Coefficient Functions Research Articles

A Method for Efficiently Predicting the Radial Distribution Function and Osmotic Coefficients of Aqueous Electrolyte Solutions.

The prediction of the structural and thermodynamic properties of electrolyte solutions is critical for a huge range of practical situations where these solutions play a vital role. Theoretical models, such as the continuum solvent model, attempt to explain the behavior of solutions using a coarse-grained description of the interactions of species in the solution, whereas molecular simulations aim to directly compute the behavior of the solution, including the interactions between all ions and molecules in the system. Both methods have limitations: theoretical models are generally less accurate because they rely on assumptions, while molecular simulations require significant computational resources, particularly if higher accuracy is desired. To address these issues, we propose an affordable and effective method that combines the advantages of the modified Poisson-Boltzmann equation (MPBE) with classical molecular dynamics (MD) simulations to predict the radial distribution functions and thermodynamic properties of electrolyte solutions. We demonstrate a method of using the MPBE to compute the short-range potential of mean force (PMF) from the radial distribution functions (RDFs) and vice versa. Furthermore, we provide insights into the relationship between the RDFs and the short-range PMF based on the MPBE. Our analysis reveals that the effective short-range PMFs can be approximately calculated using low concentration simulations but the short-range PMFs are slightly concentration-dependent in simulations at higher concentrations. Additionally, we demonstrate that for concentrated solutions, osmotic coefficients can be calculated in agreement with experiment using a virial approach. This is based on the effective short-range PMFs and RDFs obtained from the MPBE method. Our proposed MPBE can therefore accelerate the calculation of the structural and thermodynamic properties of electrolyte solutions.

Generation of cubic Hermite spline-based trochoidal milling toolpath by introducing a coefficient factor in machining curved slots

Trochoidal milling has become popular in high-speed milling due to its ability to reduce cutting force, enhance thermal dissipation, and prolong tool life. Among other toolpath patterns, the cubic Hermite spline offers significant advantages in generating trochoidal milling toolpath in machining complex and curved slots. However, determining those two different relation coefficients in the polynomial function of cubic Hermite spline remains complicated and empirical. This paper introduces a coefficient factor to simplify the determination process, which only requires the position and tangent vector parameters of each endpoint pair. The coefficient factor combines an instantaneous in-process slot width (IISW)-related width factor and a tangent vector direction-related direction factor. Two complex-curved slots are used to validate the performance of the proposed method. The results show that this method is straightforward and effective in generating trochoidal milling toolpaths for machining complex-curved slots while matching the slot geometry. Additionally, compared with a direct 5-axis trochoidal milling (one-step slotting strategy) in trochoidal milling of 3D slots on both serial and hybrid machining tools, a two-step slotting strategy (3-axis trochoidal milling and a successive 5-axis perpetual milling) is preferable in improving machining efficiency.

DFT study of optoelectronic and thermoelectric properties of pure and doped double perovskite Cs2SnI6 for solar cell applications

The global need of energy is raising day by day. To meet this demand double perovskites (DPs) can play a vital role for energy conversion devices. In this letter, we formulated and examined lead-free defect perovskites using a combination of tellurium and tin, denoted as Cs2Sn1-xTexI6 (0, 0.25), utilizing density functional theory which include structural, optical and thermal behavior. Both the optimized volume and lattice parameter exhibit a linear increase with the Te content in Cs2Sn1-xTexI6 (CsSnTeI). The negative value of formation energy for both pure and doped materials along with their Poisson and Pugh ratio confirm the stability of these materials. The band structure analysis reveals its semiconducting behavior, with the band gap expanding proportionally to the increased Te concentration with band gap value of pure (1.20 eV) and doped 91.43 eV) double perovskites. Optical characteristics, including the dielectric function and absorption coefficient, were also analyzed. To expose their potential use of Cs2Sn1-xTexI6 (0, 0.25) in thermoelectric devices, temperature dependent thermoelectric features are investigated reveal that suitability of these materials for energy conversion purposes.

Investigation of Taylor bubble behavior in upward and downward vertical and inclined pipe flows

The Taylor bubble behavior in two-phase flow is critical for various engineering applications, including momentum, heat, and mass transfer efficiencies. Therefore, understanding the dynamics of Taylor bubbles is crucial for the optimal design, operation, and safety of reactors and pipelines. In slug flow in pipes, the successive translation of liquid slugs and Taylor bubbles is characterized by the Taylor bubble velocity, which is a function of the flow distribution coefficient (C0). This study aims to compile a large database of Taylor bubble velocity and flow distribution coefficient in upward, horizontal, and downward flows, evaluate existing models, and propose a unified C0 model. As a result, it is found that the physics governing the flow distribution coefficient in downward flow is drastically different from that in upward flow, and it goes through two flow transitions, which are governed by the pipe inclination angle. This behavior is modeled by the proposed model, which predicts the Taylor bubble flow distribution coefficient through both transitions and captures its physical behavior in the downward flow. A validation study of the proposed model revealed that the proposed model outperforms the existing models as it has an average absolute percent error and standard deviation of 4.2% and 6.2%, respectively.

UNIFAC and Wilson models to correlate solubility of pharmaceutical compounds in supercritical carbon dioxide

New thermodynamic correlations were proposed for the solubility of pharmaceutical compounds in supercritical carbon dioxide based on the solid-liquid equilibrium model. The activity coefficient models proposed in the study were based on Universal quasi-chemical Functional Group Activity Coefficient and Wilson models. The pharmaceutical drug compounds used in this study belong to anticancer, antibiotics, anti-androgenic, non-steroidal anti-inflammatory drugs, anti-prostatic tumour drugs and antiemetic drugs. The correlating ability of the proposed models was discussed in terms of various statistical parameters. Further, the proposed model results were compared with six existing solid-fluid equilibrium model results. It was found that the newly proposed models were more promising than those of existing solid-fluid equilibrium models. The correlating ability of the proposed models was discussed in terms of various statistical parameters.

Augmenting machine learning of Grad–Shafranov equilibrium reconstruction with Green's functions

This work presents a method for predicting plasma equilibria in tokamak fusion experiments and reactors. The approach involves representing the plasma current as a linear combination of basis functions using principal component analysis of plasma toroidal current densities (Jt) from the EFIT-AI equilibrium database. Then utilizing EFIT's Green's function tables, basis functions are created for the poloidal flux (ψ) and diagnostics generated from the toroidal current (Jt). Similar to the idea of a physics-informed neural network (NN), this physically enforces consistency between ψ, Jt, and the synthetic diagnostics. First, the predictive capability of a least squares technique to minimize the error on the synthetic diagnostics is employed. The results show that the method achieves high accuracy in predicting ψ and moderate accuracy in predicting Jt with median R2 = 0.9993 and R2 = 0.978, respectively. A comprehensive NN using a network architecture search is also employed to predict the coefficients of the basis functions. The NN demonstrates significantly better performance compared to the least squares method with median R2 = 0.9997 and 0.9916 for Jt and ψ, respectively. The robustness of the method is evaluated by handling missing or incorrect data through the least squares filling of missing data, which shows that the NN prediction remains strong even with a reduced number of diagnostics. Additionally, the method is tested on plasmas outside of the training range showing reasonable results.

Impact of pleating on the filtration performance of fibrous air filters

Pleated fibrous air filters are widely used for aerosol filtration but the impact of pleating on its filtration performance is still lacking detailed analysis. Pleating results in changes in thickness of the filter media (indentation or bulge zone) and it is generally difficult to simulate the particle filtration process of a pleat due to large difference in scale between that of the pleat and fiber. To address this problem, a dual-scale simulation method is developed in this study by combining the fiber-scale and pleat-scale models. This is achieved by incorporating the simulation results of fiber-scale models into the pleat-scale models, using the permeability coefficient and particle capture function of the porous media in the pleat model. From the fiber-scale model, it is found that the bulge zone is beneficial to improving quality factor as it could lead to a lower pressure drop. However, for the pleat-scale model, the indentation zone has a lower pressure drop than the bulge zone, resulting in a higher quality factor. This contradicts the findings of the fiber-scale model. For real applications, it has been found that pleating with indentation is more beneficial for filter performance compared to pleating with bulges. These findings help us better understand the pleating effect on filter performance and provide guidance on optimal pleating approaches.

Computational Prediction of Structural, Optoelectronic, Thermodynamic, and Thermoelectric Response of the Cubic Perovskite RbTmCl3 via DFT‐mBJ + SOC Studies

Perovskite halides, owing to their environmental stability, non‐toxicity, and remarkable efficiency, are emerging as potential candidates for photovoltaic, solar cell, and thermodynamic applications. The electronic, optical, thermoelectric, and thermodynamic properties of cubic perovskite RbTmCl3 are studied using density functional theory (DFT). The electronic, optical, and thermoelectric properties are calculated both with and without spin‐orbit coupling (SOC) using the Tran and Blaha functional in the structure of the modified Becke Johnson (mBJ) exchange potential, while structural and mechanical properties are assessed using the exchange‐correlation functional calculated using the Perdew Burke Ernzerhof Generalized Gradient Approximation (PBE‐GGA). The negative formation energy (−592.39 KJ mol−1) and tolerance factor (1.17) for structural stability and current their existences in the cubic phase are found. Evaluation of the obtained data with and without SOC shows that the SOC effect causes the Tm‐d states to be shifted toward the level of Fermi, thereby decreasing the energy band gaps from 1.42 to 1.32 eV. Nevertheless, only the shift of the third variable peak to lower energies indicates the impact of SOC on optical properties. The effectiveness of RbTmCl3 in optical devices operating in the visible and ultraviolet regions is demonstrated by the exceptional absorption of light in these ranges. Using TB‐mBJ + SOC functional, the electronic band structure research reveals a direct semiconducting band gap of 1.32 eV in comparison to earlier calculations like LDA, PBE‐GGA, and TB‐mBJ. The absorption spectrum, reflectivity, extinction coefficient, refractive index, and dielectric function are displayed in addition to the electrical properties. Additionally, the quasi‐harmonic Debye model, which accounts for lattice vibrations, was used to study the corresponding volume, heat capacity, expansion of the heat coefficient, and Debye temperature of the RbTmCl3 crystal. We have calculated the thermoelectric parameters such as the Seebeck coefficient, thermal conductivity, electrical conductivity, and power factor as a function of temperature (100–900 K).

Selective multiple kernel fuzzy clustering with locality preserved ensemble

Multiple kernel fuzzy clustering (MKFC) has demonstrated promising performance in capturing the non-linear relationships within data. However, its effectiveness relies heavily on the appropriate selection of the fuzzification coefficient and kernel functions. To address this challenge, this paper proposes a novel clustering ensemble approach for improving the robustness and accuracy of the MKFC algorithm. The proposed method employs a multi-objective evolutionary optimization approach to heuristically select the optimal fuzzification and kernel coefficients. By employing the selected coefficients, the MKFC algorithm generates a diverse set of accurate candidate clustering results. Subsequently, a locality preserved ensemble mechanism is introduced to derive the final partition matrix, ensuring that all candidate clusterings within the Pareto non-dominated set contribute to the final consensus matrix. Moreover, this mechanism incorporates the knowledge about the locality of the dataset via a graph regularization term, thereby further enhancing the clustering performance. Comprehensive experiments conducted on widely adopted benchmark datasets demonstrate the superiority of the proposed method over the state-of-the-art approaches.

Location method of ill-conditioned microseismic source and its engineering application

Microseismic event location is one of the core parameters in microseismic monitoring, and the accuracy of localization will directly affect the effectiveness of engineering applications. However, limited by spatial factors, the geometry of the sensor installation will be close to linear, which makes the localization equation suffer from the pathological problem, and the localization accuracy is greatly reduced. To address this problem, the reasons for the pathological problem are analyzed from the perspective of the objective function residuals and coefficient matrix. The pathological problem is caused by the combined effect of the poorer sensor array and data errors, and its residual isosurface shows a conical distribution, and as the residual value decreases, the apex of the isosurface gradually extends to the far side, and the localization results do not converge. For this reason, an improved regularized Newton downhill localization algorithm is proposed. In this method, firstly, the Newtonian downhill method is improved so that the magnitudes of the seismic source parameters are the same, and the condition number of the coefficient matrix is reduced; then, the L-curve method is used to calculate the regularization factor for the pathological equations, and the coefficient matrix is improved; finally, the pathological equations are regularized, and the seismic source coordinates are obtained by the improved Newtonian downhill method. The results of engineering applications show that compared with the traditional algorithm based on automatic of P-arrival picking, the number of effective microseismic events calculated by the proposed localization algorithm is increased by 194.7 %, and the localization accuracy is substantially improved. The proposed algorithm reduces the problem of low accuracy of S-arrival picking and allows localization using only P-wave arrival. The method reduces the quality requirements of the data and significantly improves the utilization of microseismic events and positioning accuracy.

Strain-incorporated structural, electronic, and dielectric properties of cubic FAPbCl3 perovskites together with the spin-orbit coupling

The organic-inorganic perovskite materials have garnered significant interest in solar technology because of their impressive structural, electrical, and optical characteristics. This study investigated the effects of hydrostatic strains (tensile and compressive) on the structural, electronic, and optical properties and the influence of spin-orbit coupling (SOC) on the electronic properties of formamidinium lead chloride cubic (FAPbCl3) perovskite structures employing first-principles density-functional theory (DFT). FAPbCl3 without SOC showed a semiconductor perovskite structure with a direct bandgap of 2.767 eV. Compressive strain (-8–0 %) reduced bandgaps, but tensile strain only increased them by +2 %. The SOC significantly reduced the bandgap by ∼1 eV, shifting it from direct to indirect. The analysis showed that neither of compressive or tensile strains can affect the system's semiconductiveness nature under the studied strain range. The dielectric function, loss spectrum, and absorption coefficient peaks of FAPbCl3 perovskites display a blueshift under compressive strains and a redshift under tensile strains. This material can be used in solar cells, lasers, and light detectors because of its promising electrical and optical properties.

Mathematical modeling of insulating coating of thermal conductivity including body’s own radiation and non-local spatial effects

Abstract In this paper thermal conductivity analysis of insulating coating is discussed. The general partial differential thermal conductivity equation is used with consideration to nonlocal boundary effects. We relying on the Eringen’s approach involves heat-flow vector transformation using nonlocal influence function and weight coefficients. Also presented model includes own surface radiation of the coating.

Hankel and Toeplitz Determinants of Logarithmic Coefficients of Inverse Functions for the Subclass of Starlike Functions with Respect to Symmetric Conjugate Points

This paper focuses on finding the upper bounds of the second Hankel and Toeplitz determinants, whose entries are logarithmic coefficients of inverse functions for a new subclass of starlike functions with respect to symmetric conjugate points associated with the exponential function defined by subordination. Results on initial Taylor coefficients and logarithmic coefficients of inverse functions for a new subclass are also presented. This study may inspire others to focus further to the coefficient functional problems associated with the inverse functions of various classes of univalent functions.

Improving decryption quality of optical chaos communication using neural networks.

Optical chaos communication is a promising secure transmission technique because of the advantages of high speed and compatibility with existing fiber-optic systems. The deterioration of chaotic synchronization quality caused by fiber optic transmission impairments affects the quality of recovery of information, especially high-order modulated signals. Here, we demonstrate that the use of a convolutional neural network (CNN) with a bidirectional long short-term memory (LSTM) layer can reduce the decryption BER in an optical chaos communication system based on common-signal-induced semiconductor laser synchronization. The performance of a neural network is investigated as a function of network parameters and chaos synchronization coefficient. Experimental results show that the BER of 16-ary quadrature-amplitude-modulation (16QAM) signal after 100-km fiber transmission is decreased from 3.05 × 10-2 to below the soft-decision forward-error-correction (SD-FEC) threshold of 2.0 × 10-2.

3D Chaotic Nonlinear Dynamic Population-Growing Mathematical System Modeling with Multiple Controllers

Modeling, stabilization, and identification processes are significant stages in the process of developing knowledge about chaotic dynamical systems which entail the effective prediction depending on the degree of uncertainty toleration in the forecast, accuracy of the current state to be measured as well as a time scale resting on the dynamics of the system. Control of under-activated dynamical systems has been considered substantially, and it is for periods and is currently developing in various domains such as biology, data analysis, computing systems, and so forth. Dynamic systems of growing population signifies a model describing the way a population evolves over time during which population goes through major life events, split into discrete time periods. The size of the population at a given time period is determined by the rate of growth as well as other related factors. Most progress has been made in model-based control theory, which has drawbacks when the system under consideration is exceedingly complicated, and no model can be constructed. Accordingly, a 3D-discrete and dynamic human population growth system with many controllers is proposed by examining the stability and symmetry of controller system clarifications. The symmetric stability control results are presented by considering a special parametric dynamic system in its coefficients besides suggesting periodic functional coefficients in terms of sin and cos functions. The controllers have the ability to reduce population growth rate unpredictability or enhance system stability under various external conditions. The unique and very effective strategies in relevant domains could provide a deeper understanding of their impact as well as the theoretical or technological innovations thereof. These controllers are capable of reducing population growth rate unpredictability or improving system stability under various external conditions, and applicable strategies in the relevant domains can provide profound comprehension over the impact along with the theoretical as well as technological advancements.

Theoretical study of optoelectronic, elastic and mechanical properties of gallium modified barium titanate (Ba1-xGaxTiO3) perovskite ceramics by DFT

In this report, we study the gallium modified Barium titanate (Ga-BTO) Ba1-xGaxTiO3 with (x = 50 %) perovskite ceramics, which have not been synthesized experimentally to date. The density functional theory-based full potential linear augmented plane wave has been employed to determine the optoelectronic, elastic, and mechanical properties of pure and modified BaTiO3. Completely relaxed 2 × 2× 2 of primitive five atoms’ cells has been used for calculations. A large cut-off energy of 120 Ry has been used to eliminate the basics of plane wave-indicating wave functions. The influence of substitution of Ga on electronic structure and optical parameters such as reflectivity, refractivity, dielectric function, optical conductivity, extinction, and absorption coefficients are analyzed and elucidated with TDOS and PDOS. The Ga modified BTO manifests a forbidden energy gap of 1.84 eV. The static dielectric constant Ɛ1(o) has values of 8.8 and 100 for pure and modified compounds respectively. Ga dopant improved the polarizing power of BTO and for the doped BTO imaginary part, Ɛ2(ω)showed a prominent peak located at 3.9 eV. The optical properties show that perovskite compound substituted with Ga is a good absorber in the ultraviolet region and the modified material has a small value of reflectivity and static refractive index 12.2. The general profiles of optical spectra of modified BaTiO3 display highly desirable features like the prominent peaks of optical conductivity obtained at 4.2 and 5.8 eV for modified BTO that make it a promising candidate for optoelectronic and spintronics. Ga-modified BTO has higher ductility, modulus (169.96 GPa), and anisotropy (2.949) than pure BTO found by analyzing their elastic and mechanical properties.

Effect of electroacupuncture of "Zusanli" (ST36) combined with capeOX on apoptosis and ferroptosis in nude mice with colorectal cancer.

To investigate the impact of combined treatment of colorectal cancer (CRC) with electroacupuncture (EA) and capeOX (combined administration of fluorouracil, oxaliplatin and capecitabine) on the tumor volume, weight, spleen coefficient, apoptosis and ferroptosis of tumor tissue, and liver and kidney functions in nude mice with CRC, so as to explore its mechanisms underlying inhibiting CRC and alleviating toxic reactions of capeOX. Female Balb/c nude mice were randomly assigned to 3 groups：model, capeOX, and EA+capeOX, with 8 nude mice in each group. The CRC model was established by subcutaneous injection of colon cancer cells at the right inguinal region. Nude mice of the capeOX group received intraperitoneal injection of oxaliplatin for 1 day and gavage of capecitabine from day 2 to day 7. EA (1 mA, 2 Hz/100 Hz) was applied to bilateral "Zusanli" (ST36) for 20 min, once daily for 7 days. During the interven-tion, the tumor volume and weight were measured every day, and at the end of intervention, the weight of the tumor tissue and spleen were measured, with tumor volume difference and spleen coefficient calculated. The proportion of apoptotic cells was measured by flow cytometry, and the contents of serum malondialdehyde (MDA), alanine aninotransferase (ALT), aspartate aminotransferase (AST), blood urea nitrogen (BUN), and creatinine (Cr) were detected using ELISA. The expression level of glutathione peroxidase 4 (GPX4, a key regulator for ferroptosis) protein of the tumor tissue was determined using Western blot. Compared to the model group, both the capeOX group and EA+capeOX group showed a decrease in the tumor volume (on day 3 and 4 in the capeOX group, and from day 2 to 7 in the EA+capeOX group) and body weight (P<0.05, on day 3 to 7 in the EA+capeOX group and on day 2 to 7 in the capeOX group), being evidently lower in the tumor volume on day 7 in the EA+capeOX than in the capeOX group (P<0.05), and evidently higher in the body weight on day 6 and 7 in the EA+capeOX group than in the capeOX group (P<0.05). In comparison with the model group, the tumor volume difference, tumor weight and spleen coefficient in both capeOX and EA+capeOX groups were significantly decreased (P<0.05), and MDA content in EA+capeOX group was significantly decreased (P<0.05), while the contents of ALT, BUN and Cr in the capeOX group, the proportion of apoptotic cells in both capeOX and EA+capeOX groups, and the GPX4 expression level in the EA+capeOX group were all significantly increased (P<0.05). The tumor volume difference, tumor weight, and contents of MDA, ALT, AST, BUN and Cr in the EA+capeOX group were markedly lower than in the capeOX group (P<0.05), while the spleen coefficient, proportion of apoptotic cells and GPX4 expression level in the EA+capeOX group were markedly higher than those in the capeOX group (P<0.05). EA of ST36 can enhance the effect of capeOX in inhibiting colorectal cancer growth in nude mice with CRC, which may be related with its functions in promoting tumor cell apoptosis, inhibiting ferroptosis, and modulating immune tolerance. In addition, EA can lower the side effects of capeOX in hematopoietic and immune, liver, and kidney functions.

Sharp Coefficient Estimates for Analytic Functions Associated with Lemniscate of Bernoulli

The main purpose of this work is to study the third Hankel determinant for classes of Bernoulli lemniscate-related functions by introducing new subclasses of star-like functions represented by SLλ* and RLλ. In many geometric and physical applications of complex analysis, estimating sharp bounds for problems involving the coefficients of univalent functions is very important because these coefficients describe the fundamental properties of conformal maps. In the present study, we defined sharp bounds for function-coefficient problems belonging to the family of SLλ* and RLλ. Most of the computed bounds are sharp. This study will encourage further research on the sharp bounds of analytical functions related to new image domains.

Density-Based Isogeometric Topology Optimization of Shell Structures

Shell structures with high stiffness-to-weight ratios are desirable in various engineering applications. Topology optimization serves as a popular and effective tool for generating optimal shell structures. The solid isotropic material with penalization (SIMP) method is often chosen because of its simplicity and convenience. However, SIMP method is typically integrated with conventional Finite Element Analysis (FEA) which has limitations in computational accuracy. Achieving high accuracy with FEA necessitates a substantial number of elements, leading to computational burdens. In addition, the discrete representation of the material distribution function may result in rough boundaries. Owing to these limitations, this paper proposes an Isogeometric Analysis (IGA) based SIMP method for optimizing the topology of shell structures based on Reissner–Mindlin theory. This method uses Non-Uniform Rational B-Splines (NURBS) to represent both the shell structure and the material distribution function with the same basis functions, allowing for higher accuracy and smoother boundaries. The optimization model takes compliance as the objective function with a volume fraction constraint and the coefficients of the density function as design variables, resulting in an optimized shell structure defined by the material distribution function. To obtain fairing boundaries of the holes in the optimized shell structure, further process is conducted by fitting the boundaries with fair B-spline curves automatically. Furthermore, the proposed IGA-SIMP framework is applied to generate porous shell structures by imposing different local volume fraction constraints. Numerical examples are provided to demonstrate the feasibility and efficiency of the IGA-SIMP method, showing that it outperforms the FEA-SIMP method and produces smoother boundaries.

Fourier representation of geometrical imperfection for probabilistic buckling analysis

This research studies the first part of the failure of a compression member structure due to buckling. This unstable equilibrium collapse, exposes brittle failure which occurs suddenly and therefore should be avoided wherever possible. Unavoidable geometric imperfections due to structural fabrication, will weaken the structure against buckling. The behavior of bar under compression will be closely examined by taking a set of geometric imperfection data synthesized from previously available from the measurement of conical shells. Therefore, the two-dimensional surface imperfection is converted into several one-dimensional imperfection with some probability properties. In order to obtain a comparison tool for different type of imperfections, Fourier analysis is used to convert the imperfection into coefficients of trigonometric function. By examining the coefficients, geometric imperfection patterns introduced by a certain fabrication process are able to be identified. The study successfully demonstrates the applicability of Fourier analysis in representing inherent geometric imperfections as an initial step for conducting probabilistic buckling analysis. Fourier analysis has shown its capability to simultaneously characterize imperfections in two crucial parameters - the magnitude and configuration of the imperfection.