Exponential Estimates Research Articles

Effects of different calibration constraints on calibration estimators under the randomized response technique

In survey sampling, to handle the situation where human bias arises due to some personal or illegal questions, the randomized response technique developed by Warner is used [1]. The calibration technique is used to reduce the bias between the original and estimated weights in stratified random sampling. We estimate the population total, using the calibration estimation technique in which the study variable corresponds to a scrambled variable due to the randomized response technique. We have discussed the estimation procedure of population total and suggested some new estimators, such as ratio and exponential estimators, using different distance measures and calibration constraints. The simulation study has been applied tn newly developed estimators in sensitive survey sampling using a scrambled randomized response (SRR). We see that exponential estimators perform better than ratio and simple estimators of the total stratified random samplings when the variable of interest is sensitive.

Lyapunov functionals for linear neutral delay systems in the Hale's form

We construct Lyapunov functionals for linear neutral delay systems in the Hale's form. Our construction is made for no differentiable initial functions, as required for such a class of systems, but not thus considered in the existing works. The functionals depend on a special matrix-valued function which is the natural extension of the Lyapunov matrix for retarded delay systems. The constructed functionals allow deriving a converse result for exponential stability of the systems and exponential estimates for the solutions.

A New Free-matrix Inequality with application to Exponential State Estimation for Markov Jump Neural Networks

This paper is concerned with an exponential state estimation problem for a class of Markov Jump Neural Networks (MJNNs) with Time-varying Delay (TVD). Firstly, to derive a tighter bound of Exponential-type Integral Quadratic Terms (EIQTs), a new Free-matrix Exponential-type Inequality (FMEI) is provided, which includes some existing ones as its special cases. Secondly, by fully considering the information of attenuation exponent and TVD, a novel Lyapunov-Krasovskii Functional (LKF) is constructed. Then, by employing the new FMEI and other analytical techniques, a less conservative criterion guaranteeing the stochastic stability of the error system is obtained in form of Linear Matrix Inequalities (LMIs). In the end, a numerical example is given to show the effectiveness of the obtained result.

Generalized modified ratio-cum-product kind exponentially estimator of the populations mean in stratified ranked set sample

In this study, we present a proposal aimed at estimating the finite population's mean of the main variable by stratification rank set sample (S_{t}text{RSS}) through the modification made to generalized ratio-cum-product type exponential estimator. The relative bias (text{PRB}), Mean Squared Error (text{Mse}) and percentage relative efficiencies (text{PRE}) of the proposed modified estimator is obtained to the first degree of approximation. Conditions under which the proposed estimator is more efficient than the usual unbiased estimator, ratio, product type estimators, and some other estimators are obtained. Finally, the estimators' abilities are evaluated through the use of simulations, as showed that the proposed modified estimator is more efficient as compared to several other estimators.

Equivalence between exponential stabilization and observability inequality for magnetic effected piezoelectric beams with time-varying delay and time-dependent weights

<p style='text-indent:20px;'>This paper is concerned with system of magnetic effected piezoelectric beams with interior time-varying delay and time-dependent weights, in which the beam is clamped at the two side points subject to a single distributed state feedback controller with a time-varying delay. Under appropriate assumptions on the time-varying delay term and time-dependent weights, we obtain exponential stability estimates by using the multiplicative technique, and prove the equivalence between stabilization and observability.</p>

Propagator norm and sharp decay estimates for Fokker–Planck equations with linear drift

We are concerned with the short- and large-time behavior of the $L^2$-propagator norm of Fokker-Planck equations with linear drift, i.e. $\partial_t f=\mathrm{div}_{x}{(D \nabla_x f+Cxf)}$. With a coordinate transformation these equations can be normalized such that the diffusion and drift matrices are linked as $D=C_S$, the symmetric part of $C$. The main result of this paper is the connection between normalized Fokker-Planck equations and their drift-ODE $\dot x=-Cx$: Their $L^2$-propagator norms actually coincide. This implies that optimal decay estimates on the drift-ODE (w.r.t. both the maximum exponential decay rate and the minimum multiplicative constant) carry over to sharp exponential decay estimates of the Fokker-Planck solution towards the steady state. A second application of the theorem regards the short time behaviour of the solution: The short time regularization (in some weighted Sobolev space) is determined by its hypocoercivity index, which has recently been introduced for Fokker-Planck equations and ODEs (see [5, 1, 2]). In the proof we realize that the evolution in each invariant spectral subspace can be represented as an explicitly given, tensored version of the corresponding drift-ODE. In fact, the Fokker-Planck equation can even be considered as the second quantization of $\dot x=-Cx$.

Data-Based and Opportunistic Integral Concurrent Learning for Adaptive Trajectory Tracking During Switched FES-Induced Biceps Curls.

Hybrid exoskeletons, which combine functional electrical stimulation (FES) with a motorized testbed, can potentially improve the rehabilitation of people with movement disorders. However, hybrid exoskeletons have inherently nonlinear and uncertain dynamics, including combinations of discrete modes that switch between different continuous dynamic subsystems, which complicate closed-loop control. A particular complication is the uncertain muscle control effectiveness associated with FES. In this work, adaptive integral concurrent learning (ICL) motor and FES controllers are developed for a hybrid biceps curl exoskeleton, which are designed to achieve opportunistic and data-based learning of the uncertain human and electromechanical testbed parameters. Global exponential trajectory tracking and parameter estimation errors are proven through a Lyapunov-based stability analysis. The motor effectiveness is assumed to be unknown, and, to help with fatigue reduction, FES is enabled to switch between multiple electrodes on the biceps brachii, further complicating the analysis. A consequence of switching between the different uncertain subsystems is that the parameters must be opportunistically learned for each subsystem (i.e. each electrode and the motor), while that subsystem is active. Experiments were performed to validate the developed ICL controllers on twelve healthy participants. The average (± standard deviation) position tracking errors across each participant were 1.44 ± 5.32 deg, -0.25 ± 2.85 deg, and -0.17 ± 2.66 deg across biceps Curls 1-3, 4-7, and 8-10, respectively, where the average across the entire experiment was 0.28 ± 3.53 deg.

Entropic turnpike estimates for the kinetic Schrödinger problem

We investigate the kinetic Schrödinger problem, obtained considering Langevin dynamics instead of Brownian motion in Schrödinger’s thought experiment. Under a quasilinearity assumption we establish exponential entropic turnpike estimates for the corresponding Schrödinger bridges and exponentially fast convergence of the entropic cost to the sum of the marginal entropies in the long-time regime, which provides as a corollary an entropic Talagrand inequality. In order to do so, we benefit from recent advances in the understanding of classical Schrödinger bridges and adaptations of Bakry–Émery formalism to the kinetic setting. Our quantitative results are complemented by basic structural results such as dual representation of the entropic cost and the existence of Schrödinger potentials.

Switched Adaptive Concurrent Learning Control using a Stance Foot Model for Gait Rehabilitation using a Hybrid Exoskeleton

Lower-limb hybrid exoskeletons integrate powered mechanisms and functional electrical stimulation (FES) to provide assistive forces and activate muscles for restoring gait function after a neurological injury. Particularly, improving the load-bearing ability is a primary rehabilitation goal. Hence, the control of the exoskeleton and FES is critical within the stance phase of walking to ensure smooth weight transfer between limbs, and achieve a sound loading response and leg propulsion. This paper develops a concurrent learning adaptive control technique to provide torque assistance about the hip and knee joints using a cable-driven exoskeleton, and activate the quadriceps and hamstrings muscle groups via FES for treadmill walking. The human-exoskeleton dynamics are modeled with phase-dependent switching dynamics to strategically update the concurrent learning controller at early stance (heel strike), late stance (toe-off), and the swing phases of walking. Thus, the adaptive switching controller compensates for the dynamic changes within the stance phase and its transition to swing, while achieving joint kinematic tracking and estimating a subset of the leg's uncertain parameters. A multiple Lyapunov function analysis is developed to demonstrate stability of the overall phase-dependent switched system requiring a dwell-time condition that guarantees exponential tracking and parameter estimation convergence.

Linear instability of Vlasov-Maxwell systems revisited-A Hamiltonian approach

<p style='text-indent:20px;'>We consider linear stability of steady states of 1<inline-formula><tex-math id="M1">\begin{document}$ \frac{1}{2} $\end{document}</tex-math></inline-formula> and 3DVlasov-Maxwell systems for collisionless plasmas. The linearized systems canbe written as separable Hamiltonian systems with constraints. By using ageneral theory for separable Hamiltonian systems, we recover the sharp linearstability criteria obtained previously by different approaches. Moreover, weobtain the exponential trichotomy estimates for the linearized Vlasov-Maxwellsystems in both relativistic and nonrelativistic cases.</p>

Dual to Ratio cum Product Type of Exponential Estimator for Population Mean in Stratified Random Sampling

Dual to Ratio cum Product Type of Exponential Estimator for Population Mean in Stratified Random Sampling

Modified Ratio-cum- Exponential Estimator for Appraising the Population Mean using Conventional and non-Conventional Parameters

Modified Ratio-cum- Exponential Estimator for Appraising the Population Mean using Conventional and non-Conventional Parameters

BOUNDEDNESS OF SOLUTIONS OF A CLASS OF LINEAR DIFFERENTIAL EQUATIONS OF THE SECOND ORDER ON THE SEMIAXIS

In this paper based on a formula for one class of second order linear differential equations with variable coefficients were established sufficient conditions of exponential estimate of solutions for one class of second order linear differential equations on the semi axis.

Random separation property for stochastic Allen-Cahn-type equations

We study a large class of stochastic p-Laplace Allen-Cahn equations with singular potential. Under suitable assumptions on the (multiplicative-type) noise we first prove existence, uniqueness, and regularity of variational solutions. Then, we show that a random separation property holds, i.e. almost every trajectory is strictly separated in space and time from the potential barriers. The threshold of separation is random, and we further provide exponential estimates on the probability of separation from the barriers. Eventually, we exhibit a convergence-in-probability result for the random separation threshold towards the deterministic one, as the noise vanishes, and we obtain an estimate of the convergence rate.

Exponential state estimation for reaction-diffusion inertial neural networks via incomplete measurement scheme

ABSTRACT In this paper, the problem of exponential state estimation for inertial neural networks with reaction-diffusion term (RDINNs) via incomplete measurement scheme is investigated. Unlike the full measurement method, this method estimates the system by measuring the state of partially available neurons. First, by constructing an appropriate variable substitution, the second-order system is transformed into a first-order one. Then, a suitable Lyapunov-krasovskii function (LKF) is constructed, and sufficient conditions for the stability of the system are obtained . Finally, the practicality and effectiveness of the proposed method is further verified by two numerical examples.

A new exponential estimators for analysis of COVID‐19 risk

AbstractCoronavirus disease 2019 (COVID‐19) is the most crucial issue of all countries worldwide as it poses a threat and risk to people in many aspects such as health and economy. Since each country's development level, economy, and infrastructure differ, countries' struggle against COVID‐19 varies. Therefore, understanding the COVID‐19 risk levels of countries can be crucial in determining possible strategies to take specific measures for those at the highest risk. Also, determining the risk level of countries can be more critical than estimates, such as the number of cases and deaths, as the level of risk alone can be an informative indicator for all such issues. Unlike most studies, this study concentrates on evaluating and estimating the COVID‐19 risk level of countries. This study proposes two families of multivariate exponential estimators using two auxiliary attributes. Theoretically, the mean square error (MSE) equations of all proposed exponential estimators are obtained and compared with existing estimators. Some exceptional cases of the multivariate exponential estimators are regarded and compared with MSE values of proposed multivariate exponential estimators. As a result, the multivariate exponential estimators provide more efficient results than all other estimators. These theoretical findings are supported by a numerical illustration using real dataset.

Semiclassical spectral analysis of the Bochner–Schrödinger operator on symplectic manifolds of bounded geometry

We study the Bochner–Schrödinger operator \(H_{p}=\frac{1}{p}\varDelta ^{L^p\otimes E}+V\) on high tensor powers of a positive line bundle L on a symplectic manifold of bounded geometry. First, we give a rough asymptotic description of its spectrum in terms of the spectra of the model operators. This allows us to prove the existence of gaps in the spectrum under some conditions on the curvature of the line bundle. Then we consider the spectral projection of such an operator corresponding to an interval whose extreme points are not in the spectrum and study asymptotic behavior of its kernel. First, we establish the off-diagonal exponential estimate. Then we state a complete asymptotic expansion in a fixed neighborhood of the diagonal.

Improved strategy for computation of population mean under double stratified sampling framework

Abstract The article contains a new technique to estimate the mean of the variate of the interest of the finite population with the help of two auxiliary variates. The technique complies well with the stratified population in which each strata proportion is predicted by taking an initial sample called the first phase sample. When the first phase sample is taken, a second sample is taken from the first sample which is called the second phase sample which is used to estimate the mean of the variate of the interest. In our study, we have considered the population which has two correlated auxiliary variates that pass almost through the origin. In such a situation ratio estimation technique and product estimation technique that provides improved estimates of the mean of the variate of the interest. Our technique considers a ratio-product type exponential estimator of which we have established efficiency theoretically as well as empirically.

Finding the parameters of exponential estimates of solutions to the Cauchy problem for some systems of linear delay differential equations

Finding the parameters of exponential estimates of solutions to the Cauchy problem for some systems of linear delay differential equations

Estimation of Finite Population Variance Under Stratified Sampling Technique

The efficiency of the study variable can be improved by incorporating the information from the known auxiliary variables. Usually two techniques ratio and regression estimation are used with the help of auxiliary information in different approaches to acquire the high precision of the estimators. Considering the very heterogeneous population to get the size of the sample it may be originating impossible to get a sufficiently accurate and precise estimate by taking the simple random sampling technique from the complete population. Occasionally taking sample issue may differ significantly in different part of the entire population. For example, under study population consists of people living in apartments, own homes, hospitals and prisons or people living in plain regions and hill regions so in such situations the stratified sampling is one of the most commonly used approach to get a representative sample in survey sampling from different cross units of the population. The present study is set out on the recommendation of generalized variance estimators for finite population variance incorporating stratified sampling scheme with the information of single and two transformed auxiliary variables. The expressions of bias and mean square error (MSE) are obtained for the advised exponential type estimators. The conditions are obtained for which the anticipated estimators are better than the usual estimator. An empirical and simulation study is conducted to prove the superiority of the recommended estimator.