Error Analysis Research Articles

Spectral approximation and error analysis for the transmission eigenvalue problem with an isotropic inhomogeneous medium

In this paper, we propose an effective Legendre-Fourier spectral method for the transmission eigenvalue problem in polar geometry with an isotropic inhomogeneous medium. The basic idea of this methodology is to rewrite the initial problem into its equivalent form by using polar coordinates and some specially designed polar conditions. A variational method and its discrete version (i.e., Legendre-Fourier spectral method) are then presented within a class of weighted Sobolev spaces. With the help of the spectral theory of compact operators and the approximation properties of some specially designed projections in non-uniformly weighted Sobolev spaces, error estimates with spectral accuracy of the Legendre-Fourier spectral method for both the eigenvalue and eigenfunction approximations are established. Numerical experiments are presented to confirm the theoretical findings and the efficiency of our algorithm.

Comprehensive Experimental Database and Model Fitting for Electric Arc Behavior in Aircraft Environments

The More Electric Aircraft (MEA) paradigm advocates weight reduction by transitioning to electrical components, resulting in increased system voltage and higher arc fault risk. This paper presents a methodology for creating a database of nearly one thousand tests simulating parallel arc faults, using Andrea's arc model for accurate representation. Objectives include the creation of a robust experimental database and the use of a simplified model to analyze arcing behavior. The setup follows the UL 746A standard and considers various pressure and electrode separation conditions. The database includes 960 experiments with PVC and PTFE materials, and model fitting ensures accurate representation of arc currents and voltages. Error analysis shows consistent model performance across materials and highlights sensitivity to electrode spacing, pressure, and source voltage. The paper concludes with a clear 30% error threshold for model validity, enhancing the understanding of the applicability of Andrea's model under various experimental scenarios in aircraft environments.

Analysis of errors at the station "Collection of complaints and anamnesis" during the primary specialized accreditation of specialists

The article highlights the criteria for assessing the quality of training of graduates of the residency faculty for the station “Collection of complaints and anamnesis” of the primary specialized accreditation of specialists. The checklists of the station “Collection of complaints and anamnesis” for the primary specialized accreditation of graduates of the residency department in the specialty “Therapy” and “Oncology” were processed on the basis of the Khanty-Mansiysk State Medical Academy for 2021-2023. Systemic errors were identified, primarily in the area of generalizing the complaints expressed by the patient. FrequeThe article highlights the criteria for assessing the quality of training of graduates of the residency faculty for the station “Collection of complaints and anamnesis” of the primary specialized accreditation of specialists. The checklists of the station “Collection of complaints and anamnesis” for the primary specialized accreditation of graduates of the residency department in the specialty “Therapy” and “Oncology” were processed on the basis of the Khanty-Mansiysk State Medical Academy for 2021-2023. Systemic errors were identified, primarily in the area of generalizing the complaints expressed by the patient. Frequent non-compliance with points related to taking care of the patient’s comfort was discovered. Some of the results of the study will be used in the preparation of residency faculty students in the specialty “Therapy” and “Oncology”.nt non-compliance with points related to taking care of the patient’s comfort was discovered. Some of the results of the study will be used in the preparation of residency faculty students in the specialty “Therapy” and “Oncology”.

Analisis Kesalahan Berbahasa dalam Makalah Mahasiswa Program Studi Pendidikan Biologi Universitas Negeri Medan

This research aims to determine and improve students' critical thinking skills in Indonesian language errors in writing papers for students in the biology education study program at Medan State University, which was carried out using a qualitative descriptive approach. Through qualitative descriptive research methods, researchers provide a real picture of what happened in the field, namely language errors in a biology education student's paper at Medan State University. The technique for collecting data used reading and note-taking techniques. In this research, the intralinguistic analysis method is used, namely an analysis method that compares linguistic elements with reference to the meaning of these language elements (linguistic properties). Includes errors ranging from the use of capital letters, italics and bold letters. The data source in this research is papers from biology education students at Medan State University. The results of this research show that the analysis of language errors contained in student papers is in the form of Indonesian spelling errors which include errors ranging from the use of capital letters, italics and bold letters. From the overall results of this research, 25 usage errors were found. Errors in the use of letters include 6 errors in the use of capital letters, 13 errors in the use of italics, and 6 errors in the use of bold letters. Errors in the use of Indonesian spelling occur because students are not careful, pay less attention and do not master the rules of good and correct writing so that there are still several errors that appear in writing papers.

Analysis of Oral English Errors and Correction Strategies for Junior Middle School Students

Analysis of Oral English Errors and Correction Strategies for Junior Middle School Students

A Proteogenomic Pipeline for the Analysis of Protein Biosynthesis Errors in the Human Pathogen Candida albicans

Candida albicans is a diploid pathogen known for its ability to live as a commensal fungus in healthy individuals, but causing both superficial infections and disseminated candidiasis in immunocompromised patients where it is associated with high morbidity and mortality. Its success in colonizing the human host is attributed to a wide range of virulence traits that modulate interactions between the host and the pathogen, such as optimal growth rate at 37 ºC, the ability to switch between yeast and hyphal forms and a remarkable genomic and phenotypic plasticity. A fascinating aspect of its biology is a prominent heterogeneous proteome that arises from frequent genomic rearrangements, high allelic variation, and high levels of amino acid misincorporations in proteins. This leads to increased morphological and physiological phenotypic diversity of high adaptive potential, but the scope of such protein mistranslation is poorly understood due to technical difficulties in detecting and quantifying amino acid misincorporation events in complex protein samples. We have developed and optimized mass spectrometry and bioinformatics pipelines capable of identifying rare amino acid misincorporation events at the proteome level. We have also analysed the proteomic profile of an engineered C. albicans strain that exhibits high level of leucine misincorporation at protein CUG sites and employed an in vivo quantitative gain-of-function fluorescence reporter system to validate our LC-MS/MS data. C. albicans misincorporates amino acids above the background level at protein sites of diverse codons, particularly at CUG, confirming our previous data on the quantification of leucine incorporation at single CUG sites of recombinant reporter proteins, but increasing misincorporation of Leucine at these sites does not alter the translational fidelity of the other codons. These findings indicate that the C. albicans statistical proteome exceeds prior estimates, suggesting that its highly plastic phenome may also be modulated by environmental factors due to translational ambiguity.

Efficient, nonparametric removal of noise and recovery of probability distributions from time series using nonlinear-correlation functions: Photon and photon-counting noise.

A preceding paper [M. Dhar, J. A. Dickinson, and M. A. Berg, J. Chem. Phys. 159, 054110 (2023)] shows how to remove additive noise from an experimental time series, allowing both the equilibrium distribution of the system and its Green's function to be recovered. The approach is based on nonlinear-correlation functions and is fully nonparametric: no initial model of the system or of the noise is needed. However, single-molecule spectroscopy often produces time series with either photon or photon-counting noise. Unlike additive noise, photon noise is signal-size correlated and quantized. Photon counting adds the potential for bias. This paper extends noise-corrected-correlation methods to these cases and tests them on synthetic datasets. Neither signal-size correlation nor quantization is a significant complication. Analysis of the sampling error yields guidelines for the data quality needed to recover the properties of a system with a given complexity. We show that bias in photon-counting data can be corrected, even at the high count rates needed to optimize the time resolution. Using all these results, we discuss the factors that limit the time resolution of single-molecule spectroscopy and the conditions that would be needed to push measurements into the submicrosecond region.

Intelligent Prediction and Application Research on Soft Rock Tunnel Deformation Based on the ICPO-LSTM Model

In tunnel construction, the prediction of the surrounding rock deformation is related to the construction safety and stability of the tunnel structure. In order to achieve an accurate prediction of the surrounding rock deformation in soft rock tunnel construction, a Long Short-Term Memory (LSTM) neural network is used to construct a prediction model of the vault settlement and the horizontal convergence of the upper conductor in soft rock tunnels. The crested porcupine optimisation (CPO) algorithm is used to realise the hyper-parameter optimisation of the LSTM model and to construct the framework of the calculation process of the CPO-LSTM model. Taking the soft rock section of the Baoshishan Tunnel as an example, the large deformation of the surrounding rock is measured and analysed in situ, and the monitoring data of arch settlement and superconducting level convergence are obtained, which are substituted into the CPO-LSTM model for calculation, and compared and analysed with traditional machine learning and optimisation algorithms. The results show that the CPO-LSTM model has an R2 of 0.9982, a MAPE of 0.8595% and an RMSE of 0.1922, which are the best among all the models. In order to further improve the optimisation capability of the CPO, some improvements were made to the CPO and an Improved Crested Porcupine Optimiser (ICPO) was proposed. The ICPO-LSTM prediction model was established, and the ZK6 + 834 section was selected as a research object for comparison and analysis with the CPO-LSTM model. The results of the error analysis show that the prediction accuracy of the improved ICPO-LSTM model has been further improved, and the prediction accuracy of the model meets the requirements of guiding construction.

Novel Approach by Shifted Fibonacci Polynomials for Solving the Fractional Burgers Equation

This paper analyzes a novel use of the shifted Fibonacci polynomials (SFPs) to treat the time-fractional Burgers equation (TFBE). We first develop the fundamental formulas of these polynomials, which include their power series representation and the inversion formula. We establish other new formulas for the SFPs, including integer and fractional derivatives, in order to design the collocation approach for treating the TFBE. These derivative formulas serve as tools that aid in constructing the operational metrics for the integer and fractional derivatives of the SFPs. We use these matrices to transform the problem and its underlying conditions into a system of nonlinear equations that can be treated numerically. An error analysis is analyzed in detail. We also present three illustrative numerical examples and comparisons to test our proposed algorithm. These results showed that the proposed algorithm is advantageous since highly accurate approximate solutions can be obtained by choosing a few terms of retained modes of SFPs.

Hybrid Haar wavelet and meshfree methods for hyperbolic double interface problems: Numerical implementations and comparative performance analysis

This paper introduces a variety of approaches for solving 2D and 3D hyperbolic double interface problems. The methods are based on the Haar wavelet method, multiquadric radial basis function method, and integrated multiquadric radial basis function method. Temporal derivatives are handled using the second central difference and the Houbolt method. Various numerical approaches based on these methods are developed, and their implementations are discussed in complete detail. The paper evaluates and compares the performances of these approaches using both linear and nonlinear 2D and 3D double interface hyperbolic problems. Error analysis, conducted using the L-infinity norm, and efficiency assessments measured through CPU times contribute to a comprehensive understanding of the applicability and comparative effectiveness of the proposed methods. This study provides valuable insights for researchers and practitioners dealing with the challenges posed by interface problems in general.

Inconsistencies in unstructured geometric volume-of-fluid methods for two-phase flows with high density ratios

Geometric flux-based Volume-of-Fluid (VOF) methods (Marić et al., 2020) are widely considered consistent in handling two-phase flows with high density ratios. However, although the conservation of mass and momentum is consistent for two-phase incompressible single-field Navier–Stokes equations without phase-change (Liu et al., 2023), discretization may easily introduce inconsistencies that result in very large errors or catastrophic failure. We apply the consistency conditions derived for the ρLENT unstructured Level Set/Front Tracking method (Liu et al., 2023) to flux-based geometric VOF methods (Marić et al., 2020), and implement our discretization into the plicRDF-isoAdvector geometrical VOF method (Roenby et al., 2016). We find that computing the mass flux by scaling the geometrically computed fluxed phase-specific volume can ensure equivalence between the mass conservation equation and the phase indicator (volume conservation) if consistent discretization schemes are chosen for the temporal and convective term. Based on the analysis of discretization errors, we suggest a consistent combination of the temporal discretization scheme and the interpolation scheme for the momentum convection term. We confirm the consistency by solving an auxiliary mass conservation equation with a geometrical calculation of the face-centered density (Liu et al., 2023). We prove the equivalence between these two approaches mathematically and verify and validate their numerical stability for density ratios within [1, 106] and viscosity ratios within [102, 105].

Groundwater inverse modeling: Physics-informed neural network with disentangled constraints and errors

This study combines a physics-informed neural network (PINN) and Karhunen-Loeve expansion (KLE) (i.e., KLE-PINN) to solve the groundwater inverse problem. The hydraulic head (u) distribution is approximated by a deep neural network (DNN), while the hydraulic conductivity (K) field is constructed by KLE with given prior geostatistical information. KLE-PINN is applied to investigate the inversion using data from a single pumping test, natural gradient flow (NG), and hydraulic tomography (HT). The results from these cases demonstrate that our inverse method is robust. Our error analysis endeavors to quantify the sources of error by using two custom reference indicators, eforward and ecoupling. Moreover, the study finds that the inversion using data from multiple pumping tests (HT) yields more accurate estimates, leads to faster training convergence, and maintains higher stability. In addition, by investigating cases with different outer boundary conditions (BCs), we find that KLE-PINN is more flexible. Precisely, in scenarios with missing BCs, our network still fits well with the observed data, and the estimates capture the approximate spatial pattern in the region where the observation wells are distributed. Even with incorrect BCs, our network still performs well because the observational data constraints are strongly enforced during training.

Analysis of Errors in Emergency Medical Care in Case of Mine Explosion Injury Based on the Results of the "Lifeline-2024" Competition

Analysis of Errors in Emergency Medical Care in Case of Mine Explosion Injury Based on the Results of the "Lifeline-2024" Competition

Dejdumrong Collocation Approach and Operational Matrix for a Class of Second-Order Delay IVPs: Error Analysis and Applications

Dejdumrong Collocation Approach and Operational Matrix for a Class of Second-Order Delay IVPs: Error Analysis and Applications

A new approximation to the first order fractional derivative in the Caputo-Fabrizio sense using Haar Wavelet integration formula

Decades ago, fractional calculus arose to generalize ordinary derivation and integration, and then became a means of modeling and interpreting many phenomena in various fields such as engineering, physics, chemistry, biology and signal processing. The definition of the fractional derivative began with a derivative with a singular kernel, such as the Riemann-Liouville and Caputo derivative. Due to the singularity of the kernel, the definition of Caputo-Fabrizio appeared, which has a non-singular kernel and mathematical properties similar to the derivative of the integer order. This last definition attracted many mathematicians and researchers to use it in modeling phenomena and obtaining historical information about the development of the studied phenomena, but usually the analytical solution does not exist, which necessitated numerical methods to find an approximate solution. These approximate methods depend on finding an approximate formula for the fractional derivative, and then the problem is transformed into a system of algebraic equations that is easy to solve. In fact, all the numerical methods that have been used have a polynomial rate of convergence, which calls for thinking about a new method that is more effective and has a better rate of convergence. For this reason, in this paper, we propose an efficient numerical method to approximate the first order fractional derivative in the Caputo-Fabrizio sense. This method develops a new quadratic formula using Haar wavelet integration method. Error analysis of our proposed method gives an exponential convergence rate of . To check the effectiveness of the proposed method, we examine some examples with different fractional orders. The quantative results demonstrated the stability and efficiency of the proposed method.

Analysis of student errors based on Polya’s steps in three-dimensional geometry

Analysis of student errors based on Polya’s steps in three-dimensional geometry

Cubic B-spline based numerical schemes for delayed time-fractional advection-diffusion equations involving mild singularities

This article presents two efficient layer-adaptive numerical schemes for a class of time-fractional advection-diffusion equations with a large time delay. The fractional derivative of order α with α ∈ (0, 1) is taken in the Caputo sense. The solution to this type of problem generally has a layer due to the mild singularity near the time t = 0. Consequently, the polynomial interpolation discretizing scheme degrades the convergence rate in the case of uniform meshes. In the presence of a singularity, the temporal fractional operator is discretized by employing the L1 technique on a layer-resolving mesh. In contrast, the cubic B-spline collocation method is used in the spatial direction. The convergence analysis and estimation of error are presented for the proposed scheme under reasonable regularity assumptions on the coefficients. The scheme achieves its optimal convergence rate (2 − α) for suitable choice of grading parameter (γ ≥ (2 − α)/α). Furthermore, we modified the proposed scheme by discretizing the fractional operator with the help of the L1-2 technique. The modified scheme gets a quadratic order convergence for γ ≥ 2/α. In addition, we extend the proposed schemes to solve the corresponding semilinear problem. Numerical examples demonstrate the efficiency and applicability of the proposed techniques.

Approximate solution of multi-term fractional differential equations via a block-by-block method

In this paper we introduce a block-by-block method for the numerical solution of multi-term fractional differential equations (MFDEs). The main idea is to convert a MFDE to a Volterra integral equation of weakly singular type, to which a well known block-by-block method is applied. We also provide the error analysis and convergence of the method. Finally, numerical examples involving Bagley–Torvik and relaxation-oscillation equations are given to confirm applications and the theoretical results.

RMS error analysis of spatial uniformity and energy stability of the laser beam in second harmonic generation

The coupling of the energy stability and spatial uniformity of the laser beam before and after second harmonic generation (SHG) was analyzed. SHG experiments were performed using a Nd:YAG nanosecond laser and LBO crystals, and images, pulse shapes, and energies were measured. The relationship between energy stability and spatial uniformity uses a formula derived from the previous study to analyze changes in energy stability and spatial uniformity of the input beam and converted beam. In addition, the measured input beam shape and energy are compared with the results of applying the SHG converting equation considering pump depletion. Both methods were similar to the experimental results when corrected by empirical factors. Through SHG, it was confirmed that there is an optimal point of energy stability and spatial uniformity of the laser beam near the critical power.

Equilibrium and kinetic modeling studies for copper and cadmium removal from aqueous solutions by ground nut husk

This study investigates the equilibrium and kinetics of copper and cadmium adsorption from aqueous solutions using groundnut husk. Characterization of the groundnut husk reveals various functional groups and a porous morphology. Isothermal studies were conducted, and different models were evaluated to fit the data. The Langmuir model exhibits the best fit for copper adsorption, while the Temkin and Dubinin–Radushkevich (DR) isotherm models show superior fitting for cadmium adsorption. Results suggest that physisorption significantly influences both cases, with mean sorption adsorption energy lower than 8000 J/mole. Error analysis reveals that the Freundlich and Temkin models demonstrate maximum adsorption capacity, with lower Chi-square (ꭓ2), Sum of Square Errors (SSE), and SAE compared to Langmuir and DR models. Additionally, intra-particle diffusion was analyzed using the Weber and Morris equation. Evaluation of model parameters indicates that cadmium demonstrates better nonlinear fitting of data compared to copper. FTIR and SEM analysis confirm the involvement of various functional groups in adsorption and reveal changes in surface distribution post-adsorption due to copper and cadmium presence.