Time-dependent Mean Research Articles

A straightforward method for estimating evolutionary power spectral density of non-stationary typhoon wind speed

PurposeA precise estimation of the evolutionary power spectral density (EPSD) of typhoon wind speed is a difficult and significant undertaking in the analysis of turbulence effects on large-expansive structures. A majority of the prevailing EPSD estimation techniques rely on complex signal processing methodologies, such as wavelet decomposition, Hilbert–Huang transformation and time-varying autoregressive moving average (ARMA) model. However, these approaches often pose challenges in terms of comprehensibility and practical implementation for engineers. In light of this issue, the present study introduces a straightforward and effective EPSD estimation method tailored specifically for typhoon wind speed, aiming to facilitate its understanding and application in engineering contexts.Design/methodology/approachFirstly, the mathematical model of a uniformly modulated non-stationary process is employed to represent the typhoon wind speed. Secondly, the reverse arrangement test serves as an auxiliary tool in conjunction with wavelet transform or empirical mode decomposition, aiding in the determination of the optimal slowly varying mean wind speed. Thirdly, Kernel regression technique is utilized to discern the time-dependent standard deviation of wind speed fluctuations. Finally, the power spectral density (PSD) of wind speed residuals is computed to facilitate the estimation of the EPSD.FindingsFirstly, the reverse arrangement test-assisted approach enables the determination of an optimal time-dependent mean from the candidate results obtained through discrete wavelet transform (DWT) and empirical mode decomposition (EMD). Secondly, the application of the Kernel regression technique facilitates accurate identification of the time-dependent variance from the fluctuating wind speed data. Thirdly, due to the influence of the extreme weather, the Gaussianity of the reduced turbulent fluctuations in typhoon wind is easily disturbed, resulting in the obvious non-Gaussian features.Originality/valueThis paper employs the mathematical model of uniformly modulated non-stationary process to characterize typhoon wind speeds and then proposes a straightforward and efficient method for estimating the EPSD of typhoon wind. The accuracy and efficacy of the presented estimation method are verified using the field-measured wind speed data from Typhoon Rammasun. The proposed EPSD estimation method for typhoon wind exhibits suitability for engineering applications owing to its simplicity and computational efficiency.

Detectability of hemodynamic oscillations in cerebral cortex through functional near-infrared spectroscopy: a simulation study.

We explore the feasibility of using time-domain (TD) and continuous-wave (CW) functional near-infrared spectroscopy (fNIRS) to monitor brain hemodynamic oscillations during resting-state activity in humans, a phenomenon that is of increasing interest in the scientific and medical community and appears to be crucial to advancing the understanding of both healthy and pathological brain functioning. Our general object is to maximize fNIRS sensitivity to brain resting-state oscillations. More specifically, we aim to define comprehensive guidelines for optimizing main operational parameters in fNIRS measurements [average photon count rate, measurement length, sampling frequency, and source-detector distance (SSD)]. In addition, we compare TD and CW fNIRS performance for the detection and localization of oscillations. A series of synthetic TD and CW fNIRS signals were generated by exploiting the solution of the diffusion equation for two different geometries of the probed medium: a homogeneous medium and a bilayer medium. Known and periodical perturbations of the concentrations of oxy- and deoxy-hemoglobin were imposed in the medium, determining changes in its optical properties. The homogeneous slab model was used to determine the effect of multiple measurement parameters on fNIRS sensitivity to oscillatory phenomena, and the bilayer model was used to evaluate and compare the abilities of TD and CW fNIRS in detecting and isolating oscillations occurring at different depths. For TD fNIRS, two approaches to enhance depth-selectivity were evaluated: first, a time-windowing of the photon distribution of time-of-flight was performed, and then, the time-dependent mean partial pathlength (TMPP) method was used to retrieve the hemoglobin concentrations in the medium. In the homogeneous medium case, the sensitivity of TD and CW fNIRS to periodical perturbations of the optical properties increases proportionally with the average photon count rate, the measurement length, and the sampling frequency and approximatively with the square of the SSD. In the bilayer medium case, the time-windowing method can detect and correctly localize the presence of oscillatory components in the TD fNIRS signal, even in the presence of very low photon count rates. The TMPP method demonstrates how to correctly retrieve the periodical variation of hemoglobin at different depths from the TD fNIRS signal acquired at a single SSD. For CW fNIRS, measurements taken at typical SSDs used for short-separation channel regression show notable sensitivity to oscillations occurring in the deep layer, challenging the assumptions underlying this correction method when the focus is on analyzing oscillatory phenomena. We demonstrated that the TD fNIRS technique allows for the detection and depth-localization of periodical fluctuations of the hemoglobin concentrations within the probed medium using an acquisition at a single SSD, offering an alternative to multi-distance CW fNIRS setups. Moreover, we offered some valuable guidelines that can assist researchers in defining optimal experimental protocols for fNIRS studies.

Cumulative Systolic Blood Pressure and Incident Stroke Type Variation by Race and Ethnicity

Stroke risk varies by systolic blood pressure (SBP), race, and ethnicity. The association between cumulative mean SBP and incident stroke type is unclear, and whether this association differs by race and ethnicity remains unknown. To examine the association between cumulative mean SBP and first incident stroke among 3 major stroke types-ischemic stroke (IS), intracerebral hemorrhage (ICH), and subarachnoid hemorrhage (SAH)-and explore how these associations vary by race and ethnicity. Individual participant data from 6 US longitudinal cohorts (January 1, 1971, to December 31, 2019) were pooled. The analysis was performed from January 1, 2022, to January 2, 2024. The median follow-up was 21.6 (IQR, 13.6-31.8) years. Time-dependent cumulative mean SBP. The primary outcome was time from baseline visit to first incident stroke. Secondary outcomes consisted of time to first incident IS, ICH, and SAH. Among 40 016 participants, 38 167 who were 18 years or older at baseline with no history of stroke and at least 1 SBP measurement before the first incident stroke were included in the analysis. Of these, 54.0% were women; 25.0% were Black, 8.9% were Hispanic of any race, and 66.2% were White. The mean (SD) age at baseline was 53.4 (17.0) years and the mean (SD) SBP at baseline was 136.9 (20.4) mm Hg. A 10-mm Hg higher cumulative mean SBP was associated with a higher risk of overall stroke (hazard ratio [HR], 1.20 [95% CI, 1.18-1.23]), IS (HR, 1.20 [95% CI, 1.17-1.22]), and ICH (HR, 1.31 [95% CI, 1.25-1.38]) but not SAH (HR, 1.13 [95% CI, 0.99-1.29]; P = .06). Compared with White participants, Black participants had a higher risk of IS (HR, 1.20 [95% CI, 1.09-1.33]) and ICH (HR, 1.67 [95% CI, 1.30-2.13]) and Hispanic participants of any race had a higher risk of SAH (HR, 3.81 [95% CI, 1.29-11.22]). There was no consistent evidence that race and ethnicity modified the association of cumulative mean SBP with first incident stroke and stroke type. The findings of this cohort study suggest that cumulative mean SBP was associated with incident stroke type, but the associations did not differ by race and ethnicity. Culturally informed stroke prevention programs should address modifiable risk factors such as SBP along with social determinants of health and structural inequities in society.

Semiparametric estimation and testing for panel count data with informative interval-censored failure event.

Panel count data and interval-censored data are two types of incomplete data that often occur in event history studies. Almost all existing statistical methods are developed for their separate analysis. In this paper, we investigate a more general situation where a recurrent event process and an interval-censored failure event occur together. To intuitively and clearly explain the relationship between the recurrent current process and failure event, we propose a failure time-dependent mean model through a completely unspecified link function. To overcome the challenges arising from the blending of nonparametric components and parametric regression coefficients, we develop a two-stage conditional expected likelihood-based estimation procedure. We establish the consistency, the convergence rate and the asymptotic normality of the proposed two-stage estimator. Furthermore, we construct a class of two-sample tests for comparison of mean functions from different groups. The proposed methods are evaluated by extensive simulation studies and are illustrated with the skin cancer data that motivated this study.

Changes in cardiac and vascular haemodynamics as potential mediators of improvements in cardiovascular and kidney outcomes with empagliflozin in type 2 diabetes

AimsEvaluate changes in haemodynamic markers as mediators of cardiovascular (CV) and kidney benefits with empagliflozin. MethodsPost-hoc analysis of EMPA-REG OUTCOME in patients with type 2 diabetes (T2D) and established CV disease receiving empagliflozin (10 and 25 mg) or placebo. Outcomes were CV death, hospitalisation for heart failure [HF], HF death, incident/worsening nephropathy, new onset macroalbuminuria, and the composite of sustained estimated glomerular filtration rate decline ≥40 % from baseline, renal replacement therapy or renal death. To be considered a mediator, changes in variable (pulse pressure, mean arterial pressure and cardiac workload) over time had to be (1) affected by active treatment, (2) associated with the outcome, and (3) adjustment for changes over time must reduce treatment effect versus an unadjusted analysis. Variables were evaluated in Cox regression analyses. ResultsPulse pressure, mean arterial pressure and cardiac workload were significantly reduced by empagliflozin vs placebo. Using change from baseline to Week 12 or sensitivity analyses (time-dependent updated mean and current change from baseline) of these CV parameters, only small impacts on empagliflozin effect on CV and kidney outcomes were shown. ConclusionsImprovements in haemodynamic parameters did not substantially mediate empagliflozin benefits on CV and kidney outcomes in patients with T2DM and established CV disease.

Behavior in time of solutions of a Keller–Segel system with flux limitation and source term

In this paper we consider radially symmetric solutions of the following parabolic–elliptic cross-diffusion system ut=Δu-∇·(uf(|∇v|2)∇v)+g(u),0=Δv-m(t)+u,∫Ωvdx=0,u(x,0)=u0(x),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} u_t = \\Delta u - \ abla \\cdot (u f(|\ abla v|^2 )\ abla v) + g(u), &{} \\\\ 0= \\Delta v -m(t)+ u, \\quad \\int _{\\Omega }v \\,dx=0, &{} \\\\ u(x,0)= u_0(x), &{} \\end{array}\\right. } \\end{aligned}$$\\end{document}in Omega times (0,infty ), with Omega a ball in {mathbb {R}}^N, Nge 3, under homogeneous Neumann boundary conditions, where g(u)= lambda u - mu u^k, lambda>0, mu >0, and k >1, f(|nabla v|^2 )= k_f(1+ |nabla v|^2)^{-alpha }, alpha >0, which describes gradient-dependent limitation of cross diffusion fluxes. The function m(t) is the time dependent spatial mean of u(x, t) i.e. m(t):= frac{1}{|Omega |} int _{Omega } u(x,t) ,dx. Under smallness conditions on alpha and k, we prove that the solution u(x, t) blows up in L^{infty }-norm at finite time T_{max} and for some p>1 it blows up also in L^p-norm. In addition a lower bound of blow-up time is derived. Finally, under largeness conditions on alpha or k, we prove that the solution is global and bounded in time.

Scaling laws for single-file diffusion of adhesive particles.

Single-file diffusion refers to the Brownian motion in narrow channels where particles cannot pass each other. In such processes, the diffusion of a tagged particle is typically normal at short times and becomes subdiffusive at long times. For hard-sphere interparticle interaction, the time-dependent mean squared displacement of a tracer is well understood. Here we develop a scaling theory for adhesive particles. It provides a full description of the time-dependent diffusive behavior with a scaling function that depends on an effective strength of adhesive interaction. Particle clustering induced by the adhesive interaction slows down the diffusion at short times, while it enhances subdiffusion at long times. The enhancement effect can be quantified in measurements irrespective of how tagged particles are injected into the system. Combined effects of pore structure and particle adhesiveness should speed up translocation of molecules through narrow pores.

Fokker–Planck equation for the particle size distribution function in KJMA transformations

The Fokker–Planck (FP) equation has been derived for describing the temporal evolution of the particle size probability density function (PDF) for KJMA (Kolmogorov–Johnson–Mehl–Avrami) transformations. The classical case of transformations with constant rates of both nucleation and growth, in 3D space, has been treated. Integration of the equation shows that the PDF is given by the superposition of one-parameter gamma distributions with time-dependent mean size given by the KJMA theory. The asymptotic behavior of the FP solution offers a demonstration of the conjecture, previously proposed by Pineda et al. (2004), according to which the set of crystals formed at the same time are gamma-distributed, with the parameter depending on crystal birth time. Computer simulations of the transformation with constant nucleation and growth rates show that the temporal evolution of the PDF, in volume domain, is in good agreement with the Johnson–Mehl PDF. The approach based on the FP equation provides a particle size PDF that exhibits such behavior.

Side Chain Dynamics of Poly(norbornene)-g-Poly(propylene oxide) Bottlebrush Polymers.

The segmental dynamics of the side chains of poly(norbornene)-g-poly(propylene oxide) (PNB-g-PPO) bottlebrush polymer in comparison to PPO is studied by quasi-elastic neutron scattering. Having experimental time and length scale information simultaneously allows to extract spatial information in addition to relaxation time. Tethering one end of the PPO side chain onto a stiff PNB backbone slows down the segmental relaxation, over the length and time scales investigated. The power law dependence of the relaxation time on the momentum transfer, Q, indicates a more heterogeneous relaxation pattern for the bottlebrush polymer, whereas the linear PPO has less deviations from a homogenous relaxation. Similar conclusions can be drawn from the time dependent mean square displacement, 〈r2 (t)〉, and the non-Gaussian parameter, α2 (t). Herein, the bottlebrush polymer shows a more restricted dynamics, whereas the linear PPO reaches 〈r2 (t)〉∝t0.5 at the highest temperature. The deviations from Gaussian behavior are evident at the α2 (t). Both samples show a decaying α2 (t). The non-Gaussian parameter supports the results from the power law dependence of the relaxation times, with lower α2 (t) values for PPO compared to those for PNB-g-PPO, pointing to less deviations from Gaussian behavior.

Policy Iteration Method for Time-Dependent Mean Field Games Systems with Non-separable Hamiltonians

We introduce two algorithms based on a policy iteration method to numerically solve time-dependent Mean Field Game systems of partial differential equations with non-separable Hamiltonians. We prove the convergence of such algorithms in sufficiently small time intervals with Banach fixed point method. Moreover, we prove that the convergence rates are linear. We illustrate our theoretical results by numerical examples, and we discuss the performance of the proposed algorithms.

Dynamics of Rossby solitary waves with time-dependent mean flow via Euler eigenvalue model

Dynamics of Rossby solitary waves with time-dependent mean flow via Euler eigenvalue model

Local equilibrium properties of ultraslow diffusion in the Sinai model

We perform numerical studies of a thermally driven, overdamped particle in a random quenched force field, known as the Sinai model. We compare the unbounded motion on an infinite 1-dimensional domain to the motion in bounded domains with reflecting boundaries and show that the unbounded motion is at every time close to the equilibrium state of a finite system of growing size. This is due to time scale separation: inside wells of the random potential, there is relatively fast equilibration, while the motion across major potential barriers is ultraslow. Quantities studied by us are the time dependent mean squared displacement, the time dependent mean energy of an ensemble of particles, and the time dependent entropy of the probability distribution. Using a very fast numerical algorithm, we can explore times up top 1017 steps and thereby also study finite-time crossover phenomena.

Evolution of energy and magnetic moment of a quantum charged particle in power-decaying magnetic fields

We consider a quantum spinless nonrelativistic charged particle moving in the xy plane under the action of a homogeneous time-dependent magnetic field B(t)=B0(1+t/t0)−1−g, directed along the z-axis and described by means of the vector potential A(t)=B(t)[−y,x]/2. Assuming that the particle was initially in the thermal equilibrium state with a negligible coupling to a reservoir, we obtain exact formulas for the time-dependent mean values of the energy and magnetic moment in terms of the Bessel functions. Different regimes of the evolution are discovered and illustrated in several figures. The energy goes asymptotically to a finite value if g>0 (“fast” decay), while it goes asymptotically to zero if g≤0 (“slow” decay). The dependence on parameter t0 practically disappears when 1+g is close to zero value (“superslow” decay). The mean magnetic moment goes to zero for g>1, while it grows unlimitedly if g<1.

Passive tracer advection in the equatorial Pacific region: statistics, correlations and a model of fractional Brownian motion

Abstract. Evaluating passive tracer advection is a common tool to study flow structures, particularly Lagrangian trajectories ranging from molecular scales up to the atmosphere and oceans. Here we report on numerical experiments in the region of the tropical Pacific (20∘ S–20∘ N), where 6600 tracer parcels are advected from a regular initial configuration (along a meridional line at 110∘ W between 15∘ S and 15∘ N) during periods of 1 year for 25 years altogether. We exploit AVISO surface flow fields and solve the kinematic equation for passive tracer movement in the 2D advection tests. We demonstrate that the strength of the advection defined by mean monthly westward displacements of the tracer clouds exhibit surprisingly large inter- and intra-annular variabilities. Furthermore, an analysis of cross-correlations between advection strength and the El-Niño and Southern Oscillation (SOI) indices reveal a significant anticorrelation between advection intensity and ONI (the Oceanic Niño Index) and a weaker positive correlation with SOI, both with a time lag of about 3 months (the two indices are strongly anticorrelated near real time). The statistical properties of advection (time-dependent mean squared displacement and first passage time distribution) suggest that the westward-moving tracers can be mapped into a simple 1D stochastic process, namely fractional Brownian motion. We fit the model parameters and show by numerical simulations of the fractional Brownian motion model that it is able to reproduce the observed statistical properties of the tracers' trajectories well. We argue that a traditional explanation based on the superposition of ballistic drift and a diffusion term yields different statistics and is incompatible with our observations.

Provider Altruism in Incentives Contracts: Medicare’s Quality Race

The paper analyzes the impact of provider altruism and motivation on the out ­comes of pay-for-performance reimbursement in healthcare, where a fixed price contract on quantity is supplemented with a relative performance contract on quail­ty. We develop a theoretical model which forecasts the crowding out of most alt­ruistic providers. Using the example of Medicare’s nationwide natural experiment with a relative performance contract on quality (the data for 3000 acute care hospitals in 2004-2017, with the incentives contract implemented since 2013), we conduct an empirical test of the model predictions. We assume that altruism is hete­rogeneous across hospitals and the values of altruism in each hospital are higher for quality measures which are strongly associated with the patient’s benefit. The analysis employs dynamic panel data estimations to account for «habit-formation» and we exclude pre-reform and post-reform «regression-to-the-mean» effects by modeling the time-dependent long-term mean as a function of hospital characteristics. We focus on highest-quality hospitals and discover a deterioration of quality measures, which may be linked to the patient’s benefit (communication of patients with medical personnel and ability to receive help promptly). It may be interpreted as an illustration of the fact that relative performance incentive contracts may be associated with crowding altruistic providers out of the healthcare market.

Energy and Magnetic Moment of a Quantum Charged Particle in Time-Dependent Magnetic and Electric Fields of Circular and Plane Solenoids.

We consider a quantum spinless nonrelativistic charged particle moving in the plane under the action of a time-dependent magnetic field, described by means of the linear vector potential , with two fixed values of the gauge parameter : (the circular gauge) and (the Landau gauge). While the magnetic field is the same in all the cases, the systems with different values of the gauge parameter are not equivalent for nonstationary magnetic fields due to different structures of induced electric fields, whose lines of force are circles for and straight lines for . We derive general formulas for the time-dependent mean values of the energy and magnetic moment, as well as for their variances, for an arbitrary function . They are expressed in terms of solutions to the classical equation of motion , with . Explicit results are found in the cases of the sudden jump of magnetic field, the parametric resonance, the adiabatic evolution, and for several specific functions , when solutions can be expressed in terms of elementary or hypergeometric functions. These examples show that the evolution of the mentioned mean values can be rather different for the two gauges, if the evolution is not adiabatic. It appears that the adiabatic approximation fails when the magnetic field goes to zero. Moreover, the sudden jump approximation can fail in this case as well. The case of a slowly varying field changing its sign seems especially interesting. In all the cases, fluctuations of the magnetic moment are very strong, frequently exceeding the square of the mean value.

Association of Ambient Fine Particulate Matter Air Pollution With Kidney Transplant Outcomes

Increased levels of ambient fine particulate matter (PM2.5) air pollution are associated with increased risks for detrimental health outcomes, but risks for patients with kidney transplants (KTs) remain unknown. To investigate the association of PM2.5 exposure with KT outcomes. This retrospective cohort study was conducted using data on patients who received KTs from 2004 to 2016 who were identified in the national US transplant registry and followed up through March 2021. Multiple databases were linked to obtain data on PM2.5 concentration, KT outcomes, and patient clinical, transplant, and contextual factors. Data were analyzed from April 2020 through July 2021. Exposures included post-KT time-dependent annual mean PM2.5 level (in 10 μg/m3) and mean PM2.5 level in the year before KT (ie, baseline levels) in quartiles, as well as baseline annual mean PM2.5 level (in 10 μg/m3). Acute kidney rejection (ie, rejection within 1 year after KT), time to death-censored graft failure, and time to all-cause death. Multivariable logistic regression for kidney rejection and Cox analyses with nonlinear assessment of exposure-response for death-censored graft failure and all-cause death were performed. The national burden of graft failure associated with PM2.5 levels greater than the Environmental Protection Agency recommended level of 12 μg/m3 was estimated. Among 112 098 patients with KTs, 70 522 individuals (62.9%) were older than age 50 years at the time of KT, 68 117 (60.8%) were men, and the median (IQR) follow-up was 6.0 (3.9-8.9) years. There were 37 265 Black patients (33.2%), 17 047 Hispanic patients (15.2%), 48 581 White patients [43.3%]), and 9205 patients (8.2%) of other race or ethnicity. The median (IQR) baseline PM2.5 level was 9.8 (8.3-11.9) μg/m3. Increased baseline PM2.5 level, compared with quartile 1 baseline PM2.5 level, was not associated with higher odds of acute kidney rejection for quartile 2 (adjusted odds ratio [aOR], 0.99; 95% CI, 0.92-1.06) but was associated with increased odds for quartile 3 (aOR, 1.11; 95% CI, 1.04-1.20) and quartile 4 (aOR, 1.13; 95% CI, 1.05-1.23). Nonlinear assessment of exposure-response for graft failure and death showed no evidence for nonlinearity. Increased PM2.5 levels were associated with increased risk of death-censored graft failure (adjusted hazard ratio [aHR] per 10 μg/m3 increase, 1.17; 95% CI, 1.09-1.25) and all-cause death (aHR per 10 μg/m3 increase, 1.21; 95% CI, 1.14-1.28). The national burden of death-censored graft failure associated with PM2.5 above 12 μg/m3 was 57 failures (95% uncertainty interval, 48-67 failures) per year among patients with KTs. This cohort study found that PM2.5 level was an independent risk factor associated with acute rejection, graft failure, and death among patients with KTs. These findings suggest that efforts toward decreasing levels of PM2.5 concentration may be associated with improved outcomes after KT.

AN M/M/1 QUEUE SUBJECT TO DIFFERENTIATED VACATION WITH PARTIAL INTERRUPTION AND CUSTOMER IMPATIENCE

ABSTRACT In this work, we consider an M/M/1 queueing model subject to differentiatedvacations. When the server is idle, he takes a break for a specific duration(termed Type I vacation). Upon completing the vacation, he takes another short break if the server stillfinds an empty system (termed as Type IIvacation). Both types of vacation follow an exponential distribution. As TypeI vacation persists for an extended period, it is reasonable to assume that theservice continues during this vacation by an alternate server at a slower rate,unlike the Type II vacation. Also, Type II being a shorter duration vacation,we assume that the Type II vacation is interrupted when the queue size reaches a predefined threshold value. Further, the arriving customers willwait only for a fixed duration of time, and if the service is incomplete by then,they leave the system permanently. We obtain an exact analytical expressionfor the transient state probabilities using Laplace transform, generatingfunctions, and continued fraction methodology. We also derive some useful measuresof effectiveness like the time-dependent mean and variance andillustrate their variations graphically.

Illusiveness of the Problem of Time

The essay is devoted to the problem of time in the context of quantum cosmology, which acquires a philosophical level to date. At an example of the minisuperspace model, we show that this problem is illusive because it does not prevent calculating the mean values of the operators over the quantum state of the universe. Contrariwise, the different approaches to the description of these time-dependent mean values give similar results.

Magnetization dynamics of a harmonically confined quantum charged particle in time dependent magnetic fields inside a circular solenoid

I consider a quantum spinless charged particle moving in the xy plane under the action of a time-dependent magnetic field described by means of the linear vector potential in the ‘circular’ gauge A = B(t)(−y, x)/2 and in the presence of the confining potential . This potential guarantees finite values of all second-order moments in the initial equilibrium (thermal) state. Time-dependent mean values of the energy and magnetic moment are expressed in terms of solutions to the classical equation of motion . Simple approximate results are found in the cases of the sudden jump of magnetic field, the parametric resonance, and the adiabatic evolution. They are compared with exact solutions in terms of the hypergeometric function for the specific function B(t) of the Epstein–Eckart type. A great amplification of the magnetic moment is discovered for the initial high-temperature states, both for the fast and slow monotonous variations of the magnetic field. Classical adiabatic invariants do not exist if the magnetic field slowly goes to zero or changes its sign.