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The two-dimensional laser-based Ångstrom method measures the in-plane thermal properties for anisotropic film-like materials. It involves periodic laser heating at the center of a suspended film sample and records its transient thermal response by infrared imaging. These spatiotemporal temperature data must be analyzed to extract the unknown thermal conductivity values in the orthotropic directions, an inverse parameter fitting problem. Previous demonstration of the metrology technique used a least-squares fitting method that relies on numerical differentiation to evaluate the second-order partial derivatives in the differential equation describing transient conduction in the physical system. This fitting approach is susceptible to measurement noise, introducing high uncertainty in the extracted properties when working with noisy data. For example, when noise of a signal-to-noise ratio of 10 is added to simulated amplitude and phase data, the error in the extracted thermal conductivity can exceed 80%. In this work, we introduce a new alternative inverse parameter fitting approach using physics-informed neural networks (PINNs) to increase the robustness of the measurement technique for noisy temperature data. We demonstrate the effectiveness of this approach even for scenarios with extreme levels of noise in the data. Specifically, the PINN-approach accurately extracts the properties to within 5% of the true values even for high noise levels (a signal-to-noise ratio of 1). This offers a promising avenue for improving the robustness and accuracy of advanced thermal metrology tools that rely on inverse parameter fitting of temperature data to extract thermal properties. [ABSTRACT FROM AUTHOR]

Paramagnetic molecules with a metal ion as an electron spin center are promising building blocks for molecular qubits and high-density memory arrays. However, fast spin relaxation and decoherence in these molecules lead to a rapid loss of magnetization and quantum information. Nonadiabatic coupling (NAC), closely related to spin-vibrational coupling, is the main source of spin relaxation and decoherence in paramagnetic molecules at higher temperatures. Predicting these couplings using numerical differentiation requires a large number of computationally intensive ab initio or crystal field electronic structure calculations. To reduce computational cost and improve accuracy, we derive and implement analytical NAC and state-specific energy gradient for the ab initio parametrized crystal field Hamiltonian describing single-ion molecular magnets. Our implementation requires only a single crystal field calculation. In addition, the accurate NACs and state-specific energy gradients can be used to model spin relaxation using sophisticated nonadiabatic molecular dynamics, which avoids the harmonic approximation for molecular vibrations. To test our implementation, we calculate the NAC values for three lanthanide complexes. The predicted values support the relaxation mechanisms reported in previous studies. [ABSTRACT FROM AUTHOR]

Mathematical foundations and algorithms for recognizing bichromatic two-dimensional graphic codes, regardless of their type (QR codes, DataMatrix, GridMatrix, etc.) are presented. The stages of achieving the result include detecting the code, localizing it within an arbitrary quadrilateral, transforming the quadrilateral to a canonical square, constructing a grid of elements (modules) of the square code, and filling it with a sequence of bits. It is shown that perspective transformation formulas make it possible to transform localized quadrangular regions to canonical squares with an acceptable error level for further processing. A flat grid of square code elements is formed based on the search for extrema of the derivatives of the pixel intensity distribution of the square image along the axes x and y. The algorithm for filling grid cells (code modules) with a sequence of zeros and ones uses information about the average intensity of each such cell. At the end of the paper, the algorithms are tested on a variety of real images of two-dimensional codes, and the limitations of the proposed algorithms are examined. [ABSTRACT FROM AUTHOR]

The problem of optimal recovering high-order mixed derivatives of bivariate functions with finite smoothness is studied. Based on the truncation method, an algorithm for numerical differentiation is constructed, which is order-optimal both in the sense of accuracy and in terms of the amount of involved Galerkin information. Numerical examples are provided to illustrate the fact that our approach can be implemented successfully. [ABSTRACT FROM AUTHOR]

BACKGROUND: Graphic type differentiation of cell count data of synovial aspirates is a new method for the diagnosis of early and late periprosthetic joint infection. OBJECTIVE: The aim of the study was to analyse if the same 6 LMNE-types can be differentiated in the new Yumizen H500 cell counter as it was the case for the old cell counter ABX Pentra XL 80 of previous publications, to verify if the erythrocyte and thrombocyte curves of the new device give additional information and to calculate the difference of cell count in LMNE-type I and III (with abrasion) in the cell counter and in the manual counting chamber (Neubauer improved). METHODS: 450 aspirates of 152 total hip arthroplasties and 298 knee arthroplasties obtained for the diagnosis of periprosthetic joint infection were analysed with the Yumizen H500. RESULTS: All LMNE-matrices of the 450 aspirates could assigned to one of the six LMNE-types. There were 76 LMNE-type I, 72 LMNE-type II, 14 LMNE-type III, 241 LMNE-type IV, 36 LMNE-type V and 12 LMNE-type VI. The erythrocyte and thrombocyte distribution curves were very helpful for differentiation of hematoma and infection. The cell count in the manual counting procedure was lower than in the cell counter: for the LMNE-type I (abrasion type) the median of the difference was 925/ μ L (median) and for the LMNE-type III (combined type of infection and abrasion) 3570/ μ L (median). CONCLUSION: The described graphic type differentiation is a new and helpful method for differentiation of hematoma and early PJI as well as abrasion and late PJI. [ABSTRACT FROM AUTHOR]

The conventional average acceleration approach relies on K-band range observation, containing an unknown bias, which leads to possible degradation of the precision of Earth's gravity field modelling. It also suffers from correlated errors caused by three-point numerical differentiation. In this study, an improved approach is proposed that makes use of K-band range-rate observations instead and overcoming the influence of correlated errors by introducing a whitening filter. GRACE-Follow On data spanning the period from January 2019 to December 2022 were processed by the proposed approach and a series of time-varying gravity field models was derived, referred to as SSM-AAA-GFO in this paper. This model series is compared comprehensively with three official model series. Results demonstrate that all model series are highly coincident below degree 30 and reflect similar time-varying gravity field signals in both large and small basins. After filtering, SSM-AAA-GFO shows uncertainty, in the form of equivalent water height below 2.5 cm, which is comparable with three official model series. The comparison results confirm the effectiveness of the proposed approach for precisely modelling a time-varying gravity field based on K-band range-rate observations. [ABSTRACT FROM AUTHOR]

Due to their several attractive properties, BDF-type multistep methods are usually the method-of-choice for solving stiff initial value problems (IVPs) of ordinary differential equations (ODEs) and differential-algebraic equations (DAEs). Recently, a class of BDF-type methods based on linear barycentric rational interpolants (LBRIs), referred to as RBDF methods, was introduced for solving ODEs. In the present paper, we are going to introduce a new family of LBRIs-based BDF-type formulas for the numerical solution of ODEs and DAEs with desirable stability properties and smaller error constants than those of the RBDF methods. Numerical experiments of the proposed methods on some well-known IVPs for ODEs and DAEs with index ≤ 3 illustrate the efficiency and capability of the methods in solving such problems. [ABSTRACT FROM AUTHOR]

The periodic sinc function interpolation offers a compelling solution to address the issue of noise in the analysis of thermogravimetric analysis (TGA) data, thereby enhancing the outcomes of differential techniques such as the Friedman isoconversional method. In this study, we introduce a novel approach that leverages the periodic sinc function interpolation to directly obtain smooth reaction rates from TGA data, eliminating the reliance on numerical differentiation methods. The efficacy of this method has been confirmed through its application to noisy experimental data derived from the thermal decomposition of various polymers, showcasing its robustness. Readers are provided with the corresponding code for Gnu Octave, serving as a free alternative to MATLAB. Additionally, the activation energies calculated from the experimental data using both the Friedman method and periodic sinc function interpolation closely align with those determined by the integral Vyazovkin method, emphasizing the validity and reliability of this new approach. [ABSTRACT FROM AUTHOR]

Abstract A new method for estimating the first and second derivatives of discrete audio signals intended to achieve higher computational precision in analyzing the performance and characteristics of digital audio systems is presented. The method could find numerous applications in modeling nonlinear audio circuit systems, e.g., for audio synthesis and creating audio effects, music recognition and classification, time-frequency analysis based on nonstationary audio signal decomposition, audio steganalysis and digital audio authentication or audio feature extraction methods. The proposed algorithm employs the ordinary 7 point-stencil central-difference formulas with improvements that minimize the round-off and truncation errors. This is achieved by treating the step size of numerical differentiation as a regularization parameter, which acts as a decision threshold in all calculations. This approach requires shifting discrete audio data by fractions of the initial sample rate, which was obtained by fractional delay FIR filters designed with modified 11-term cosine-sum windows for interpolation and shifting of audio signals. The maximum relative error in estimating first and second derivatives of discrete audio signals are respectively in order of $$10^{-13}$$ 10 - 13 and $$10^{-10}$$ 10 - 10 over the entire audio band, which is close to double-precision floating-point accuracy for the first and better than single-precision floating-point accuracy for the second derivative estimation. Numerical testing showed that this performance of the proposed method is not influenced by the type of signal being differentiated (either stationary or nonstationary), and provides better results than other known differentiation methods, in the audio band up to 21 kHz.

The accurate calculation of reaction rates from experimental data is crucial for understanding and characterizing chemical processes. However, the presence of noise in experimental data can introduce errors in rate calculations. In this study, we introduced a novel approach that utilizes the Legendre series expansion method to directly derive smooth reaction rates from noisy experimental data, eliminating the need for numerical differentiation methods. This approach proves to be highly effective in handling noisy thermogravimetric analysis (TGA) data obtained from the thermal decomposition of specific polymers. We demonstrated the robustness and reliability of this method and provided Gnu Octave codes as a free alternative to MATLAB, making the implementation more accessible. Furthermore, the smooth reaction rates obtained were used to evaluate the activation energy using the Friedman isoconversional method. The results showed excellent agreement with those obtained using the Vyazovkin integral method. Additionally, the proposed method can be applied to obtain smooth derivative thermogravimetric (DTG) curves using noisy TGA data set. [ABSTRACT FROM AUTHOR]

Sound fields above porous layers or rough acoustically hard boundaries may include airborne surface waves. The surface wave properties depend on the effective surface admittance. Analytical expressions for surface wave speeds are derived from models for the acoustical properties of rigid porous media. Surface wave effects on measurements of level difference spectra over porous asphalt are investigated and predictions of phase, group speeds, and vertical attenuation of the surface waves over externally reacting hard backed layers corresponding to a porous asphalt are compared. Predictions of surface wave characteristics above an identical vertical slit medium are compared with data obtained over arrays of parallel aluminum strips on an acoustically hard surface. Group speeds of surface waves over lattices, parallel regularly spaced strips, and snow obtained by numerical differentiation of the phase speed spectra corresponding to admittance spectra deduced from complex excess attenuation are found to compare well with those estimated from time domain data. An effective admittance, deduced from a boundary element method simulation of the excess attenuation spectrum over regularly spaced ribs so that the frequency of the peak corresponds with that in the measured spectrum, is used to estimate the group speed of the associated surface wave. [ABSTRACT FROM AUTHOR]

Investigating subsurface shear wave velocity (v s) structures using surface wave dispersion data involves minimizing a misfit function that is commonly solved through gradient-based optimization. Sensitivity kernels for model updates are commonly estimated using numerical differentiation, variational methods or implicit functions which however, may involve numerical instability and computational challenges when dealing with complex velocity models and large data sets. In this study, we propose a novel surface wave inversion framework in which error-free gradients are calculated by automatic differentiation (AD) and forward modelling is implemented by convenient computational graphs in the state-of-the-art deep learning framework. The AD-based inversion approach is first validated using two synthetic data sets. Then, the subsurface structures at three distinct locations, namely the Great Plains and the Long Beach in the US and Tong Zhou in China, are also derived using this method with seismic ambient noise data, which show nice consistency with those obtained using traditional methods. With the significantly improved computational efficiency, a great number of initial models can be inverted simultaneously to mitigate the impact of local minima and to estimate the uncertainty in the invert models. We have developed a new surface wave inversion package named ADsurf based on automatic differentiation and computational graphs in the deep learning framework, and its computational efficiency is also compared with the traditional finite-difference-based gradient estimation approach. While a great number of intriguing studies on the geophysical inverse problems have been conducted recently using deep learning for end-to-end mapping, the use of AD provided in the in the deep learning frameworks to assist and expedite the gradient computations are still underexploited in geophysics. Thus, it is expected that various geophysical inverse problems in many different areas beyond the surface wave inversion can also be tackled with this new paradigm in the future. [ABSTRACT FROM AUTHOR]

The accurate measurement of cable force plays an important role in the structural form, structural behavior, and safety evaluation of large-span cable-stayed bridges. A fast cable-force estimation method is proposed based on the alignment recognition method and smartphone-captured video. This method can realize real-time, non-contact, and non-destructive force measurements. Videos of bridge cables were collected using a portable smartphone, and an alignment recognition method was employed to obtain the lateral displacement along the cable, followed by numerical differentiation of the lateral displacement of the cable to acquire the acceleration time history. Based on the fundamental dynamic equation of beam elements, a unified explicit formula for cable-force calculation considering service characteristics was established in the frequency method. Finally, the method was applied to the cable-force measurement during the operation and maintenance stage of the Gengcun Dou Bridge. The measured cable forces were compared with the values obtained from the on-site monitoring system. The results show that the proposed method can accurately identify the acceleration time history at different positions of the cable, and the corresponding frequency components are essentially consistent. The average relative deviation of the measured cable forces from the reference method is within 3%. The proposed cable-force measurement method is simple as regards equipment, convenient for operation, and high in efficiency, providing a new method for the measurement of cable forces in large-span cable-stayed bridges, which can offer reliable measured cable-force data for structural analysis during bridge operation and the maintenance stage. [ABSTRACT FROM AUTHOR]

A new method for estimating the first and second derivatives of discrete audio signals intended to achieve higher computational precision in analyzing the performance and characteristics of digital audio systems is presented. The method could find numerous applications in modeling nonlinear audio circuit systems, e.g., for audio synthesis and creating audio effects, music recognition and classification, time-frequency analysis based on nonstationary audio signal decomposition, audio steganalysis and digital audio authentication or audio feature extraction methods. The proposed algorithm employs the ordinary 7 point-stencil central-difference formulas with improvements that minimize the round-off and truncation errors. This is achieved by treating the step size of numerical differentiation as a regularization parameter, which acts as a decision threshold in all calculations. This approach requires shifting discrete audio data by fractions of the initial sample rate, which was obtained by fractional delay FIR filters designed with modified 11-term cosine-sum windows for interpolation and shifting of audio signals. The maximum relative error in estimating first and second derivatives of discrete audio signals are respectively in order of 10 - 13 and 10 - 10 over the entire audio band, which is close to double-precision floating-point accuracy for the first and better than single-precision floating-point accuracy for the second derivative estimation. Numerical testing showed that this performance of the proposed method is not influenced by the type of signal being differentiated (either stationary or nonstationary), and provides better results than other known differentiation methods, in the audio band up to 21 kHz. [ABSTRACT FROM AUTHOR]

Numerical differentiation of functions with large gradients is considered. It is assumed that the original function of one variable can be decomposed into the sum of a regular component with bounded derivatives up to a certain order and a boundary layer component, which has large gradients and is known up to a factor. In particular, this decomposition is relevant for solution of a singularly perturbed boundary value problem, since the application of classical polynomial formulas of numerical differentiation to functions with large gradients can lead to significant errors. The error of numerical differentiation formulas is estimated for constructed formulas exact on the boundary layer component of the original function. The results of numerical experiments, consistent with the error estimates obtained, are presented. [ABSTRACT FROM AUTHOR]

Coherent optical phonon (COP) and coherent acoustic phonon (CAP) play vital roles in probing ultrafast dynamics across various materials and systems. Their applications range from metamaterial diagnosis to manipulating ferroelectric polarization and self-trapping of carriers in perovskites. However, the inherently small signals associated with COP and CAP pose challenges in extracting valuable information. In this paper, we address this challenge by presenting a numerical differential method for signal amplification and deduction of phonon dephasing time. In addition, application of this method has been demonstrated in a superlattice system, establishing the validity and effectiveness of this technique. [ABSTRACT FROM AUTHOR]

Function approximation is a fundamental process in a variety of problems in computational mechanics, structural engineering, as well as other domains that require the precise approximation of a phenomenon with an analytic function. This work demonstrates a unified approach to these techniques, utilizing partial sums of the Taylor series in a high arithmetic precision. In particular, the proposed approach is capable of interpolation, extrapolation, numerical differentiation, numerical integration, solution of ordinary and partial differential equations, and system identification. The method employs Taylor polynomials and hundreds of digits in the computations to obtain precise results. Interestingly, some well-known problems are found to arise in the calculation accuracy and not methodological inefficiencies, as would be expected. In particular, the approximation errors are precisely predictable, the Runge phenomenon is eliminated, and the extrapolation extent may a priory be anticipated. The attained polynomials offer a precise representation of the unknown system as well as its radius of convergence, which provides a rigorous estimation of the prediction ability. The approximation errors are comprehensively analyzed for a variety of calculation digits and test problems and can be reproduced by the provided computer code. [ABSTRACT FROM AUTHOR]

Numerical algorithms for calculating the left- and right-sided Riemann–Liouville fractional integrals and the left- and right-sided fractional derivatives in the Caputo sense using spline interpolation techniques are derived. The spline of the fifth degree (the so-called quintic spline) is mainly taken into account, but the linear and cubic splines are also considered to compare the quality of the developed method and numerical calculations. The estimation of errors for the derived approximation algorithms is presented. Examples of the numerical evaluation of the fractional integrals and derivatives are executed using 128-bit floating-point numbers and arithmetic routines. For each derived algorithm, the experimental orders of convergence are calculated. Also, an illustrative computational example showing the action of the considered fractional operators on the symmetric function in the interval is presented. [ABSTRACT FROM AUTHOR]

In article algorithms of numerical differentiation of output signals of digital devices in a time domain on the basis of exponential functions are considered. The main requirements to algorithms of numerical differentiation for the solution of a problem of the return digital transformation to management theories are formulated. Results of researches of authors of algorithms of numerical differentiation in frequency area on the basis of estimates of derivative interpolation polynomial are stated. The most effective is chosen for the solution of a problem of the return digital transformation algorithm of numerical differentiation. [ABSTRACT FROM AUTHOR]

Numerical differentiation of a function over the unit interval of the real axis, which is contaminated with noise, by inverting the simple integration operator J mapping in L 2 is discussed extensively in the literature. The complete singular system of the compact operator J is explicitly given with singular values σ n ⁢ (J) asymptotically proportional to 1 n . This indicates a degree one of ill-posedness for the associated inverse problem of differentiation. We recall the concept of the degree of ill-posedness for linear operator equations with compact forward operators in Hilbert spaces. In contrast to the one-dimensional case, there is little specific material available about the inverse problem of mixed differentiation, where the d-dimensional analog J d to J, defined over unit d-cube, is to be inverted. In this note, we show for that problem that the degree of ill-posedness stays at one for all dimensions d ∈ ℕ . Some more discussion refers to the two-dimensional case in order to characterize the range of the operator J 2 . [ABSTRACT FROM AUTHOR]

FE2-approaches are chosen to incorporate micro-mechanical information into macroscopical computations. Although the approach is applied for several years, there are some open theoretical aspects in the algorithmic description. In this contribution the entire concept is brought into the method of vertical lines to solve initial boundary-value problems based on the homogenization concept under consideration. This requires a clear description of known and unknown quantities in the space and time discretization procedure. Dependent on the constitutive model, systems of algebraic or differential-algebraic equations (DAE) arise. Here, we concentrate on models of evolutionary type. Frequently, the resulting DAE-systems are solved using the Backward-Euler method, which are embedded in the more general class of diagonally, implicit Runge-Kutta methods. After the time discretization the resulting systems of non-linear equations have to be solved. This can be done using various schemes to obtain more efficient computations. Here, the three-level Newton algorithm, the Newton-Raphson-Schur method, numerical differentiation and a Chord-version are derived in the context of FE2 and discussed. Finally, step-size-controlled FE2-problems are solved, highlighting the importance of time-adaptive computations, particularly, in combination with the non-linear solution scheme, where the Multilevel-Newton-Chord method shows high efficiency. This contributes to an additional reduction of the overall computational times. [ABSTRACT FROM AUTHOR]

Standard algorithms for numerical integration are defined for simple integrals. Formulas for computation of repeated integrals and derivatives for equidistant domain partition based on modified Newton-Cotes formulas are derived. We compare usage of the new formulas with the classical quadrature formulas and discuss possible application of the results to solving higher order differential equations. [ABSTRACT FROM AUTHOR]

Abstract The presence of minor components represents a challenging problem in spectrophotometric analysis of pharmaceuticals. If one component has a low absorptivity or present in a low concentration compared to the other components, this will hinder its quantitation by spectrophotometric methods. Continuous Wavelet Transform (CWT) as a signal processing technique was utilized to figure out a solution to such a problem. A comparative study was established between traditional derivative spectrophotometry (Numerical Differentiation, ND) and CWT to indicate the advantages and limitations of each technique and possibility of solving the problem of minor components. A mixture of ibuprofen (IBU) and phenylephrine (PHE) with its degradation products forming a ternary mixture was used for comparing the two techniques. The two techniques were applied on raw spectral data and on ratio spectra data resulting in four methods, namely ND, CWT, Derivative Ratio-Zero Crossing (DRZC) and Continuous Wavelet Transform Ratio-Zero Crossing (CWTR-ZC) methods. By comparing the results in laboratory prepared mixtures, CWT technique showed advantages in analysis of mixtures with minor components than ND. The proposed methods were validated according to the ICH guideline Q2(R1), where their linearity was established with correlation coefficient ranging from 0.9995 to 0.9999. The linearity was in the range 3–40 μg/mL for PHE in all methods, while for IBU it was 20–180 and 30–180 μg/mL in CWT and ND methods, respectively. The CWT methods were applied for quantitative determination of the drugs in their dosage form showing the ability of the methods to quantitate minor components in pharmaceutical formulations.

This research aims to apply an output filtering method to conduct the system parameter identification of an unstable wheel-driven pendulum system. First, the nonlinear dynamic model of the system is established by utilizing the Lagrangian dynamic theorem. Next, the Least-Square (LS) is introduced for system parameter identification formulation. Nevertheless, considering the real scenario, the wheel displacement is acquired from encoders subject to quantization errors. The pitch angle of the pendulum cart is also accompanied by Gaussian noise. Therefore, using numerical differentiation for angular acceleration in the LS estimations directly would induce incorrect state information seriously. To address this practical issue, an output filtering method is considered. The developed parameter identification algorithm could attenuate the influence of the quantization effect as well as noisy data and thus obtain much more accurate parameter identification results. Comparative simulation reveals that the output filtering method has a superior parameter estimation performance than the direct numerical difference method. [ABSTRACT FROM AUTHOR]

The presence of minor components represents a challenging problem in spectrophotometric analysis of pharmaceuticals. If one component has a low absorptivity or present in a low concentration compared to the other components, this will hinder its quantitation by spectrophotometric methods. Continuous Wavelet Transform (CWT) as a signal processing technique was utilized to figure out a solution to such a problem. A comparative study was established between traditional derivative spectrophotometry (Numerical Differentiation, ND) and CWT to indicate the advantages and limitations of each technique and possibility of solving the problem of minor components. A mixture of ibuprofen (IBU) and phenylephrine (PHE) with its degradation products forming a ternary mixture was used for comparing the two techniques. The two techniques were applied on raw spectral data and on ratio spectra data resulting in four methods, namely ND, CWT, Derivative Ratio-Zero Crossing (DRZC) and Continuous Wavelet Transform Ratio-Zero Crossing (CWTR-ZC) methods. By comparing the results in laboratory prepared mixtures, CWT technique showed advantages in analysis of mixtures with minor components than ND. The proposed methods were validated according to the ICH guideline Q2(R1), where their linearity was established with correlation coefficient ranging from 0.9995 to 0.9999. The linearity was in the range 3–40 μg/mL for PHE in all methods, while for IBU it was 20–180 and 30–180 μg/mL in CWT and ND methods, respectively. The CWT methods were applied for quantitative determination of the drugs in their dosage form showing the ability of the methods to quantitate minor components in pharmaceutical formulations. [ABSTRACT FROM AUTHOR]

Electrically actuated linear motion of a water droplet over PDMS-coated single active electrode is analyzed from detailed experiments and modeling. In an experiment, continuous motion of the droplet is achieved when it is located over an active electrode with a horizontal ground wire placed just above in an open-electrowetting-on-dielectric configuration. Using a CCD camera, the instantaneous centroid position of the droplet is determined and its velocity is inferred by numerical differentiation. The edge-detected image is also used to determine the advancing and receding contact angles of the moving drop relative to the substrate. Motion of 2, 6, and 10 µ l water droplets for voltages in the range of 170–270 V DC is examined to investigate the effect of drop volume and voltage on drop deformation and velocity. The motion of the droplet is initiated by Young-Lippmann spreading at the three-phase contact line, followed by a nonuniform electric force field distributed between the active electrode and the ground wire localized at the droplet-air interface. Simulations carried out using COMSOL© Multiphysics with full coupling between the electric field and hydrodynamics are in conformity with experiments. A contact angle model with pinning and friction leads to close agreement between simulations and drop motion over a bare PDMS layer, particularly in terms of the relevant timescales. When contact line friction is neglected, the fully coupled numerical solution shows a good match with experimentally determined drop movement over a silicone oil-coated PDMS layer. Over both surfaces, continuous motion of the water droplet is seen to be achieved in three stages, namely, initial spreading, acceleration, and attainment of constant speed. [ABSTRACT FROM AUTHOR]

The high stiffness actuator (HSA), applied to each joint of an electrical driven humanoid robot, can directly affect the motion performance of the torque-controlled humanoid robots. For high control performance of HSA, a high-precision dynamic torque control (HDTC) is proposed. The HDTC consists of two phases: (1) A novel dynamic current control is used to linearize high stiffness actuator torque control system, which can estimate and compensate the nonlinear coupling parts; (2) An enhanced internal model control is designed to ensure high tracking accuracy in the system containing noisy torque signal and even numerical differentiation signals. Benefitting from dynamic current control and the enhanced internal model control, the proposed HDTC is accurate and adaptable. Finally, the superiority of the HDTC is verified with comparative experiments. • A novel dynamic current control (DCC) is proposed to linearize the high stiffness actuator (HSA) torque control system, which can estimate and compensate the nonlinear coupling parts. • An enhanced internal model control (EIMC) is proposed to effectively eliminate negative influence of the filter, which can improve the accuracy of the torque control system. • The effectiveness of the proposed method is tested by comparison experiments in the HSA experimental test platform. [ABSTRACT FROM AUTHOR]

Our objective is to calculate the derivatives of data corrupted by noise. This is a challenging task as even small amounts of noise can result in significant errors in the computation. This is mainly due to the randomness of the noise, which can result in high-frequency fluctuations. To overcome this challenge, we suggest an approach that involves approximating the data by eliminating high-frequency terms from the Fourier expansion of the given data with respect to the polynomial-exponential basis. This truncation method helps to regularize the issue, while the use of the polynomial-exponential basis ensures accuracy in the computation. We demonstrate the effectiveness of our approach through numerical examples in one and two dimensions. [ABSTRACT FROM AUTHOR]

We investigate minimax testing for detecting local signals or linear combinations of such signals when only indirect data are available. Naturally, in the presence of noise, signals that are too small cannot be reliably detected. In a Gaussian white noise model, we discuss upper and lower bounds for the minimal size of the signal such that testing with small error probabilities is possible. In certain situations we are able to characterize the asymptotic minimax detection boundary. Our results are applied to inverse problems such as numerical differentiation, deconvolution and the inversion of the Radon transform. [ABSTRACT FROM AUTHOR]

Function approximation is a fundamental process in a variety of problems in computational mechanics, structural engineering, as well as other domains that require the precise approximation of a phenomenon with an analytic function. This work demonstrates a unified approach to these techniques, utilizing partial sums of the Taylor series in a high arithmetic precision. In particular, the proposed approach is capable of interpolation, extrapolation, numerical differentiation, numerical integration, solution of ordinary and partial differential equations, and system identification. The method employs Taylor polynomials and hundreds of digits in the computations to obtain precise results. Interestingly, some well-known problems are found to arise in the calculation accuracy and not methodological inefficiencies, as would be expected. In particular, the approximation errors are precisely predictable, the Runge phenomenon is eliminated, and the extrapolation extent may a priory be anticipated. The attained polynomials offer a precise representation of the unknown system as well as its radius of convergence, which provides a rigorous estimation of the prediction ability. The approximation errors are comprehensively analyzed for a variety of calculation digits and test problems and can be reproduced by the provided computer code.

Detailed derivation of the analytical, reciprocal-space approach of Hessian calculation within the self-consistent-charge density-functional based tight-binding framework (SCC-DFTB) is presented. This approach provides an accurate and efficient way for obtaining the SCC-DFTB Hessian of periodic systems. Its superiority with respect to the traditional numerical force differentiation method is demonstrated for doped graphene, graphene nanoribbons, boron–nitride nanotubes, bulk zinc-oxide, and other systems. [ABSTRACT FROM AUTHOR]

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The problem of correction of inconsistent systems of linear inequalities with a matrix block structure is relevant in the modern theory of correction of inconsistent models, such as incorrect constraint models in mathematical programming problems. The study of methods for correcting incorrect models is a relatively new branch of theoretical computer science. The purpose of this study is to consider a gradient approach to solving problems of inconsistent systems of linear inequalities with a matrix block structure, based on the criteria of the minimal sum of quadratic norms. For the gradient approach to solving problems of inconsistent systems of linear inequalities with a matrix block structure, correction (approximation) methods are used, and numerical differentiation methods are used to calculate the gradient of functions. As a result of the study, the effectiveness of the proposed gradient approach for solving problems of inconsistent systems of linear inequalities with the matrix block structure, the accuracy and reliability of solutions of systems of linear inequalities of the specified method were shown. The obtained results showed the multi-stage practice of solving applied problems, the inconsistency of the initial description of the object, the reduction of errors in experimental data, rounding errors, contradictions and ambiguity of information to incorrect models. They also demonstrated the accuracy and convergence speed of the proposed method, which indicated its effectiveness and practical applicability in various areas, where the accuracy and reliability of solutions to systems of linear inequalities are important, and the study of optimisation by researchers and the use of various solutions in solving the problem of incompatibility between systems of linear equations and inequalities. The practical significance of this study lies in the ability to solve problems of correction of inconsistent systems of linear inequalities with the matrix block structure in real economic, technical, and other applied problems. [ABSTRACT FROM AUTHOR]

The design of continuous thickeners and clarifiers is commonly based on the solid flux theory. Batch sedimentation experiments conducted with solid concentrations still provide useful information for their application. The construction of models for the velocity of settling allows the estimation of the flux of solids throughout time, which can, in turn, be used to find the area of the units required to achieve a given solid concentration in the clarified stream. This paper addresses the numerical treatment of data obtained from batch sedimentation experiments of calcium carbonate particles. We propose a systematic framework to fit a model that is capable of representing the process features that involve (i) the numerical differentiation of data to generate initial estimates for the instantaneous velocity of settling; (ii) the integration of a differential equation to fit the model for the velocity of settling; and (iii) the assessment of the quality of the fit using common statistical indicators. The model used for demonstration has a theoretical basis combined with an empirical component to account for the effect of the particle concentrations and their state of aggregation. The values of the numerical parameters obtained are related to the characteristic dimensions of the aggregates and their mass-length fractal dimensions. [ABSTRACT FROM AUTHOR]

Background: The contact force–deformation relation between two spheres is one fundamental property which determines the mechanical response of the colliding bodies. In general, collisions of solid spheres occur in a very short time, so a highly accurate and a non-contact measurement method is required. Objective: In this study, a concept for the experimental determination of contact force–deformation curves of two colliding spheres is presented. Thereby two identical steel spheres collide by using a pendular setup and the sphere's velocities are measured by Laser Doppler Vibrometers (LDVs). Methods: The displacements are obtained by integrating the velocity with time and the accelerations are obtained by differentiating the velocity with time. From these values, the deformation and the contact forces can be calculated. Then, the elastic results are compared with the Hertzian theory of impact and experiments are conducted with elasto-plastic spheres. Results: Although the deformations are in good agreement with these analytical values, the influence of numerical differentiation is observed in the contact force. However, it is shown that the elastic contact force–deformation curve of two colliding spheres can be obtained with reasonable accuracy by using LDVs. For elasto-plastic spheres the coefficient of restitution became smaller than for the purely elastic case, and the force–deformation curve with hysteresis are measured. Conclusions: The results of this study are considered to be reasonable as far as comparisons with those for elastic collisions. Therefore, more detailed verification by numerical analysis, such as finite element analysis, is desirable. [ABSTRACT FROM AUTHOR]

This study presents the development of a design method that has been extended to the design of radial-inflow turbines operating in organic Rankine cycles (ORC). Both the conventional design method and the circulation method available in the literature have been reviewed. The two main limitations of the current circulation method that make it not suitable for the ORC turbine design are the lack of real gas capability and 3D blades with high stresses. Using the circulation method, the flow field is decomposed into a potential part and a rotational part. The mean velocity field and the periodic velocity field are solved separately. To model the thermodynamic properties of the real gas, NIST REFPROP or CoolProp are used. The blade geometry is then solved iteratively by assuming that the velocity vector is parallel to the blade surface. The blade boundary condition is modified to force the blade camber to be radial-fibred, which is helpful to reduce the centrifugal bending stress on the blade. All the formulations are derived step by step, and the numerical treatments, including grid generation, numerical differentiation, computational scheme, and convergence, are discussed in detail. This method is validated by designing a R245fa ORC turbine rotor. The performance of the rotor design is predicted by CFD and FEA simulations, and it is compared to the results using other methodologies in the literature. [ABSTRACT FROM AUTHOR]

The depth of the aerosol layer at the Villum Research Station at Station Nord in the high Arctic is analysed based on 8 years of observations from a ceilometer and one full year from a wind lidar. The layer is of particular interest for aerosol process modelling and atmospheric chemistry studies. The depth of the aerosol layer is assigned to the inflection point in the attenuated backscatter profile by two methods; one is based on polynomial approximation of the profile and the other is direct numerical differentiation. The analysis is based on two types of hourly profiles; one consists of averaging the attenuated backscatter observations and the other by computing the median. Due to sporadic occurrence of outliers in the ranges around 50 m in the ceilometer observations, this part of the profile is not used in this study. Restricting the observations to heights above 100 m, the depths of the aerosol layer are found to be typically ≈230 m. It varies little between winter and summer, but the spread in the depth is larger during the winter as compared to summer. To extend the study of the aerosol‐layer depth below 100 m, a method is applied that combines the ceilometer measurements with the carrier‐to‐noise ratio from the wind lidar. The results are available for 2018 only, and they show aerosol‐layer depths below ≈80 m as well as depths around 230 m and they show few observations between ≈80 and ≈230 m. Near the ground, the observed backscatter exhibits a pronounced seasonal variation, having low values during the summer and high values during the winter. The strength of the seasonal variability decreases with height, especially above the aerosol‐layer depth, and is virtually absent at 1 km. [ABSTRACT FROM AUTHOR]

Anomaly detection is a significant issue that has attracted considerable research. The artificial immune system offers strong pattern recognition ability, adaptability and dynamic characteristics; therefore, it has been extensively used for anomaly detection. However, the boundary between normal and abnormal data patterns is difficult to define, which reduces the anomaly detection precisions of artificial immune approaches. Biological macrophages have a strong ability to identify various abnormalities, therefore, this study proposes a novel numerical differentiation-based artificial macrophage detection model (NDAMM) for anomaly detection. In particular, numerical differentiation is introduced in signal extraction, which can perceive signal changes more clearly and perform signal mapping. Furthermore, we design an artificial macrophage algorithm to determine weights based on input data and identify abnormalities using a signal fusion process. Finally, the proposed approach is implemented in anomaly detection. Through implementations on 20 anomaly detection datasets, the results of these experiments demonstrate that the NDAMM surpasses state-of-the-art anomaly detection methodologies. Ablation studies, parametric analysis, and statistical analysis are used to validate the effectiveness of our model. [ABSTRACT FROM AUTHOR]

This work proposes a new parameter identification method based on the Wentzel-Kramers-Brillouin (WKB) approximation for slow linear time-varying (LTV) dynamic systems. The considered time period is divided into a series of short time windows. In each time window, the assumption of "short time linearly varying" parameters is employed, and a nonlinear optimization problem is solved using the WKB results for the slow LTV dynamic system. A search algorithm is developed to find the optimal solution. In the identification process, only one type of response signal (displacement, velocity or acceleration) is required. Thus, numerical differentiation or integration of the measured signal, which leads to truncation or cumulative errors in noise environment, is avoided. The accuracy and robustness of the new identification method are validated by applying it to a particular LTV system with time-varying stiffness. [ABSTRACT FROM AUTHOR]

The ease of estimating the uncertainties of numerical simulations in metal forming is of particular interest. This uncertainty arises from, for example, material parameter identification, geometric dimensions, external loads, and contact conditions. In this paper, we aim to address this issue with the extension to geometric influences, and boundary as well as friction conditions. The uncertainty quantification from material parameter identification – here, in terms of sensitivities of the resulting simulations based on the confidence interval of the parameters – is transferred from the literature and the individual proportions are quantified and compared, respectively. For material parameter identification, experiments on steel and glass fiber reinforced plastic are used and the confidence intervals are determined. Particularly in the case of sequential determination of the parameters, the uncertainties are estimated with the aid of Gaussian error propagation. This concept can also be transferred to geometric dimensions or loads. The application of numerical differentiation for the sensitivities within the Gaussian error propagation leads to a concept where the finite element program can be treated as a black‐box. Here, all uncertainties of the simulation results are obtained, leading to the result that geometric influences in the deep drawing process used as well as the friction coefficient have the largest effect. This provides a very simple procedure for the uncertainty quantification of all individual influencing variables in any finite element simulation. [ABSTRACT FROM AUTHOR]

Effective pressures at the boundaries of the outer-, inner-, and shell-confined H atoms are investigated using the numerical differentiation technique in which the system energies are solved by employing the generalized pseudospectral method. High-precision inner and outer effective pressures are obtained, and the results are compared with previous predictions based on a formula from the virial equation of state. We find that although the virial-based formula is well-suited for estimating the pressure on outer-confined atoms, it is generally not applicable for the inner- and shell-confined situations. The asymptotic laws of effective pressures on the outer- and inner-confined H atoms as the confinement radius approaches zero and infinity, respectively, are analyzed. For the shell-confined H atom, the distinct behaviors of effective pressures at the inner and outer boundaries as well as for the s and non-s bound states are attributed to the different confinement effect of boundaries on both the system eigenenergies and wave functions. [ABSTRACT FROM AUTHOR]

In this paper, we modify the classical cubic B-spline quasi-interpolant Q 3 by utilizing multi-node higher-order expansions and the P t -exact A-discretization approach. The proposed quartic spline quasi-interpolant Q 3 1 possesses higher accuracy with small computational complexity increased, while requires no additional information of the target function. Applying the proposed quasi-interpolant, we develop high accuracy quadrature formulas and the differentiation matrices. All corresponding error estimates with explicit coefficients are provided. Numerical experiments illustrate the effectiveness of the proposed quasi-interpolants. [ABSTRACT FROM AUTHOR]

An elegant, well-established effective data filter concept, proposed originally by Abraham Savitzky and Marcel J.E. Golay, is undoubtedly a very effective tool, however not free from limitations and drawbacks. Despite the latter, over the years it has become a "monopolist" in many fields of spectra processing, claiming a "commercial" superiority over alternative approaches, which would potentially allow to obtain equivalent or in some cases even more reliable results. In order to show that basic operations performed on spectral datasets, like smoothing or differentiation, do not have to be equated to the application of the one particular single algorithm, several of such alternatives are briefly presented within this paper and discussed with regard to their practical realization. A special emphasis is put on the fast Fourier methodology (FFT), being widespread in the general domain of signal processing. Finally, a user-friendly Matlab routine, in which the outlined algorithms are implemented, is shared, so that one can select and apply the technique of spectral data processing more adequate for their individual requirements without the need to code it prior to use. [ABSTRACT FROM AUTHOR]

On principle of equivalent tensions of adhesive connection, the estimation of remaining tensions is conducted in coverage. For this purpose, graphic differentiation of experimental dependence of adhesive durability is utilized from the thickness of coverage. Functional connection is set between adhesive durability, thickness of coverage, critical deformation of basis and remaining tensions. It allows choosing the parameter When using electrical contact treatment after gas-thermal spraying of coatings, positive results were obtained: the value of residual stresses does not exceed 200 MPa; parts with gas-thermal coatings that have undergone electrocontact treatment are characterized by a smooth change in hardness from the coating to the surface of the part and an increased diffusion zones of the system «basis-coverage» at finish treatment and exploitation. the amount of pressure and the technological scheme of the electrical contact treatment process significantly affect the level of residual stresses and the adhesion strength of the coatings. From this point of view, preference should be given to the application of coatings by gas-thermal spraying with subsequent electrical contact treatment with a gradual increase in temperature and pressure of the process. The method of graphic differentiation of the experimental dependence of the adhesive strength on the thickness of the coating was used to determine the residual stresses in the coatings obtained by gas-thermal spraying with subsequent electrical contact treatment. According to the principle of equivalence of residual stresses and critical deformation of the part during peeling of the coating, a functional relationship of adhesive strength, thickness of the coating, critical deformation of the part and residual stresses was established. This dependence allows you to determine the characteristics of coated parts for their operating conditions. [ABSTRACT FROM AUTHOR]

In this paper, a second-order, unconditionally stable numerical scheme for the Cahn–Hilliard–Hele–Shaw system is constructed, which combines the backward differentiation formula in time and the mixed finite element method in space. Based on the Lagrange multiplier approach, we develop the linear discretization for the nonlinear term, which is extremely efficient. By rigorous proofs and calculations, we prove the unique solvability of the numerical solution, the unconditional stability and the error estimates of the proposed scheme. Furthermore, some numerical tests are given on the temporal and spatial convergence rates, spinodal decomposition and energy decay, which are consistent with the results of the theoretical analysis. [ABSTRACT FROM AUTHOR]