Normal Derivative Research Articles

Selecting near-infrared reflection spectroscopy pretreatment methods by chemical components valid and invalid absorption wavebands

An appropriate pretreatment method is crucial to establish a reliable model for near-infrared spectroscopy. It is usually selected by comparing model performance with different pretreatment methods, such as the root mean square error of internal cross-validation or the root mean square error of prediction. When the statistical indexes of the models are similar and indistinguishable, or the results of model calibration and cross-validation are too different, indicating an overfitting situation, it may be due to the selection of an inappropriate pretreatment. In this paper, an approach is proposed to select appropriate pretreatment methods by chemical components valid and invalid absorption wavebands. Due to there being no chemical groups absorption in wavebands of 1250–1350 nm and 1780–1850 nm, they are considered chemical invalid absorption bands. The root mean square distance in the invalid wavebands characterizes the difference in samples’ physical state, such as particle size and instrument error. Except for the invalid absorption band, other wavebands in the near-infrared region are considered chemical valid absorption wavebands. The root mean square distance in the valid absorption wavebands mainly characterize the chemical composition differences between different samples. So, we define effective information rate of spectra. The larger effective information rate of spectra, the more favorable it is to establish a quantitative model. In this study, spectra of 60 wheat powder samples were used to compare seven pretreatment methods. Moreover, the average effective information rates of spectra were calculated for the seven pretreatment methods respectively. The results show that the combination of standard normal variate transformation and first derivative with Savitzky–Golay smoothing is the best. Finally, a comparison of model prediction with different pretreatment methods was made. The results are the same as the former. Therefore, using the effective information rate of spectra for selecting an appropriate near-infrared spectroscopy pretreatment method is practical and can be used not only for near-infrared modeling but also for designing near-infrared instruments as an index for optimization.

An inequality for the normal derivative of the Lane–Emden ground state

Abstract We consider Lane–Emden ground states with polytropic index 0 ≤ q - 1 ≤ 1 {0\leq q-1\leq 1} , that is, minimizers of the Dirichlet integral among L q {L^{q}} -normalized functions. Our main result is a sharp lower bound on the L 2 {L^{2}} -norm of the normal derivative in terms of the energy, which implies a corresponding isoperimetric inequality. Our bound holds for arbitrary bounded open Lipschitz sets Ω ⊂ ℝ d {\Omega\subset\mathbb{R}^{d}} , without assuming convexity.

On the application of the Fourier method to solve the problem of correction of thermographic images

The work is devoted to the construction of computational algorithms implementing the method of correction of thermographic images. The correction is carried out on the basis of solving some ill-posed mixed problem for the Laplace equation in a cylindrical region of rectangular cross-section. This problem corresponds to the problem of the analytical continuation of the stationary temperature distribution as a harmonic function from the surface of the object under study towards the heat sources. The cylindrical region is bounded by an arbitrary surface and plane. On an arbitrary surface, a temperature distribution is measured (and thus is known). It is called a thermogram and reproduces an image of the internal heat-generating structure. On this surface, which is the boundary of the object under study, convective heat exchange with the external environment of a given temperature takes place, which is described by Newton’s law. This is the third boundary condition, which together with the first boundary condition corresponds to the Cauchy conditions - the boundary values of the desired function and its normal derivative. The problem is ill-posed. In this paper, using the Tikhonov regularization method, an approximate solution of the problem was obtained, stable with respect to the error in the Cauchy data, and which can be used to build effective computational algorithms. The paper considers algorithms that can significantly reduce the amount of calculations.

Deformation-induced coupling of the generalized external actions in third-gradient materials

In this study, diverse typologies of external actions are outlined, which turn out to be admissible for the third-gradient modeling of elastic materials. It is shown how such loading, when prescribed over the boundary surface, along the border edges and at the wedges of a deformable body in the Eulerian configuration, can be transformed into the Lagrangian description generating multiple interactions, with a surprising deformation-induced coupling. Such a phenomenon becomes more and more important at increasing the order of the beta -forces, specified by duality as covectors expending work on the beta th normal derivative of the virtual displacements, being herein at most beta =2. Insights are provided into the true nature of such generalized forces, resting on the differential geometric features of the deformation process.

Hybrid High-Order and Weak Galerkin Methods for the Biharmonic Problem

We devise and analyze two hybrid high-order (HHO) methods for the numerical approximation of the biharmonic problem. The methods support polyhedral meshes, rely on the primal formulation of the problem, and deliver $O(h^{k+1})$ $H^2$-error estimates when using polynomials of order $k\ge0$ to approximate the normal derivative on the mesh (inter)faces. Both HHO methods hinge on a stabilization in the spirit of Lehrenfeld--Sch\"oberl for second-order PDEs. The cell unknowns are polynomials of order $(k+2)$ that can be eliminated locally by means of static condensation. The face unknowns approximating the trace of the solution on the mesh (inter)faces are polynomials of order $(k+1)$ in the first HHO method which is valid in dimension two and uses an original stabilization involving the canonical hybrid finite element, and they are of order $(k+2)$ for the second HHO method which is valid in arbitrary dimension and uses only $L^2$-orthogonal projections in the stabilization. A comparative discussion with the weak Galerkin methods from the literature is provided, highlighting the close connections and the improvements proposed herein. Additionally, we show how the two HHO methods can be combined with a Nitsche-like boundary-penalty technique to weakly enforce the boundary conditions. An originality in the devised Nitsche's technique is to avoid any penalty parameter that must be large enough. Finally, numerical results showcase the efficiency of the proposed methods and indicate that the HHO methods can generally outperform discontinuous Galerkin methods and even be competitive with $C^0$-interior penalty methods on triangular meshes.

Skeleton-stabilized divergence-conforming B-spline discretizations for incompressible flow problems of high Reynolds number

We consider a stabilization method for divergence-conforming B-spline discretizations of the incompressible Navier–Stokes problem wherein jumps in high-order normal derivatives of the velocity field are penalized across interior mesh facets. We prove that this method is pressure robust, consistent, and energy stable, and we show how to select the stabilization parameter appearing in the method so that excessive numerical dissipation is avoided in both the cross-wind direction and in the diffusion-dominated regime. We examine the efficacy of the method using a suite of numerical experiments, and we find the method yields optimal L2 and H1 convergence rates for the velocity field, eliminates spurious small-scale structures that pollute Galerkin approximations, and is effective as an Implicit Large Eddy Simulation (ILES) methodology.

Study on Characteristic Raman Shift Screening Method Based on MPA for Raman Spectrum of Mine Water Inrush Source

In the process of mine water inrush disaster prevention, accurate and rapid identification of water inrush source type is of great significance to coal mine safety production. However, traditional hydrochemical methods have shortcomings such as time-consuming and complex detection. Therefore, a new idea of identifying mine water inrush source by Raman spectroscopy is proposed. Goaf water, roof sandstone fissure water, Ordovician limestone water, Taiyuan limestone water, and surface water as well as their mixed water samples are selected as research objects, and Raman spectral data of different water samples are collected by the Raman spectroscopy system. To eliminate the influence of laser power fluctuation and spectrometer system noise in Raman spectrum acquisition, detrend correction (DC), multiplicative scatter correction (MSC), standard normal variate transformation (SNV), first derivative (FD), and mean centering (MC) were used to preprocess the raw Raman spectra. Due to the large dimension and long analysis time of Raman spectrum data, the marine predator algorithm (MPA) is used to screen the characteristic Raman shifts of the Raman spectrum of water samples, and the characteristic Raman shift information that can best characterize the mine water samples is obtained. Finally, to verify the feasibility of MPA screening the characteristic Raman shifts of Raman spectrum of mine water inrush source, the selected characteristic Raman displacement information is used as input to construct BP neural network (BP), k-nearest neighbor algorithm (KNN), support vector machine (SVM), and decision tree (DT) classification models, respectively. Experiments show that SNV has the best preprocessing effect on the raw Raman spectrum, which can effectively eliminate part of the noise in the Raman spectrum data and improve the accuracy of Raman spectrum identification. Using MPA, 226 characteristic Raman shifts can be screened from 2048 Raman data points, reducing the number of Raman shifts to 11.04%, and the modeling accuracy of characteristic Raman shift information screened by MPA is higher than that of full Raman data. In addition, the average analysis speed of BP, KNN, SVM, and DT water source identification models is 7.61 times faster than that of all Raman data. The results show that MPA is adopted to screen the characteristic Raman displacement of mine water source Raman spectrum, which can effectively reduce the redundancy of Raman spectral data and greatly improve the speed of Raman spectral analysis, which is of great significance to ensure the real-time detection of the mine water source.

Uniqueness for fractional nonsymmetric diffusion equations and an application to an inverse source problem

In this article, we discuss a solution to time‐fractional diffusion equation with the homogeneous Dirichlet boundary condition, where an elliptic operator is not necessarily symmetric. We prove that the solution is identically zero if its normal derivative with respect to the operator vanishes on an arbitrarily chosen subboundary of the spatial domain over a time interval. The proof is based on the Laplace transform and the spectral decomposition for a nonsymmetric elliptic operator. As a direct application, we prove the uniqueness result for an inverse problem on determining the spatial component in the source term by Neumann boundary data on subdoundary.

A novel form of imperfect contact laws in flexoelectricity

Flexoelectric materials exhibit coupling between electric polarization and mechanical strain gradient. At the nano-scale and micro-scale, these materials offer a promising potential for the development of mechanical transducers and energy harvesters. This work proposes a new imperfect interface model simulating the behavior of a thin flexoelectric layer (adhesive) of vanishing thickness, embedded between two flexoelectric media (adherents). The adhesive is assumed to be mechanically compliant and electrically lowly-conducting. The interface model is obtained by using the asymptotic analysis. The contact laws, expressed in terms of the jumps and means values of the displacements, normal derivatives of the displacements, and electric potential across the interface, represent a formal generalization of the soft elastic and piezoelectric interface conditions. A simple application, considering a one-dimensional three-layer composite micro-bar subject to electro-mechanical loads, is developed in order to analytically/numerically assess the asymptotic model. Nonlocal phenomena and end-effects, related to a flexoelectric length-scale parameter, are highlighted. The example illustrates the usefulness of the proposed approach toward the design of thin nano- and microscale devices exploiting the flexoelectric effect.

Finite energy Navier-Stokes flows with unbounded gradients induced by localized flux in the half-space

For the Stokes system in the half space, Kang [Math. Ann. 331 (2005), pp. 87-109] showed that a solution generated by a compactly supported, Hölder continuous boundary flux may have unbounded normal derivatives near the boundary. In this paper we first prove explicit global pointwise estimates of the above solution, showing in particular that it has finite global energy and its derivatives blow up everywhere on the boundary away from the flux. We then use the above solution as a profile to construct solutions of the Navier-Stokes equations which also have finite global energy and unbounded normal derivatives due to the flux. Our main tool is the pointwise estimates of the Green tensor of the Stokes system proved by us in an earlier paper. We also examine the Stokes flows generated by dipole bumps’ boundary flux, and identify the regions where the normal derivatives of the solutions tend to positive or negative infinity near the boundary.

Rectifiability of divergence-free fields along invariant 2-tori

We find conditions under which the restriction of a divergence-free vector field B to an invariant toroidal surface S is rectifiable; namely constant in a suitable global coordinate system. The main results are similar in conclusion to Arnold’s Structure Theorems but require weaker assumptions than the commutation [B,nabla times B] = 0. Relaxing the need for a first integral of B (also known as a flux function), we assume the existence of a solution u: S rightarrow {mathbb {R}} to the cohomological equation Bvert _S(u) = partial _n B on a toroidal surface S mutually invariant to B and nabla times B. The right hand side partial _n B is a normal surface derivative available to vector fields tangent to S. In this situation, we show that the field B on S is either identically zero or nowhere zero with Bvert _S/Vert BVert ^2 vert _S being rectifiable. We are calling the latter the semi-rectifiability of B (with proportionality Vert BVert ^2 vert _S). The nowhere zero property relies on Bers’ results in pseudo-analytic function theory about a generalised Laplace-Beltrami equation arising from Witten cohomology deformation. With the use of de Rham cohomology, we also point out a Diophantine integral condition where one can conclude that Bvert _S itself is rectifiable. The rectifiability and semi-rectifiability of Bvert _S is fundamental to the so-called magnetic coordinates, which are central to the theory of magnetically confined plasmas.

Modified Problems for Euler–Darboux Equations with Parameters with Absolute Values Equal to $$ \frac{1}{2} $$

We consider the Euler–Darboux equation with parameters such that their absolute values are equal to \( \frac{1}{2} \). Since the Cauchy problem in the classical formulation is ill-posed for such values of parameters, we propose formulations and solutions of modified Cauchy-type problems with the following values of parameters:(a) \( \alpha =\beta =\frac{1}{2}; \)(b) \( \alpha =-\frac{1}{2}\ \mathrm{and}\ \beta =\frac{1}{2}; \)(c) \( \alpha =\beta =-\frac{1}{2}. \)In case (a), the modified Cauchy problem is solved by the Riemann method. We use the obtained result to formulate the analog of the problem Δ1 in the first quadrant with translated boundary conditions on axes and nonstandard conjunction conditions on the singularity line of the coefficients of the equation y = x. The first condition is gluing normal derivatives of the solution and the second one contains limit values of a combination of the solution and its normal derivatives. The problem is reduced to a uniquely solvable system of integral equations.

The application of statistical preprocessing on spectral data does not always guarantee the improvement of the predictive quality of multivariate models: Case of soil spectroscopy applied to Moroccan soils

Statistical preprocessing methods are widely used to improve the predictive performance of chemometric models especially on MIR and NIR spectral database. The main role of pretreatments in the case of soil spectroscopy is to remove/reduce the scattering impact, thus, highlight the part of the signal related to the chemical, physical or biological properties of interest. Nevertheless, given the complexity of the soil as a matrix, together the pure absorption data and the information hidden in the scattering are of key interest as they cooperatively describe the physicochemical state of the soil samples. Consequently, pretreatment methods that removes/ reduces the scattering material in the spectroscopic data may lower the predictive quality of the multivariate models. The purpose of this study was to explore the effect of preprocessing methods on FTIR spectra of soil samples and test the hypothesis that the use of scatter correction techniques as pretreatment (Viz. standard normal variate, Savitzky-Golay 1st and 2nd derivatives, and the multiplicative scatter correction) does not guarantee the improvement of the predictive performance of partial least squares regression models for the prediction of total carbon, organic carbon and total nitrogen in soil samples. The obtained results showed that among all the multivariate calibrations, the PLS models set-up on the unprocessed spectral data led to similar/better predictive qualities for the estimation of selected soil properties especially for total carbon with an R2 of 0.92 and RMSECV of 0.129, either because the information contained in the scattering background is important for the predictions or there is no scatter in the spectral data. Hence, FTIR spectroscopy as a simple, fast and nondestructive analytical method that in many cases will not require a supplementary effort when performing chemometric modeling by avoiding the preprocessing step could be highly recommended for soil health indicators prediction.

Radial solutions for a Neumann problem with mean curvature on exterior domain

We are concerned with the Neumann problem with Minkowski-curvature operator (P) where Ω is an exterior domain in and denotes the normal interior derivative on . We have shown the existence of radial positive solutions of (P) when g satisfies some suitable assumptions. The proof of our main result is based upon the shooting method.

NUMERICAL METHOD FOR SOLVING THE DIRICHLET BOUNDARY VALUE PROBLEM FOR NONLINEAR TRIHARMONIC EQUATION

In this work, we consider the Dirichlet boundary value problem for nonlinear triharmonic equation. Due to the reduction of the problem to operator equation for the pair of the right hand side function and the unknown second normal derivative of the function to be sought, we design an iterative method at both continuous and discrete levels for numerical solution of the problem. Some examples demonstrate that the numerical method is of fourth order convergence. When the right hand side function does not depend on the unknown function and its derivatives, the numerical method gives more accurate results in comparison with the results obtained by the interior method of Gudi and Neilan.

Rapid and Low-Cost Detection of Millet Quality by Miniature Near-Infrared Spectroscopy and Iteratively Retaining Informative Variables.

Traditional chemical methods for testing the fat content of millet, a widely consumed grain, are time-consuming and costly. In this study, we developed a low-cost and rapid method for fat detection and quantification in millet. A miniature NIR spectrometer connected to a smartphone was used to collect spectral data from millet samples of different origins. The standard normal variate (SNV) and first derivative (1D) methods were used to preprocess spectral signals. Variable selection methods, including bootstrapping soft shrinkage (BOSS), the variable iterative space shrinkage approach (VISSA), iteratively retaining informative variables (IRIV), iteratively variable subset optimization (IVSO), and competitive adaptive reweighted sampling (CARS), were used to select characteristic wavelengths. The partial least squares regression (PLSR) algorithm was employed to develop the regression models aimed at predicting the fat content in millet. The results showed that the proposed 1D-IRIV-PLSR model achieved optimal accuracy for fat detection, with a correlation coefficient for prediction (Rp) of 0.953, a root mean square error for prediction (RMSEP) of 0.301 g/100 g, and a residual predictive deviation (RPD) of 3.225, by using only 18 characteristic wavelengths. This result highlights the feasibility of using this low-cost and high-portability assessment tool for millet quality testing, which provides an optional solution for in situ inspection of millet quality in different scenarios, such as production lines or sales stores.

Totally geodesic discs in bounded symmetric domains

In this paper, we characterize $$C^2$$ -smooth totally geodesic isometric embeddings $$f:\Omega \rightarrow \Omega '$$ between bounded symmetric domains $$\Omega $$ and $$\Omega '$$ which extend $$C^1$$ -smoothly over some open subset in the Shilov boundaries and have nontrivial normal derivatives on it. In particular, if $$\Omega $$ is irreducible, there exist totally geodesic bounded symmetric subdomains $$\Omega _1$$ and $$\Omega _2$$ of $$\Omega '$$ such that $$f = (f_1, f_2)$$ maps into $$\Omega _1\times \Omega _2\subset \Omega $$ where $$f_1$$ is holomorphic and $$f_2$$ is anti-holomorphic totally geodesic isometric embeddings. If $$\text {rank}(\Omega ')<2\text {rank}(\Omega )$$ , then either f or $${\bar{f}}$$ is a standard holomorphic embedding.

Photoacoustic inversion formulas using mixed data on finite time intervals* *Funding: This work has been supported by the Austrian Science Fund (FWF), Project P 30747-N32.

We study the inverse source problem in photoacoustic tomography (PAT) for mixed data, which denote a weighted linear combination of the acoustic pressure and its normal derivative on an observation surface. We consider in particular the case where the data are only available on finite time intervals, which accounts for real-world usage of PAT where data are only feasible within a certain time interval. Extending our previous work, we derive explicit formulas up to a smoothing integral on convex domains with a smooth boundary, yielding exact reconstruction for circular or elliptical domains. We also present numerical reconstructions of our new exact inversion formulas on finite time intervals and compare them with the reconstructions of our previous formulas for unlimited time wave measurements.

An unfitted finite element method using level set functions for extrapolation into deformable diffuse interfaces

We explore a new way to handle flux boundary conditions imposed on level sets. The proposed approach is a diffuse interface version of the shifted boundary method (SBM) for continuous Galerkin discretizations of conservation laws in embedded domains. We impose the interface conditions weakly and approximate surface integrals by volume integrals. The discretized weak form of the governing equation has the structure of an immersed boundary finite element method. That is, integration is performed over a fixed fictitious domain. Source terms are included to account for interface conditions and extend the boundary data into the complement of the embedded domain. The calculation of these extra terms requires (i) construction of an approximate delta function and (ii) extrapolation of embedded boundary data into quadrature points. We accomplish these tasks using a level set function, which is given analytically or evolved numerically. A globally defined averaged gradient of this approximate signed distance function is used to construct a simple map to the closest point on the interface. The normal and tangential derivatives of the numerical solution at that point are calculated using the interface conditions and/or interpolation on uniform stencils. Similarly to SBM, extrapolation of data back to the quadrature points is performed using Taylor expansions. Computations that require extrapolation are restricted to a narrow band around the interface. Numerical results are presented for elliptic, parabolic, and hyperbolic test problems, which are specifically designed to assess the error caused by the numerical treatment of interface conditions on fixed and moving boundaries in 2D.

Variational Approach for a Robin Problem Involving Non Standard Growth Conditions

In this manuscript, we study a Robin problem driven by the p(x)-Laplacian with two parameters. −div(|∇w|p(x)−2∇w)=λV(x)|w|q(x)−2w,x∈Q, |∇w|p(x)−2∂w∂n+θ(x)|w|p(x)−2=βV1(x)|w|r(x)−2w.x∈∂Q. Here, Q is a regular bounded domain in RN, λ,β&gt;0, p,q are continuous functions on Q¯, ∂w∂n is the outer unit normal derivative on ∂Q, θ∈L∞(∂Q), such that essinfx∈∂Qθ(x)&gt;0,V is an indefinite function in Ls(x)(Q) and V1 is a non-negative one in Ls1(x)(∂Q). Using variational tools, we show the existence of a non-trivial solution.