Widely-used Formula Research Articles

Theoretical Analysis of Random Scattering Induced by Microlensing

Theoretical investigations into the deflection angle caused by microlenses offer a direct path to uncovering principles of the cosmological microlensing effect. This work specifically concentrates on the the probability density function (PDF) of the light deflection angle induced by microlenses. We have made several significant improvements to the widely used formula from Katz et al. First, we update the coefficient from 3.05 to 1.454, resulting in a better fit between the theoretical PDF and our simulation results. Second, we developed an elegant fitting formula for the PDF that can replace its integral representation within a certain accuracy, which is numerically divergent unless arbitrary upper limits are chosen. Third, to facilitate further theoretical work in this area, we have identified a more suitable Gaussian approximation for the fitting formula.

Evaluation of Four eGFR Calculating Formulae in Predicting Postoperative Acute Kidney Injury in Adult Patients Undergoing Open‐Heart Surgery with Cardiopulmonary Bypass

Objective There are four widely-used formulae to calculate the perioperative glomerular filtration rate (GFR) of patients undergoing cardiac surgery. We assessed the predictive values of these formulae in the occurrence of postoperative acute kidney injury (AKI). Methods Patients who underwent open-heart valvular surgery with cardiopulmonary bypass from January 2015 to October 2017 were enrolled in this retrospective study. Demographic data and perioperative serum creatinine levels were collected. The estimated GFR (eGFR) was calculated using four formulae: Cockcroft Gault (CG), body surface area (BSA) corrected CG, simplified modification of diet in renal disease (MDRD), and chronic kidney disease-epidemiology collaboration (CKD-EPI) formula. The incidence of post-operative AKI was calculated and the predictive capability of these formulae was evaluated. Results A total of 290 patients were included. 136 patients (46.90%) developed AKI after surgery. The eGFR in the AKI group was significantly lower than those in the non-AKI group at all investigated time points. In addition, the eGFR in the non-AKI group increased temporarily on the day of operation, then decreased on the following days, and returned to preoperative level about one week after surgery. However, in the AKI group, the eGFR decreased from the day of operation, which still did not recover to the preoperative level by the end of the first week after surgery. The eGFR calculated by the BSA-standardized CG formula had the highest AUC ROC curves of 0.699 and 0.774 before operation and on the day of operation, respectively. While eGFR calculated by CKD-EPI formula had the highest AUC ROC of 0.874 and 0.887 at the first and second postoperative day. Conclusions The eGFR formula is a powerful tool for perioperative renal function assessment. The BSA-corrected CG and CKD-EPI formula have better performance in predicting postoperative AKI after cardiopulmonary bypass than serum creatinine level and other formulae.

A new dimensionless number for thermoelectric generator performance

Thermoelectric generators (TEG) convert heat into electricity and offer a green option for renewable energy generation. The Thomson effect has been proven to impact the overall performance of TEGs negatively but is ignored in the widely-used formula for TEG maximum efficiency calculation. In this study, a new dimensionless number called the Ury number (Ur) is introduced to capture the influence of the Thomson effect on TEG power output and maximum efficiency. Ur is defined as the ratio between the Thomson coefficient and the Seebeck coefficient. It is found that as Ur increases both TEG output power and maximum efficiency decrease. Ur combined with the device figure of merit, ZT, can better dictate the overall performance of a TEG.

Physically based coefficient for streamflow estimation in ice-covered channels

Measuremental and methodological problems arising from the presence of ice cover on waterways have seriously hindered the progress in predicting timely and accurate streamflow, which is a prerequisite for the effective management of ice-covered waterways (e.g. streams, rivers, channels, and canals). Among various methods in the attempt to address this issue, the most attractive is the physically-based method, which does not require abundant, and often unobtainable, data. This paper develops a physically based K-factor method, in the form of a generalized Manning equation, for discharge estimations in ice-covered channels. The main advantage of this method is that it enables the prediction of streamflow in ice-affected seasons by a K-factor adjustment of the stage-discharge relationship developed in the open-water seasons. The results of parameter estimation show that the K-factor is only related to the boundary resistance ratio, which is parameterized by two exponents of the two-power law for describing the vertical streamwise velocity profile. Experimental measurements collected from the literature were used to test the applicability of the proposed formula, and to compare its relative accuracy with the Sabaneev and Larsen formulae. The comparative results indicate that the proposed method is more accurate in estimating streamflow of ice-covered channels than these two widely-used formulae based on traditional two-layer hypothesis.

A parabolic model of drag coefficient for storm surge simulation in the South China Sea

Drag coefficient (Cd) is an essential metric in the calculation of momentum exchange over the air-sea interface and thus has large impacts on the simulation or forecast of the upper ocean state associated with sea surface winds such as storm surges. Generally, Cd is a function of wind speed. However, the exact relationship between Cd and wind speed is still in dispute, and the widely-used formula that is a linear function of wind speed in an ocean model could lead to large bias at high wind speed. Here we establish a parabolic model of Cd based on storm surge observations and simulation in the South China Sea (SCS) through a number of tropical cyclone cases. Simulation of storm surges for independent Tropical cyclones (TCs) cases indicates that the new parabolic model of Cd outperforms traditional linear models.

On the geometric autocorrelation function of polycrystalline materials

Herein we derive an expression for direct determination of the geometric autocorrelation function W of a polycrystalline material from images of its grain boundary network (e.g., those delivered by orientation imaging microscopy). We also obtain an identity that relates the mean linear intercept function to a directional derivative of the geometric autocorrelation function. These formulae were applied to examine whether a widely-used formula for W, particularly in theoretical studies of attenuation of elastic waves in polycrystalline media, would be valid for the grain boundary structure of a commercial aluminum alloy. The conclusion was negative.

A general probability formula of the number of location areas' boundaries crossed by a mobile between two successive call arrivals

Mobility management is a challenging topic in mobile computing environment. Studying the situation of mobiles crossing the boundaries of location areas is significant for evaluating the costs and performances of various location management strategies. Hitherto, several formulae were derived to describe the probability of the number of location areas' boundaries crossed by a mobile. Some of them were widely used in analyzing the costs and performances of mobility management strategies. Utilizing the density evolution method of vector Markov processes, we propose a general probability formula of the number of location areas' boundaries crossed by a mobile between two successive calls. Fortunately, several widely-used formulae are special cases of the proposed formula.