First Approximation Research Articles

Mean Estimators for Sensitive Variables Using Scrambled Response Models

The most popular methods for sensitive study are regression estimation methods that use standard regression coefficients. When it comes to survey researchers, mean estimation is an important issue. The vast majority of population mean estimation methods that can be found in sampling theory are meant only to be utilized with non-sensitive data-sets. When the variable that is important is sensitive, such as drug usage, illegal income, abortion, exam cheating, the amount of income tax due, employee rule-breaking, etc., these estimation methods cannot operate efficiently. In this article, we propose a novel method for computing the population mean using exponential technique of robust-type regression estimators for scrambled response model (SRM) under simple random sampling (SRS). The mean square error (MSE) equation is generated using a first-order approximation and examined with existing estimating techniques in order to evaluate the efficacy of the new approach. Additionally, the proposed estimator's percentage relative efficiency (PRE) is determined compared to other estimators. The effectiveness of the proposed method is demonstrated using real data sets. According to the results, the suggested estimator performs better than other estimators in the literature.

Performance of range-separated long-range SOPPA short-range density functional theory method for vertical excitation energies.

In this paper, benchmark results are presented on the calculation of vertical electronic excitation energies using a long-range second-order polarization propagator approximation (SOPPA) description with a short-range density functional theory description based on the Perdew-Burke-Ernzerhof (PBE) functional. The excitation energies are investigated for 132 singlet states and 71 triplet states across 28 medium-sized organic molecules. The results show that overall SOPPA-srPBE always performs better than PBE and that SOPPA-srPBE performs better than SOPPA for singlet states, but slightly worse than SOPPA for triplet states when CC3 results are the reference values.

Consistent Second-Order Treatment of Spin-Orbit Coupling and Dynamic Correlation in Quasidegenerate N-Electron Valence Perturbation Theory.

We present a formulation and implementation of second-order quasidegenerate N-electron valence perturbation theory (QDNEVPT2) that provides a balanced and accurate description of spin-orbit coupling and dynamic correlation effects in multiconfigurational electronic states. In our approach, the energies and wave functions of electronic states are computed by treating electron repulsion and spin-orbit coupling operators as equal perturbations to the nonrelativistic complete active-space wave functions, and their contributions are incorporated fully up to the second order. The spin-orbit effects are described using the Breit-Pauli (BP) or exact two-component Douglas-Kroll-Hess (DKH) Hamiltonians within spin-orbit mean-field approximation. The resulting second-order methods (BP2- and DKH2-QDNEVPT2) are capable of treating spin-orbit coupling effects in nearly degenerate electronic states by diagonalizing an effective Hamiltonian expanded in a compact non-relativistic basis. For a variety of atoms and small molecules across the entire periodic table, we demonstrate that DKH2-QDNEVPT2 is competitive in accuracy with variational two-component relativistic theories. BP2-QDNEVPT2 shows high accuracy for the second- and third-period elements, but its performance deteriorates for heavier atoms and molecules. We also consider the first-order spin-orbit QDNEVPT2 approximations (BP1- and DKH1-QDNEVPT2), among which DKH1-QDNEVPT2 is reliable but less accurate than DKH2-QDNEVPT2. Both DKH1- and DKH2-QDNEVPT2 hold promise as efficient and accurate electronic structure methods for treating electron correlation and spin-orbit coupling in a variety of applications.

Generalized covariant entropy bound in Einstein gravity with quadratic curvature corrections

We explore the generalized covariant entropy bound in the theory where Einstein gravity is perturbed by quadratic curvature terms, which can be viewed as the first-order quantum correction to Einstein gravity. By replacing the Bekenstein-Hawking entropy with the holographic entanglement entropy of this theory and introducing two reasonable physical assumptions, we demonstrate that the corresponding Generalized Covariant Entropy Bound is satisfied under a first-order approximation of the perturbation from the quadratic curvature terms. Our findings suggest that the entropy bound and the Generalized Second Law of black holes are satisfied in the Einstein gravity under the first-order perturbation from the quadratic curvature corrections, and they also imply that the generalized covariant entropy bound may still hold even after considering the quantum correction of gravity, but in this case, we may need to use holographic entanglement entropy as the formula for gravitational entropy.

Analytical Study of Damped Standing Longitudinal MHD Waves in Flowing Coronal Loops: The Effect of Thermal Conduction

The longitudinal magnetohydrodynamic (MHD) oscillations of coronal loops have been investigated in dissipative flowing loops. Thermal conduction has been considered as the damping mechanism of the wave. We aim to construct the damped longitudinal waves by superposing two propagating waves that propagate in opposite directions. The two propagating components must have the same oscillation frequencies and damping rates, which has been described impossible by some authors, but we have used a technique to overcome this difficulty. The equations of motion are combined to obtain a differential equation for the velocity perturbation. Using the weak damping condition, the perturbation method is used to solve the dispersion relation. In the leading order approximation, the oscillation frequency of the standing waves is determined. In the first-order approximation, we let both the oscillation frequency and wavelength of the propagating waves be perturbed due to the presence of thermal conduction, which enables us to determine the damping rate of the standing waves. Our results show that the plasma flow is an essential parameter in determining the effectiveness of the damping mechanism. Also, the exact solutions of the dispersion relation have been determined without using weak damping assumption. Interestingly the two solutions are the same. Introducing plasma flow to the coronal loop causes the period ratio of the fundamental mode to the first overtone to decrease more.

Unveiling the Power of Implicit Six-Point Block Scheme: Advancing numerical approximation of two-dimensional PDEs in physical systems.

In the era of computational advancements, harnessing computer algorithms for approximating solutions to differential equations has become indispensable for its unparalleled productivity. The numerical approximation of partial differential equation (PDE) models holds crucial significance in modelling physical systems, driving the necessity for robust methodologies. In this article, we introduce the Implicit Six-Point Block Scheme (ISBS), employing a collocation approach for second-order numerical approximations of ordinary differential equations (ODEs) derived from one or two-dimensional physical systems. The methodology involves transforming the governing PDEs into a fully-fledged system of algebraic ordinary differential equations by employing ISBS to replace spatial derivatives while utilizing a central difference scheme for temporal or y-derivatives. In this report, the convergence properties of ISBS, aligning with the principles of multi-step methods, are rigorously analyzed. The numerical results obtained through ISBS demonstrate excellent agreement with theoretical solutions. Additionally, we compute absolute errors across various problem instances, showcasing the robustness and efficacy of ISBS in practical applications. Furthermore, we present a comprehensive comparative analysis with existing methodologies from recent literature, highlighting the superior performance of ISBS. Our findings are substantiated through illustrative tables and figures, underscoring the transformative potential of ISBS in advancing the numerical approximation of two-dimensional PDEs in physical systems.

Modified correlated measurement errors model for estimation of population mean utilizing auxiliary information

The existence of measurement errors cannot be avoided in practice. It is a prominent fact that the existence of measurement errors diminishes conventional properties of the estimators. A modified correlated measurement errors model has been proposed. Shalabh and Tsai (Commun Stat Simul Comput 46(7):5566–5593. 10.1080/03610918.2016.1165845, 2017) correlated measurement errors model is a particular member of the suggested modified model. In this article, we have tackled the estimation of population mean utilizing auxiliary information under modified correlated measurement errors model. We have developed ratio and product estimators and studied their properties in case of simple random sampling without replacement (SRSWOR) up to first order of approximation. It has been illustrated that suggested ratio and product estimators are more efficient than the conventional unbiased estimator as well as Shalabh and Tsai (Commun Stat Simul Comput 46(7):5566–5593. 10.1080/03610918.2016.1165845, 2017) ratio and product estimators under very realistic situations. An empirical study has also been performed to demonstrate the merits of the recommended estimators over other estimators.

Brownian motion in the Hilbert space of quantum states and the stochastically emergent Lorentz symmetry: A fractal geometric approach from Wiener process to formulating Feynman’s path-integral measure for relativistic quantum fields

This paper aims to provide a consistent, finite-valued, and mathematically well-defined reformulation of Feynman’s path-integral measure for quantum fields obtained by studying the Wiener stochastic process in the infinite-dimensional Hilbert space of quantum states. This reformulation will undoubtedly have a crucial role in formulating quantum gravity within a mathematically well-defined framework. In fact, this study is fundamentally different from any previous research on the relationship between Feynman’s path-integral and the Wiener stochastic process. In this research, we focus on the fact that the classic Wiener measure is no longer applicable in infinite-dimensional Hilbert spaces due to fundamental differences between displacements in low and extremely high dimensions. Thus, an analytic norm motivated by the role of the fractal functions in the Wilsonian renormalization approach is worked out to properly characterize Brownian motion in the Hilbert space of quantum states on a compact flat manifold. This norm, the so-called fractal norm, pushes the rougher functions (physically the quantum states with higher energies) to the farther points of the Hilbert space until the fractal functions as the roughest ones are moved to infinity. Implementing the Wiener stochastic process with the fractal norm, results in a modified form of the Wiener measure called the Wiener fractal measure, which is a well-defined measure for Feynman’s path-integral formulation of quantum fields. Wiener fractal measure has a complicated formula of non-local terms but produces the Klein–Gordon action at the first order of approximation. Using complex integrals to compensate for the removal of non-local terms appearing in higher orders of approximation, the Wiener fractal measure turns into a complex measure and generates Feynman’s path-integral formulation of scalar quantum fields. This brings us to the main objective of this study. Finally, some various significant aspects of quantum field theory (such as renormalizability, RG flow, Wick rotation, regularization, etc.) are revisited by means of the analytical aspects of the Wiener fractal measure.

Almost Unbiased Estimators for Population Coefficient of Variation Using Auxiliary Information

The objective of the paper is to propose an almost unbiased ratio estimator for the finite coefficient of variation (CV). In this paper, we have proposed an exponential ratio type and log ratio type estimators for estimating population coefficient of variation. Two real data sets and one simulation study is carried out in support of the theoretical results. Mean squared error and Percent relative efficiency criteria is used to assess the performance of the estimators. It has been shown that the proposed class of estimators are almost unbiased up to the first order of approximation. Also proposed estimators are better in efficiency to other estimators consider in this study.

Evolving Improved Sampling Protocols for Dose–Response Modelling Using Genetic Algorithms with a Profile-Likelihood Metric

Practical limitations of quality and quantity of data can limit the precision of parameter identification in mathematical models. Model-based experimental design approaches have been developed to minimise parameter uncertainty, but the majority of these approaches have relied on first-order approximations of model sensitivity at a local point in parameter space. Practical identifiability approaches such as profile-likelihood have shown potential for quantifying parameter uncertainty beyond linear approximations. This research presents a genetic algorithm approach to optimise sample timing across various parameterisations of a demonstrative PK-PD model with the goal of aiding experimental design. The optimisation relies on a chosen metric of parameter uncertainty that is based on the profile-likelihood method. Additionally, the approach considers cases where multiple parameter scenarios may require simultaneous optimisation. The genetic algorithm approach was able to locate near-optimal sampling protocols for a wide range of sample number (n = 3–20), and it reduced the parameter variance metric by 33–37% on average. The profile-likelihood metric also correlated well with an existing Monte Carlo-based metric (with a worst-case r > 0.89), while reducing computational cost by an order of magnitude. The combination of the new profile-likelihood metric and the genetic algorithm demonstrate the feasibility of considering the nonlinear nature of models in optimal experimental design at a reasonable computational cost. The outputs of such a process could allow for experimenters to either improve parameter certainty given a fixed number of samples, or reduce sample quantity while retaining the same level of parameter certainty.

A well-balanced second-order finite volume approximation for a coupled system of granular flow

A well-balanced second-order finite volume scheme is proposed and analyzed for a 2×2 system of non-linear partial differential equations which describes the dynamics of growing sandpiles created by a vertical source on a flat, bounded rectangular table in multiple dimensions. To derive a second-order scheme, we combine a MUSCL type spatial reconstruction with strong stability preserving Runge-Kutta time stepping method. The resulting scheme is ensured to be well-balanced through a modified limiting approach that allows the scheme to revert to a well-balanced first-order scheme if it fails to capture or preserve the discrete steady state solution during its time evolution. Further, the scheme maintains the second-order accuracy away from the steady state. The well-balance property of the scheme is proven analytically in one dimension and demonstrated numerically in two dimensions. Additionally, numerical experiments reveal that the second-order scheme reduces finite time oscillations and provides better resolutions of the physical properties of the state variables, as compared to the existing first-order schemes of the literature.

Tunable spatial and angular spin splitting of reflected vortex-beam off hyperbolic metasurface

This paper investigated the spin splitting of the reflected beam with a vortex beam irradiating on a hyperbolic metasurface. This metasurface structure is the periodically rectangular-groove array on the surface of a hyperbolic crystal with the optical axis normal to its surface, where the array is a sub-wavelength biaxially hyperbolic grating with its periodical direction situated in the surface plane. Based on the hexagonal nitride boron crystal, we theoretically and numerically analyzed the spatial and angular spin splittings of the reflected beam, containing those produced by the transverse and longitudinal shifts. We found that the spatial shifts or spin splitting can be obtained in the first-order approximation of beam deflection, but the angular shifts can be described in the second-order approximation so that it enormously varies with the beam waist. Due to the hyperbolic anisotropy of the metasurface and the intrinsic orbital angular momentum in the incident beam, the expressions of spatial and angular shifts are effectively modified, and a unique double reflection effect occurs on the surface. Near certain reflective intensity distribution singularities, extremely asymmetrical spin splitting can be observed. These shifts and spin splitting can be tuned by varying the orientation of the incident plane and the angle of incidence.

Estimating neutrosophic finite median employing robust measures of the auxiliary variable

Our study explores neutrosophic statistics, an extension of classical and fuzzy statistics, to address the challenges of data uncertainty. By leveraging accurate measurements of an auxiliary variable, we can derive precise estimates for the unknown population median. The estimators introduced in this research are particularly useful for analysing unclear, vague data or within the neutrosophic realm. Unlike traditional methods that yield single-valued outcomes, our estimators produce ranges, suggesting where the population parameter is likely to be. We present the suggested generalised estimator's bias and mean square error within a first-order approximation framework. The practicality and efficiency of these proposed neutrosophic estimators are demonstrated through real-world data applications and the simulated data set.

Estimation of population mean using ranked set sampling in the presence of measurement errors

Ranked set sampling is widely acknowledged for its superior efficiency compared with simple random sampling. Only a small amount of work has been conducted using ranked set sampling when measurement errors are present. This study introduces innovative estimators utilizing ranked set sampling to assess the population mean when faced with both correlated and uncorrelated measurement errors. The expressions for the bias and mean squared error of the proposed estimators are derived up to the first-order approximation, revealing their superior performance compared to the other examined estimators. The efficacy of the suggested estimators in handling measurement errors was demonstrated through numerical illustration and simulation study investigations. The recommended estimators are further compared to the existing ones using the percentage relative efficiency and mean squared error, and the impact of measurement errors on the results is highlighted through the percentage computation of measurement errors. The innovative estimators suggested were formulated by judiciously incorporating ratio, exponential, and log estimators. Numerical examples involving expenditure and income, as well as simulated data generated from a normal population using R software, affirm the superior performance of the proposed estimators over existing ones such as the usual mean estimator and those proposed by Vishwakarma and Singh (2022), as evidenced by the higher percent relative efficiency and lower mean squared error. The evaluation of the percentage contribution of measurement error values confirms the impact of measurement errors on the properties of the estimators.

Multiple scales method for analyzing a forced damped rotational pendulum oscillator with gallows

This study reports the analytical solution for a generalized rotational pendulum system with gallows and periodic excited forces. The multiple scales method (MSM) is applied to solve the proposed problem. Several types of rotational pendulum oscillators are studied and talked about in detail. These include the forced damped rotating pendulum oscillator with gallows, the damped standard simple pendulum oscillator, and the damped rotating pendulum oscillator without gallows. The MSM first-order approximations for all the cases mentioned are derived in detail. The obtained results are illustrated with concrete numerical examples. The first-order MSM approximations are compared to the fourth-order Runge–Kutta (RK4) numerical approximations. Additionally, the maximum error is estimated for the first-order approximations obtained through the MSM, compared to the numerical approximations obtained by the RK4 method. Furthermore, we conducted a comparative analysis of the outcomes obtained by the used method (MSM) and He-MSM to ascertain their respective levels of precision. The proposed method can be applied to analyze many strong nonlinear oscillatory equations.

Hybrid class of robust type estimators for variance estimation using mean and variance of auxiliary variable

This study is completed for the estimation of unknown population variance for the variable of mean and variance of interest. To accomplish this task, a new generalized class of robust kind of variance estimators proposed utilizing known descriptives of auxiliary variable, for example, Mid-range, Hodges-Lehmann Mean, Tri-mean, deciles mean, coefficient of skewness, interquartile range, first quartile, coefficient of kurtosis, semi-interquartile average, inter decile range and Mean, etc. These conventional measures of auxiliary variable improve the accuracy of the suggested class under simple random sampling without replacement (SRSWOR) scheme. The properties such as the bias, mean square errors (MSE), and least MSE of the suggested class are derived up to first order of approximation. The superiority conditions of the developed class of estimators over existing estimators are also made out theoretically. Finally, numerical representation is also completed for the motivations behind the article. The usual variance estimator is considered as a benchmark for comparing all considered estimators in numerical illustration. The results have been indicated that the suggested class is performing better than the usual variance estimator and all other thoughts about existing estimators.

Development of improved estimators of finite population mean in simple random sampling with dual auxiliaries and its application to real world problems

In general, the incorporation of supplementary information reduces the Mean Square Error (MSE) and, consequently, enhances the precision of estimating a population parameter. This improvement relies on the appropriate application of a suitable function, with careful consideration. This study introduces two innovative families of estimators for the finite population mean, both of which exhibit superior performance in scenarios involving dual auxiliary information in simple random sampling. Expressions up to the first-order approximation, for bias, and Mean Square Error were derived, and the conditions under which these proposed families surpassed the existing estimators. Our evaluation involved the use of both real and simulated data to compute the Mean Square Error and Percent Relative Efficiency (PRE) of the estimators. A comparative analysis revealed that under the specified conditions, both proposed families yielded more precise results.

POWER MEDIAN-BASED ESTIMATORS OF FINITE POPULATION MEAN

In this paper, median based mean estimators for estimating finite population mean are proposed. The proposed estimators were obtained by transforming estimators in literature utilizing mean of auxiliary variable into median based estimators with the aim of obtaining estimators with higher efficiency. The mean square error of the proposed estimators was obtained up to the first order of approximation using Taylor series approach and the optimum values of the unknown of the estimators were obtained by means of partial derivative of the mean square error and equating to zero. A Numerical study was carried out to support the fact that the proposed estimators are more efficient as compared to the existing ones, as the proposed estimators have the least mean squared error at optimum values of the unknown constants and have higher percentage relative efficiency (PRE). This implies that the proposed estimators are more efficient than the traditional ones considered in the study.

An approximated exponentially tilted empirical likelihood estimator of moment condition models

. This study proposes an approximated estimator for moment condition models. Our estimator approximates the exponentially tilted empirical likelihood (ETEL) estimator in Schennach (2007) by using a second-order approximation of implied probabilities (IPs) for the exponential tilting (ET). The resulting approximated ETEL estimator generates positive IPs. As the nested optimization of ETEL and EL is avoided, it is computationally simple. It is second-order asymptotically equivalent to the EL estimator. In particular, it has the same O(n −1) bias as EL. It also has the same O(n −2) variance as EL. Like ET and ETEL estimators, it is n convergent under model misspecification, while the EL estimator may not be.

Inversion of fracture weakness parameters based on 3C vertical seismic profile data — Part I: Horizontal transversely isotropic media composed of a single fracture set in the isotropic background media

Natural fractures in oil and gas reservoirs are a crucial factor that cannot be ignored because they significantly influence the reservoir’s petrophysical properties and hydrocarbon development. A horizontal transversely isotropic (HTI) medium composed of a single fracture set in an isotropic background is a typical anisotropic medium. Meanwhile, the S-wave splitting is a sensitive response of such anisotropic media, resulting in the generation of fast and slow S waves. The normal and tangential fracture weaknesses are crucial parameters that characterize the anisotropy of fractured media. We develop an inversion method for fracture weakness based on 3C vertical seismic profiling (3CVSP) data. First, assuming weak anisotropy and an HTI medium containing a single fracture set, we derive a first-order linear approximation of the traveltimes of the converted fast and slow S waves (PS1 and PS2 waves) with respect to fracture weakness parameters in the phase velocity domain. By solving for the horizontal projection of the slowness vector, approximate equations of the traveltimes of the PS1 and PS2 waves are converted from the phase velocity domain to the group velocity domain. Furthermore, we devise an inversion workflow consisting of three primary steps: (1) preprocessing the VSP data to derive the traveltimes and azimuth of the HTI medium, (2) constructing a forward model with undetermined fracture weakness parameters, and (3) following the establishment of the objective function, conducting the inversion for the fracture weakness parameters. We demonstrate the reliability of the method through numerical examples and synthetic 3CVSP data. The inversion errors are primarily influenced by the azimuth angle, with minimal influence from the receiver depth. Furthermore, a collective set of inverted results derived from all geophones is more stable and accurate than individual geophones. The application to actual 3CVSP data further confirms the effectiveness of our approach.