal Impulsive Control Problems Based On Feedback Necessary Optimality Conditions

We propose and compare three numeric algorithms for optimal control of state-linear impulsive systems. The algorithms rely on the standard transformation of impulsive control problems through the discontinuous time rescaling, and the so-called “feedback”, direct and dual, maximum principles. The feedback maximum principles are variational necessary optimality conditions operating with feedback controls, which are designed through the usual constructions of the Pontryagin’s Maximum Principle (PMP); though these optimality conditions are formulated completely in the formalism of PMP, they essentially strengthen it. All the algorithms are non-local in the sense that they are aimed at improving non-optimal extrema of PMP (local minima), and, therefore, show the potential of global optimization.

Long time error analysis of the fourth‐order compact finite difference methods for the nonlinear Klein–Gordon equation with weak nonlinearity

AbstractWe present the fourth‐order compact finite difference (4cFD) discretizations for the long time dynamics of the nonlinear Klein–Gordon equation (NKGE), while the nonlinearity strength is characterized by ϵp with a constant p ∈ ℕ+ and a dimensionless parameter ϵ ∈ (0, 1]. Based on analytical results of the life‐span of the solution, rigorous error bounds of the 4cFD methods are carried out up to the time at O(ϵ−p). We pay particular attention to how error bounds depend explicitly on the mesh size h and time step τ as well as the small parameter ϵ ∈ (0, 1], which indicate that, in order to obtain ‘correct’ numerical solutions up to the time at O(ϵ−p), the ϵ‐scalability (or meshing strategy requirement) of the 4cFD methods should be taken as: h = O(ϵp/4) and τ = O(ϵp/2). It has better spatial resolution capacity than the classical second order central difference methods. By a rescaling in time, it is equivalent to an oscillatory NKGE whose solution propagates waves with wavelength at O(1) in space and O(ϵp) in time. It is straightforward to get the error bounds of the oscillatory NKGE in the fixed time. Finally, numerical results are provided to confirm our theoretical analysis.

The challenge of stochastic Størmer-Verlet thermostats generating correct statistics.

In light of the recently developed complete GJ set of single random variable stochastic, discrete-time Størmer-Verlet algorithms for statistically accurate simulations of Langevin equations [N. Grønbech-Jensen, Mol. Phys. 118, e1662506 (2020)], we investigate two outstanding questions: (1) Are there any algorithmic or statistical benefits from including multiple random variables per time step and (2) are there objective reasons for using one or more methods from the available set of statistically correct algorithms? To address the first question, we assume a general form for the discrete-time equations with two random variables and then follow the systematic, brute-force GJ methodology by enforcing correct thermodynamics in linear systems. It is concluded that correct configurational Boltzmann sampling of a particle in a harmonic potential implies correct configurational free-particle diffusion and that these requirements only can be accomplished if the two random variables per time step are identical. We consequently submit that the GJ set represents all possible stochastic Størmer-Verlet methods that can reproduce time step-independent statistics of linear systems. The second question is thus addressed within the GJ set. Based on numerical simulations of complex molecular systems, as well as on analytic considerations, we analyze apparent friction-induced differences in the stability of the methods. We attribute these differences to an inherent, friction-dependent discrete-time scaling, which depends on the specific method. We suggest that the method with the simplest interpretation of temporal scaling, the GJ-I/GJF-2GJ method, be preferred for statistical applications.

Assessing Goodness-of-Fit in Marked Point Process Models of Neural Population Coding via Time and Rate Rescaling.

Marked point process models have recently been used to capture the coding properties of neural populations from multiunit electrophysiological recordings without spike sorting. These clusterless models have been shown in some instances to better describe the firing properties of neural populations than collections of receptive field models for sorted neurons and to lead to better decoding results. To assess their quality, we previously proposed a goodness-of-fit technique for marked point process models based on time rescaling, which for a correct model produces a set of uniform samples over a random region of space. However, assessing uniformity over such a region can be challenging, especially in high dimensions. Here, we propose a set of new transformations in both time and the space of spike waveform features, which generate events that are uniformly distributed in the new mark and time spaces. These transformations are scalable to multidimensional mark spaces and provide uniformly distributed samples in hypercubes, which are well suited for uniformity tests. We discuss the properties of these transformations and demonstrate aspects of model fit captured by each transformation. We also compare multiple uniformity tests to determine their power to identify lack-of-fit in the rescaled data. We demonstrate an application of these transformations and uniformity tests in a simulation study. Proofs for each transformation are provided in the appendix.