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In previous investigations, the demarcation of capture zones within a specific research area predominantly relied on a singular method, leading to pronounced limitations and uncertainties. To address this challenge, an extensive field survey was conducted, focusing on the systematic classification of water sources in the Linfen City region. Building upon this classification, an intricate fusion of a hydrogeological analysis and formulaic methodology was employed. This integrated approach, coupled with independent numerical simulation methods, was applied to delineate recharge areas for both alluvial fan pore water in piedmont regions and exposed karst water in small- to medium-sized water sources. Simultaneously, precise spatial interpolation was carried out on water quality monitoring data from supply wells within the water source area for the year 2020. This meticulous analysis facilitated the determination of the spatial distribution characteristics of hydrochemical elements. To assess the water quality within the capture zone, the class III groundwater quality standards of China were employed as a pivotal tool for validating the results of the delineation of water source recharge areas. In the final analysis, a comparative study between the integrated coupling method and numerical simulation outcomes revealed the successful delineation of the boundaries for the water supply areas of Tumen and Caojiapo in Linfen City, covering areas of 5.5 km2 and 22.29 km2, respectively. Simultaneously, the combination of the three methods accurately outlined the boundary of the Hexi water supply area, encompassing an area of 2.5224 km2. These results vividly illustrate that the amalgamation of various methodologies proves more beneficial for the precise delineation of capture zones, particularly in diverse types and scales of groundwater sources. The synergy exhibited by these three methods underscores their collective efficacy, providing a more comprehensive and intuitive delineation of the recharge areas for small- to medium-sized water sources. Consequently, these findings significantly enhance the practical application value of the study and hold promise in making substantial contributions to local groundwater security and management initiatives. [ABSTRACT FROM AUTHOR]

The Dirac equation is solved using the Formula method. A non-relativistic mapping gives energy eigenvalues, in this limit, for a Hamiltonian including the Morse potential. The energies of the H2, LiH, HCl, CO, ScH, ScN, ScF and I2 molecules for both low and high-lying states are computed. Our results are found to be consistent with the literature. Also, new high-lying states for the Morse potential are also reported in the present study. The results of this study indicate that for the energy Eigen values of the H2 molecule, the deviation from the literature becomes more pronounced for the states (n, l) = (5, 20) and (6, 20).

We obtain solutions of Schrödinger equation for the modified Mobius square plus Eckart (MMPSE) potential via the formula method. Numerical results are reported. In addition, the partition function Z and other thermodynamic properties such as vibrational free energy, F, vibrational internal energy, U, vibrational entropy, S, and vibrational specific heat, C are presented. We also discuss special cases of this potential. Our result is consistent with previous studies in the literature.

The study presents approximate analytical solutions of the Schrödinger equation with a newly proposed potential model called Inversely Quadratic Hellmann-Kratzer potential (IQHKP). This potential is a superposition of Inversely Quadratic Hellman potential and Kratzer potential. The energy eigenvalues and corresponding wavefunction are calculated via the formula method. We applied our results to evaluate thermodynamic functions such as vibrational free energy, F, vibrational internal energy, U, vibrational entropy, S, and vibrational specific heat, C. We also reported special cases of importance.

By using Formula method for bound state problems, we have obtained the approximate analytical solution of the radial Schrödinger equation for Varshni potential model. The energy eigenvalues and the corresponding eigenfunctions are calculated in closed forms. In the process, eigen values for a = b =1, a = −1, b = −2, and b = −1, a = −2 for 2p, 3p, 3d, 4p and 4f states with three different values of the potential range were generated using the Varshni potential. Special cases of the potential are equally studied. Some numerical results have been presented which showed a good agreement with previous results obtained from another method. Keywords: Varshni potential, Schrödinger equation, Formula method, Bound state

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By using Formula method for bound state problems, we have obtained the approximate analytical solution of the radial Schrödinger equation for Varshni potential model. The energy eigenvalues and the corresponding eigenfunctions are calculated in closed forms. In the process, eigen values for a = b =1, a = −1, b = −2, and b = −1, a = −2 for 2p, 3p, 3d, 4p and 4f states with three different values of the potential range were generated using the Varshni potential. Special cases of the potential are equally studied. Some numerical results have been presented which showed a good agreement with previous results obtained from another method. [ABSTRACT FROM AUTHOR]

Background: Central venous catheters (CVCs) are inserted in many critically ill patients, but there is no gold standard in estimating their approximate depth of insertion. Many techniques have been described in literature. In this study, we compare the topographic method with the standard formula technique. Materials and Methods: 260 patients, in whom central venous catheterization was warranted, were randomly assigned to either topographic method or formula method (130 in each group). The position of the CVC tip in relation to carina was measured on a postprocedure chest X-ray. The primary endpoint was the need for catheter repositioning. Results: The majority of the CVCs tips positioned by the formula method were situated below the carina, and 68% of these catheters required repositioning after obtaining postprocedure chest X-ray (P < 0.001). Conclusion: The topographic method is superior to formula approach in estimating the depth of insertion of right internal jugular CVCs.

The formation of natural gas hydrate is harmful to the exploitation and transportation of natural gas, so it is important to accurately predict the formation conditions of gas hydrate. At present, three prediction methods named thermodynamics model calculation, formula method and image method are mainly included. The formula method has got much attention with the character of easy calculation which is applied conveniently in engineering. In view of this, eight calculation equations are presented. Through the known pressure calculate temperature and the known temperature calculate the pressure, the calculated values of hydrate formation conditions are compared with the experimental values for the sweet natural gas, sour gas with the high component of CO2 or H2S. When the temperature of hydrate formation conditions is calculated by the pressure, for the sweet natural gas, the minimum average relative deviation is got by Hammer and is recommended to calculate hydrate formation conditions. For sour gas with the high component of CO2, Bahadori method is recommended to calculate hydrate formation conditions. Zahedi I method is recommended to calculate sour gas with the high component of H2S. When the pressure of hydrate formation conditions is calculated by the temperature, it is found that small average relative deviation is got only by Towerl method, Bahadori method and Hammer method for the sweet natural gas. Neither of the methods is suitable for sour gas with the high component of CO2 or H2S. [ABSTRACT FROM AUTHOR]

Background and Aims: Subclavian central venous catheterisation (CVC) is employed in critically ill patients requiring long-term central venous access. There is no gold standard for estimating their depth of insertion. In this study, we compared the landmark topographic method with the formula technique for estimating depth of insertion of right subclavian CVCs. Methods: Two hundred and sixty patients admitted to Intensive Care Unit requiring subclavian CVC were randomly assigned to either topographic method or formula method (130 in each group). Catheter tip position in relation to the carina was measured on a post-procedure chest X-ray. The primary endpoint was the need for catheter repositioning. Mann–Whitney test and Chi-square test was performed for statistical analysis using SPSS for windows version 18.0 (Armonk, NY: IBM Corp). Results: Nearly, half the catheters positioned by both the methods were situated >1 cm below the carina and required repositioning. Conclusion: Both the techniques were not effective in estimating the approximate depth of insertion of right subclavian CVCs.

Background: Central venous catheters (CVCs) are inserted in many critically ill patients, but there is no gold standard in estimating their approximate depth of insertion. Many techniques have been described in literature. In this study, we compare the topographic method with the standard formula technique. Materials and Methods: 260 patients, in whom central venous catheterization was warranted, were randomly assigned to either topographic method or formula method (130 in each group). The position of the CVC tip in relation to carina was measured on a postprocedure chest X-ray. The primary endpoint was the need for catheter repositioning. Results: The majority of the CVCs tips positioned by the formula method were situated below the carina, and 68% of these catheters required repositioning after obtaining postprocedure chest X-ray (P < 0.001). Conclusion: The topographic method is superior to formula approach in estimating the depth of insertion of right internal jugular CVCs. [ABSTRACT FROM AUTHOR]

We derive the relativistic energy spectrum for the modified Dirac equation by adding a harmonic oscillator potential where the coordinates and momenta are assumed to obey the commutation relation . In the nonrelativistic (NR) limit, our results are in agreement with the ones obtained previously. Furthermore, the extension to the construction of creation and annihilation operators for the harmonic oscillators with minimal length uncertainty relation is presented. Finally, we show that the commutation relation of the algebra is satisfied by the operators and . [ABSTRACT FROM AUTHOR]

Spectroscopic techniques are very essential tools in studying electronic structures, spectroscopic constants and energetic properties of diatomic molecules. These techniques are also required for parametrization of new method based on theoretical analysis and computational calculations. In this research, we apply the proper quantization rule in spectroscopic study of some diatomic molecules by solving the Schrödinger equation with two solvable quantum molecular potentials; Tietz-Wei and shifted Deng-Fan potential models for their approximate nonrelativistic energy states via an appropriate approximation to the centrifugal term. We show that the energy levels can be determined from its ground state energy. The beauty and simplicity of the method applied in this study is that, it can be applied to any exactly as well as approximately solvable models. The validity and accuracy of the method is tested with previous techniques via numerical computation for $$\hbox {H}_2$$ and CO diatomic molecules. Our result also include energy spectrum of five different electronic states of NO and two different electronic state of ICl. [ABSTRACT FROM AUTHOR]

Abstract: Many methods are considered for calculating long-period fiber gratings (LPFGs) transmission spectra such as integral method, formula method and transfer matrix method. In this paper, these methods are analyzed based on the coupled-mode theory and their characteristics and the conditions of application are pointed out. Exact solution for multi-mode transmission spectrum can be obtained directly by integral method in a wide wavelength range, however, the computation process of this method is complicated. The formula method is convenient for calculating and expressing, but it can only solve single mode transmission spectrum directly in a narrow wavelength range. The transfer matrix method is suitable for calculating transmission spectra of non-uniform fiber gratings, it has higher calculation accuracy and faster speed. Through the analysis and comparing with the three methods, we will provide a best method with high precision and simple calculation for long-period fiber gratings transmission spectra in theory, which is important for the research and application of LPFGs. [Copyright &y& Elsevier]

Background and Aims: Subclavian central venous catheterisation (CVC) is employed in critically ill patients requiring long-term central venous access. There is no gold standard for estimating their depth of insertion. In this study, we compared the landmark topographic method with the formula technique for estimating depth of insertion of right subclavian CVCs. Methods: Two hundred and sixty patients admitted to Intensive Care Unit requiring subclavian CVC were randomly assigned to either topographic method or formula method (130 in each group). Catheter tip position in relation to the carina was measured on a post-procedure chest X-ray. The primary endpoint was the need for catheter repositioning. Mann-Whitney test and Chi-square test was performed for statistical analysis using SPSS for windows version 18.0 (Armonk, NY: IBM Corp). Results: Nearly, half the catheters positioned by both the methods were situated >1 cm below the carina and required repositioning. Conclusion: Both the techniques were not effective in estimating the approximate depth of insertion of right subclavian CVCs. [ABSTRACT FROM AUTHOR]

We obtain solutions of Schrödinger equation for the modified Mobius square plus Eckart (MMPSE) potential via the formula method. Numerical results are reported. In addition, the partition function Z and other thermodynamic properties such as vibrational free energy, F , vibrational internal energy, U , vibrational entropy, S , and vibrational specific heat, C are presented. We also discuss special cases of this potential. Our result is consistent with previous studies in the literature., Competing Interests: The authors declare no conflict of interest., (© 2022 The Author(s).)

Derives a formulation for spatial stress tensors and spatial material tensors of hyperelastic material. Looks at a class of materials with the strain energy function decomposed into a volumetric and a deviatoric part. Separate terms formulate the strain energy with respect to the invariants of the left Cauchy-Green tensor. Stress and material tensors, which play a crucial role in the solution process of the finite element formulation, are derived solely in the current configuration. Applies the described framework to several different constitutive models based on phenomenologically and physically motivated material descriptions. Proposes a formulation for the finite element implementation of van der Waals material. Compares numerical results with experimental investigations given in the literature. For three-dimensional finite element computations standard elements and mixed elements, based on a three-field variational principle where displacements, the hydrostatic pressure and the dilatations are independent variables, are used. [ABSTRACT FROM AUTHOR]