Product Of Exponentials Research Articles

Cytotoxicity, phytotoxicity, and photocatalytic assessment of biopolymer cellulose-mediated silver nanoparticles

The appearance of silver nanoparticles (Ag NPs), their exponential production and the consumer products that contain them have raised concerns about their potential risks. Hence, the monitoring of Ag NPs in environmental and biological systems is of vital importance. Furthermore, until now, the information about the toxicity of Ag NPs synthesized from green sources rather than polluting methods is scarce. Hence, in the current work, a simple and straightforward synthesis of Ag NPs mediated by cellulose biopolymer and the evaluation of their cytotoxicity, phytotoxicity, and photocatalytic activity were described for the first time. Initially, the green synthesized Ce-Ag NPs were fully physicochemically characterized. Ce-Ag NPs were sphere-shaped, had high crystallinity, and had a mean diameter of 5–100 nm. The in vitro cytotoxicity evaluation of the green synthesized Ce-Ag NPs was carried out in human normal keratinocyte (HaCaT) and human breast cancer (MCF-7) cells by MTT assay. After the 24 h-treatment with Ce-Ag NPs (0.1–500 µg/mL), the viability of both cell lines was significantly inhibited. Furthermore, the Ce-Ag NPs exhibited dose-dependent cytotoxicity with a median inhibitory concentration (IC 50 ) of 5.4 μg/mL and 4.9 μg/mL against HaCaT and MCF-7 cells, respectively. Moreover, in the phytotoxicity assay, the Ce-Ag NPs did not significantly affect seed germination, roots, and shoots length of Mung ( Vigna radiata ) but affect seed germination and shoots length of Wheat ( Triticum aestivum ). Besides, the photocatalytic activity of Ce-Ag NPs was evaluated by the decomposition of methylene blue (MB), detecting a great degradation from 120 min • Biopolymer cellulose was used for the green synthesis of silver NPs. • Ce-AgNPs shown dose-dependent cytotoxicity on human normal keratinocyte (HaCaT) cells. • Ce-AgNPs shown dose-dependent cytotoxicity on human breast cancer (MCF-7) cells. • Ce-AgNPs did not affect seed germination nor lengths of shoots and roots of Mung. • Ce-AgNPs affected seed germination and lengths of shoots of Wheat. • Ce-AgNPs exhibited a significant photocatalytic activity.

High-order numerical solutions to the shallow-water equations on the rotated cubed-sphere grid

A novel numerical approach to solving the shallow-water equations on the sphere using high-order numerical discretizations in both space and time is proposed. A space-time tensor formalism is used to express the equations of motion covariantly and to describe the geometry of the rotated cubed-sphere grid. The spatial discretization is done with the direct flux reconstruction method, which is an alternative formulation to the discontinuous Galerkin approach. The equations of motion are solved in differential form and the resulting discretization is free from quadrature rules. It is well known that the time step of traditional explicit methods is limited by the phase velocity of the fastest waves. Exponential integration is employed to enable integrations with significantly larger time step sizes and improve the efficiency of the overall time integration. New multistep-type exponential propagation iterative methods of orders 4, 5 and 6 are constructed and applied to integrate the shallow-water equations in time. These new schemes enable time integration with high-order accuracy but without significant increases in computational time compared to low-order methods. The exponential matrix functions-vector products used in the exponential schemes are approximated using the complex-step approximation of the Jacobian in the Krylov-based KIOPS (Krylov with incomplete orthogonalization procedure solver) algorithm. Performance of the new numerical methods is evaluated using a set of standard benchmark tests.

A New Variational Bayesian-Based Kalman Filter with Unknown Time-Varying Measurement Loss Probability and Non-Stationary Heavy-Tailed Measurement Noise.

In this paper, a new variational Bayesian-based Kalman filter (KF) is presented to solve the filtering problem for a linear system with unknown time-varying measurement loss probability (UTVMLP) and non-stationary heavy-tailed measurement noise (NSHTMN). Firstly, the NSHTMN was modelled as a Gaussian-Student’s t-mixture distribution via employing a Bernoulli random variable (BM). Secondly, by utilizing another Bernoulli random variable (BL), the form of the likelihood function consisting of two mixture distributions was converted from a weight sum to an exponential product and a new hierarchical Gaussian state-space model was therefore established. Finally, the system state vector, BM, BL, the intermediate random variables, the mixing probability, and the UTVMLP were jointly inferred by employing the variational Bayesian technique. Simulation results revealed that in the scenario of NSHTMN, the proposed filter had a better performance than current algorithms and further improved the estimation accuracy of UTVMLP.

Powers of Elliptic Scator Numbers

Elliptic scator algebra is possible in 1+n dimensions, n∈N. It is isomorphic to complex algebra in 1 + 1 dimensions, when the real part and any one hypercomplex component are considered. It is endowed with two representations: an additive one, where the scator components are represented as a sum; and a polar representation, where the scator components are represented as products of exponentials. Within the scator framework, De Moivre’s formula is generalized to 1+n dimensions in the so called Victoria equation. This novel formula is then used to obtain compact expressions for the integer powers of scator elements. A scator in S1+n can be factored into a product of n scators that are geometrically represented as its projections onto n two dimensional planes. A geometric interpretation of scator multiplication in terms of rotations with respect to the scalar axis is expounded. The powers of scators, when the ratio of their director components is a rational number, lie on closed curves. For 1 + 2 dimensional scators, twisted curves in a three dimensional space are obtained. Collecting previous results, it is possible to evaluate the exponential of a scator element in 1 + 2 dimensions.

An improved minimal error model for the robotic kinematic calibration based on the POE formula

SummaryThe conventional product of exponentials $\left(\rm POE\right)$ -based methods dissatisfy the parametric minimality for the kinematic calibration of serial robots due to overlooking the magnitude and pitch constraints. Thus, the minimal kinematic model is presented to solve this problem, which can be developed further. This paper puts forward an improved algorithm for the minimal parameter calibration. An actual kinematic model with the minimal parameters $\left(\rm MP\right)$ is constructed according to the geometric properties of actual joint twists in the auxiliary frames established on the basis of the nominal joint axes. Then, the initial pose error is defined in the tool coordinate frame, which is expressed as the exponential map of the twist, and all twist descriptions are unified, so as to give a unified kinematic model in mathematics. By differentiating the kinematic model, a minimal error model is derived in explicit form. Subsequently, we propose a novel parameter identification method, which identifies the orientation error and position error parameters separately by the iterative least-squares method and updates the MP uniformly. Finally, the simulations and experiments on the different serial robots are conducted to verify the correctness and effectiveness of the proposed algorithm. The simulation results show our calibration algorithm outperforms the existing ones in the accuracy aspect, and the experiment result shows that the absolute pose accuracy of the UR5 industrial robot is upgraded about 9 times under a statistics sense after the calibration.

Molecular Characterization of Increased Amplicon Lengths in SARS-CoV-2 Reverse Transcription Loop-Mediated Isothermal Amplification Assays.

Loop-mediated isothermal amplification (LAMP) is a power tool for the amplification of specific RNA and DNA targets. Much like PCR, LAMP requires primers that surround a target amplification region and generates exponential product through a unique highly specific daisy-chain single-temperature amplification reaction. However, until recently, attempts to amplify targets of greater than 200 base pairs (bp) have been mostly unsuccessful and limited to short amplicon targets of less than 150 bp. Although short amplicons have the benefit of a rapid detection (<40 min), they do not allow for the prediction of RNA integrity based on RNA length and possible intactness. In this study, 8 primer sets were developed using 2 LAMP primer-specific software packages against the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) nucleocapsid gene with insert lengths ranging from 262 to 945 bp in order to amplify and infer the integrity of viral RNA. Because these amplification lengths are greater than the current methods that use an insert length of 130 or less, they require a longer incubation, modified primer and temperature strategies, and the addition of specific adjuncts to prevent nonspecific amplification. This proof of concept study resulted in successful reverse transcription LAMP reactions for amplicon targets of 262, 687, 693, and 945 bp using a clinical nasopharyngeal NP sample, purified SARS-CoV-2 RNA, and crude lysate containing inactivated virus.

Theoretical formulations and simulations of one-dimensional inhibition kinetics of ethanologenic microorganisms in batch fermenters

The utilization of bio-based technology for energy has piqued researchers' curiosity around the world. As a result, bioethanol fermentation has been a hot topic of research for many scientists since it uses less energy and chemicals, produces fewer harmful by-products and emissions, and has environmentally favorable applications. The modeling and simulations of one-dimensional product and substrate inhibitions for sorghum, maize, and cassava extracts are discussed in this paper. Because it provides an edge over other methodologies, mechanistic modeling techniques are used. Models of substrate and product inhibitions in one dimension (1-D) are constructed. These 1-D models are then confirmed using parameter estimates before being employed in the work's simulations. For each dynamic model constructed, model fitness coefficients (α) are calculated. For the product, the exponential inhibition model, sorghum extract data has the best model fitness coefficient (α = 0.4088), for product sudden stop inhibition model and cassava extract data gives the best model fitness coefficient (alpha = 0.4417) for product exponential model. The projected yield increases for substrate exponential inhibition with sorghum extract data, substrate linear inhibition with maize extract data, and substrate linear inhibition with cassava extract data are 74%, 27%, and 25%, respectively. This unique framework has offered the industry a wide choice of kinetics models to choose from to alleviate inhibitions in fermentation systems and maximize yield and productivity in the bioethanol fermentation process.Article HighlightsThe modeling of inhibitions namely linear, sudden stop, and exponential in batch fermentation processes are presented in this article.Model fitness coefficient analysis showed the product as a primary inhibitor and substrate as a secondary inhibitor during the processThe cassava and maize processes described linear inhibition model and sorghum fermentation showed exponential product inhibition model.

The method of comprehensive performance optimization design for trimaran unmanned vehicle

The comprehensive performance optimization design of the trimaran unmanned vehicle can be summarized as a complex optimization design problem with multiple objective functions under multiple constraints. First, this paper uses the towing resistance test data of a certain ship model to establish a ship type and resistance database using the response surface fitting method. Then, a single total objective function is used in the form of power exponential product to normalize the objective functions of the trimaran unmanned craft’s speed, maneuverability and seakeeping. Secondly, the constraint conditions are the upper and lower limits of the design variables, the penalty function is composed of multiple equality and inequality constraints involved, the fitness function is established, and a more complete comprehensive optimization mathematical model is constructed. Finally, the overall performance of the trimaran unmanned vehicle is optimized by genetic algorithm, particle swarm algorithm and two improved optimization algorithms. Discussed the pros and cons of each optimization algorithm and the sensitivity of the influence factor under the parallel strategy, compared and analyzed the calculation results of each optimization algorithm, selected the best optimization calculation plan, and obtained the best optimization calculation result, which is trimaran how to ensure the comprehensive and optimal navigation performance during the design of unmanned boats provides a practical way and method.

Quantum analysis on the convergence speed of exponential product formulas — differential-subtraction and exchange-integration method on concise norm bounds

A general method to evaluate rigorously concise norm bounds on the difference between the original exponential operators and their corresponding exponential product formulas is proposed, in order to evaluate the convergence speed of exponential product formulas for a new kind of exponential operator, exp(x2[A, B]). One of the remarkable results on this issue is given by the following formula: e[A,B] is equal to the n → ∞ limit of the product (exp(iA/n)exp(−iB/n)exp(−iA/n)exp(iB/n))n2 for the Hermitian operators A and B. The convergence speed of this formula is proved rigorously to be O(1/n) even for unbounded operators A and B under the condition that the third-order free Lie elements of A and B should be bounded in norm.

A Novel Analytical Inverse Kinematics Method for SSRMS-Type Space Manipulators Based on the POE Formula and the Paden-Kahan Subproblem

Space manipulators which have a similar symmetrical structure with seven revolute joints, such as the space station remote manipulator system (SSRMS), can be called SSRMS-type space manipulators. The analytical inverse kinematics of an SSRMS-type manipulator can be solved by locking a single joint; the locked joint (joint 1, 2, 6, or 7) can be determined by configuration analysis. Although widely used in establishing the kinematics of SSRMS-type manipulators, the Denavit-Hartenberg (DH) method has a singular problem when two adjacent joint axes are nearly parallel. To avoid this problem, this paper proposes a novel analytical inverse kinematics method for SSRMS-type manipulators based on the product of exponentials (POE) formula and the Paden-Kahan subproblem. Because of the symmetrical structure, an SSRMS-type manipulator degrades to two kinds of 6-degree-of-freedom (DOF) manipulators when locking a single joint (joint 1, 2, 6, or 7). The analytical inverse kinematics of these two kinds of 6-DOF manipulators is solved by combining the Paden-Kahan subproblems and geometric and algebraic methods, respectively. The proposed approach is not only singularity free compared with the traditional DH-based methods but also more accurate than the POE-based numerical solution. The simulation results verify the efficiency of the proposed approach.

Reuse, Recycle, and Regeneration of LiFePO4 Cathode from Spent Lithium-Ion Batteries for Rechargeable Lithium- and Sodium-Ion Batteries

Rechargeable lithium-ion batteries are dominating the energy storage market with a current market value of $50 billion. However, the exponential production of lithium-ion batteries is accompanied by an increased backflow as environmentally hazardous spent/end-of-life batteries, which need to be recycled efficiently. Herein, we demonstrate the possibility of reuse, recycle, and regeneration of a spent LiFePO₄ (LFP) cathode for rechargeable lithium- and sodium-ion batteries. An approach of reusing the spent-LFP electrode in developing a new lithium-ion battery was initially explored. The refabricated LFP lithium-ion half-cells delivered excellent capacity and rate capability. Further, an unprecedented approach of using a spent-LFP electrode for sodium ion storage was investigated. The adopted low-temperature, rapid (15 min) microwave delithiation process could completely delithiate the LFP electrode, forming FePO₄. This recycled FePO₄ was tested for its sodium ion storage capability. Finally, an aspect of commercially viable, nearly zero-energy chemical lithiation/sodiation of microwave-derived FePO₄ was also explored. Thus, regenerated LiFePO₄ as lithium-ion half-cells exhibited a reversible capacity of 145 mAh/g at a current rate of 1C and 107 mAh/g at 10C and 96% capacity retention at 5C for 300 cycles. Similarly, the chemically sodiated FePO₄ sodium-ion cell demonstrated a capacity close to 138 mAh/g at C/15, which is comparable to freshly prepared NaFePO₄ (NFP) electrode’s performances. Our electrochemical results confirm the potential of these rapid, sustainable, and energy-efficient methods for reutilization, recycling, and regeneration of failed LFP cathodes recovered from 32 650 cells for their proficient usage in both lithium- and sodium-ion storage.

Kinematic calibration of a 3-PRRU parallel manipulator based on the complete, minimal and continuous error model

Kinematic error model plays an important role in the kinematic calibration of robot manipulators. In order to guarantee the best calibration result, the error model should meet the requirements of completeness, minimality and continuity. For the conventional serial robots, these issues have been intensively studied and a consensus has been reached during the past decades. However, for parallel manipulators, the problem of minimality, namely the determination of identifiable parameters mainly relies on numerical studies. There is a lack of effective methods to analyze the parameters’ identifiability in the error modeling process. Besides, there is still no agreement on the formula of the maximum number of identifiable parameters. In our previous work, a complete, minimal and continuous error modeling approach for parallel manipulators has been proposed based on product of exponential (POE) formula and the equation that the maximal number of identifiable parameters 4r+2p+6 was proved. However, the correctness and effectiveness of this method in kinematic calibration have not been verified on any parallel manipulator or robot. Therefore, in this paper a kinematic calibration is conducted on a prototype of a 3-PRRU parallel manipulator based on the proposed error model. The method of validating the completeness and minimality are respectively given by numerical simulations. Furthermore, to illustrate the advantage of the proposed error model, the kinematic parameters of the prototype are also calibrated based on the conventional error model which is derived by means of differentiating the limbs’ inverse kinematics. By comparing the calibration results, it can be concluded that the error model satisfying the three criteria can better improve the positioning accuracy. In addition, since the calibration method is described uniformly, it can be applied to most parallel manipulators by referring to the calibration procedure this paper presented.

Short-Term Time Wind Speed Forecasting Based on Spatio-Temporal Geostatistical Approach and Kriging Method

Short-term wind speed prediction is an essential task for wind resource and wind energy planning. However, most of this literature does not take into account the spatio-termporal correlation of wind data from the geographical field. For this reason, we propose an integrated spatio-temporal kriging and functional kriging strategy to exploit such spatio-temporal correlation into the wind speed prediction. First, the deterministic trend component in wind data is estimated to be removed. The residuals are used for spatio-temporal modeling and prediction. Based on the spatio-temporal kriging framework, four spatio-temporal covariance models (product-sum model, separable exponential product model, separable and nonseparable Gneiting models) are considered which describe the spatio-temporal correlation of wind data. In particular, the flexibility of using the nonseparable Gneiting model is highlighted. More specifically, four spatio-temporal random fields are modeled from the 12 wind monitoring stations over Ireland. We also use an involved weighted least squares method for estimating parameters of the four covariance models involved in the spatio-temporal kriging strategy. We apply the fitted covariance models to generate day-ahead wind speed predictions at both observed and nonobserved locations where wind station already exist but also to nearby locations. Leave-one-out cross-validation is applied to check the significance of the difference among the four models, these spatio-temporal ordinary kriging (STOK), functional ordinary kriging (FOK) and autoregressive integrated moving average (ARIMA) methods are compared for day-ahead wind speed predictions. Forecasting results indicate that the predicting accuracy is improved almost 33.5% using FOK compared with three approaches which confirm the effectiveness of the functional kriging method in the paper.

Complex Fracture Depletion Model for Reserves Estimations in Shale

Abstract Reserve estimation is a subject of continuous importance in the petroleum industry; controlling field development related decisions and providing valuation of corporations. Tight formations are usually completed with multistage hydraulic fractures and horizontal wellbores. However, these completion scheme results in heterogeneous fracture lengths and spacing. Consequently, some counterparts of the reservoir would experience boundary-dominated flow, while others are still experiencing an infinite-acting linear flow which creates a composite flow regime dubbed as complex fracture depletion (CFD). It is worth noting that the CFD flow regime might be preceded by a linear flow depending on the fracture complexity. We are proposing a unified model that integrates the flow regime analysis and the well performance analysis. Our model utilizes the derivative of the cumulative production with respect to the square root of produced time where the linear flow exhibits a horizontal line and the CFD exhibits an exponential straight line. Therefore, the onset of the CFD becomes the only variable for the regression analysis. Another consequence of utilizing an exponential fit of the flow derivative is a continuous reduction in the Arps’ “b” exponent from a “b” value of two during CFD. We also validated our model estimations to the estimations of Arap’s and stretch exponential production decline (SEPD) with recent production data from the Bone Spring formation, New Mexico, and major shale/tight reservoirs.

Non-destructive Testing Assisted by Six-Axis Manipulator Based on POE Formula

At present, aluminum plates are widely used in aerospace, machinery, and other fields because of their light weight, good manufacturability, and strong corrosion resistance. However, during long-term use, defects that will affect the structural strength such as cracks are easy to appear. Therefore, it is necessary to conduct automatic non-destructive testing of large metal plates to determine the location, size, and structural damage to their defects. Compared with manual detection, automated detection has many advantages such as high detection efficiency, high repeatability, strong flexibility, and low labor intensity. Based on the spinor theory, this paper uses the product of exponentials (POE) to model the kinematics of the Aubo-i5 manipulator and obtain the positive kinematics equations, which are compiled by analytic geometry and algebraic methods combined with Paden-Kahan subproblems and matrix theory. The inverse kinematics algorithm of the 6R manipulator is based on this model. An automated non-destructive testing system including a robotic arm and ultrasonic phased array testing equipment was built. At the same time, a robotic arm simulation platform was built based on ROS, and the inverse kinematics algorithm was used to control the robotic arm through the MoveIt!. The angle error can be controlled within 2.5°.

Production Data Analysis of Hydraulically Fractured Horizontal Wells from Different Shale Formations

A comprehensive overview and analysis of the productivity of 1216 recently abandoned multi-stage hydraulically fractured horizontal wells from five shale formations in the United States (US) is presented in this study. In this study, two decline curve analysis (DCA) methods were used to match actual production history data using least-squares fitting to find the best fit production parameters to reliably forecast production. The production history matching conducted resulted in very accurate matches (correlation coefficient of 0.99) between actual production data and the two DCA methods (Arps hyperbolic decline and stretched exponential production decline (SEPD) models). Using the outcomes from production history matching, universal averages of decline parameters for Arps hyperbolic decline and SEPD models were developed for each of the five formations. Furthermore, hindcasting was performed by matching a portion of the known production history and comparing the remaining portion of the known production history to the forecast. The Arps hyperbolic decline and SEPD methods were used to match production using only limited early production data (three months, six months, one year and two years). The main goals for fitting the DCA model to early production data was to estimate the optimum decline parameters that are then used to forecast production and estimate ultimate recovery. Production history matching using limited early production periods produced accurate production forecasts using as few as six months of production history (correlation coefficients between 0.85 and 0.94 using Arps hyperbolic decline). The main outcome of this study was a production analysis conducted on the production data of more than 1000 wells from five different shale formations to present the expected production behaviors of similar wells. Different production key performance indicators (KPIs) such as average well life, cumulative production volumes at different periods, average drop in production rate within the first year of production, average time to reach maximum flow rate, and the maximum flow rate were measured on all the wells from the five formations to provide an overview of the production performance of each formation.

A novel decline curve regression procedure for analyzing shale gas production

Accurate prediction of gas production is critical for the shale gas development. Compared with the complicated physics-based simulation, the decline curve analysis (DCA) models based on parameter linearization or direct curve regression are much more straightforward and efficient. However, the linearization method might be time consuming in obtaining the parameters, and the direct regression might lead to large errors in production forecast as it assigns equal weight to the historical data. In this paper, we develop a new DCA procedure combining a data transformation method which convert the production data into logarithmic space, and a nonlinear regression algorithm. The new procedure can effectively capture the trend of late-time production history. The efficiency of the new procedure is verified with the production data of 550 gas wells in the Barnett and Marcellus shales. Based on these field data, we also analyze the performance of seven popular DCA models, i.e. Arps, power law exponential (PLE) decline, stretched exponential production decline (SEPD), Duong, Wang , variable decline modified Arps (VDMA) and logistic growth models. The performances of these models are compared and the tuned parameters for each model are provided. It is shown that the accuracy of the production forecast by the above models can be significantly improved by our new regression method, which is beneficial to the prediction and optimization of shale gas production.

Probabilistic Decline Curve Analysis in the Permian Basin using Bayesian and Approximate Bayesian Inference

Summary In this work, a probabilistic methodology for decline curve analysis (DCA) in unconventional reservoirs is presented using several Bayesian model-fitting algorithms and deterministic models. The deterministic models considered are the power law exponential (PLE) model, the stretched exponential production decline (SEPD) model, Duong’s model, and the logistic growth analysis (LGA) model. Accurate production forecasting and uncertainty quantification were the primary objectives of this study. The Bayesian inferencing techniques described in this work utilize three sampling vehicles, namely the Gibbs sampling (implemented in OpenBUGS, an open-source software), the Metropolis-Hastings (MH) algorithm, and approximate Bayesian computation (ABC) to sample parameter values from their posterior distributions. These different sampling algorithms are applied in conjunction with DCA models to estimate DCA parameter prediction intervals. Using these prediction levels, production is forecasted, and uncertainty bounds are established. To examine its reliability, the methodology was tested on over 74 oil and gas wells located in the three main subplays of the Permian Basin, namely, the Delaware play, the Central Basin Platform, and the Midland play. Results show that the examined DCA-Bayesian models are well calibrated, result in low production errors, and narrow uncertainty bounds for the production history data sets. Also, the LGA model is best in terms of prediction errors for all algorithms except MH. The Gibbs algorithm is nearly the best algorithm in terms of prediction error for all DCA models except the Arps model. Prediction errors are the highest in the Central Basin Platform. The methodology was also successfully applied to unconventional reservoirs with as low as six months of available production history. Depending on the amount of production history data available, the probabilistic model that provides the best fit can vary. It is therefore recommended that all possible combinations of the deterministic and Bayesian model-fitting algorithms be applied to the available production history. This is to obtain more confidence in the conclusions related to the production forecasts, reserves estimate, and uncertainty bounds. The novelty of this methodology relies on using multiple combinations of DCA-Bayesian models to achieve accurate reserves estimates and narrow uncertainty bounds. This paper can help assess shale plays because some of the shale plays are in the early stages of development when productivity estimations are carried out.

Exponential ratio and product type estimators of the mean in stratified two-phase sampling

In this study, we propose exponential ratio estimators in the stratified two-phase sampling utilizing an auxiliary attribute. The expressions for the mean squared error of these exponential-type estimators under two different cases are derived and theoretical comparisons are made with competing estimators. We show that the proposed estimators have a lower mean square error than the simple mean estimator, usual stratified two-phase sampling ratio, and product estimators, usual exponential ratio and product estimators for the stratified two-phase sampling under the obtained conditions in theory. In addition, these theoretical results are supported with the aid of a numerical example.

An Inverse Problem for LTV ODE’s

It is well known that any linear combination of products of polynomials and exponentials (a.k.a. Bohl functions) are annihilated by LTI-ODE’s. In this paper, we consider the more general classes of piecewise Bohl functions and of real analytic functions and present the realization of the latter as solutions of linear differential polynomials with real analytic coefficients. In particular, we show that the minimal order depends solely on the multiplicity of the zeros of the given function. We extend the problem further to meromorphic functions without poles on the real line. We briefly comment on piecewise-defined functions and the nonhomogeneous problem.