Normal Derivative Research Articles

Predicting the Electric Conductivity and Potassium Leaching of Coffee by NIR Spectroscopy Technique

This study aimed to investigate the relationship between the experimental values of electrical conductivity and potassium leaching and near-infrared (NIR) spectra of green coffee beans. Analyses of electrical conductivity and potassium leaching were performed on two hundred fifty samples of green coffee beans. The near-infrared spectra of these samples were collected. The spectra were subjected to pretreatments of multiplicative scatter correction (MSC), standard normal variate (SNV), first derivative, and second derivative. The multivariate statistical method of partial least square (PLS) regression was used in the elaboration of calibration models, which were later applied to forecast external samples. The results obtained demonstrate that good calibration models were developed. The best model obtained for estimating electrical conductivity showed a correlation of 0.97, while for potassium leaching, the maximum correlation between experimental and predicted values was 0.88. These results demonstrate good prospects for predicting the coffee bean cell membranes integrity through spectroscopic techniques.

Exact Green's formula for the fractional Laplacian and perturbations

Let Ω be an open, smooth, bounded subset of $ \mathbb{R}^n $. In connection with the fractional Laplacian $(-\Delta )^a$ ($a>0$), and more generally for a $2a$-order classical pseudodifferential operator (ψdo) $P$ with even symbol, one can define the Dirichlet value $\gamma _0^{a-1}u$, resp. Neumann value $\gamma _1^{a-1}u$ of $u(x)$, as the trace, resp. normal derivative, of $u/d^{a-1}$ on $\partial \Omega $, where $d(x)$ is the distance from $x\in \Omega $ to $\partial \Omega $; they define well-posed boundary value problems for $P$.
 A Green's formula was shown in a preceding paper, containing a generally nonlocal term $(B\gamma _0^{a-1}u,\gamma _0^{a-1}v)_{\partial \Omega }$, where $B$ is a first-order ψdo on $\partial \Omega $. Presently, we determine $B$ from $L$ in the case $P=L^a$, where $L$ is a strongly elliptic second-order differential operator. A particular result is that $B=0$ when $L=-\Delta $, and that $B$ is multiplication by a function (is local) when $L$ equals $-\Delta $ plus a first-order term. In cases of more general $L$, $B$ can be nonlocal.

The Nonsteady Boussinesq System with Mixed Boundary Conditions including Conditions of Friction Type

In this paper, we are concerned with the nonsteady Boussinesq system under mixed boundary conditions. The boundary conditions for fluid may include Tresca slip, leak and one-sided leak conditions, velocity, static (or total) pressure, rotation, and stress (or total stress) together, and the boundary conditions for temperature may include Dirichlet, Neumann, and Robin conditions together. Relying on the relations among strain, rotation, normal derivative of velocity, and shape of the boundary surface, we get variational formulation. The formulations consist of a variational inequality for velocity due to the boundary conditions of friction type and a variational equation for temperature. For the case of boundary conditions including the static pressure and stress, we prove that if the data of the problem are small enough and compatibility conditions at the initial instance are satisfied, then there exists a unique solution on the given interval. For the case of boundary conditions including the total pressure and total stress, we prove the existence of a solution without restriction on the data and parameters of the problem.

Quadrature Formula for the Direct Value of the Normal Derivative of the Single Layer Potential

A quadrature formula for the direct value of the normal derivative of the single layer potential with smooth density defined on a closed or nonclosed surface is obtained. Single layer potentials for the Laplace and Helmholtz equations are considered. Numerical tests confirm that our quadrature formula gives considerably higher accuracy than the standard quadrature formula. It can be used when numerically solving boundary value problems for the Laplace and Helmholtz equations by the potential method and the boundary integral equation method.

Domain Identification for Inverse Problem via Conformal Mapping and Fixed Point Methods in Two Dimensions

In this paper, we present a survey of the inverse eigenvalue problem for a Laplacian equation based on available Cauchy data on a known part Γ0 and a homogeneous Dirichlet condition on an unknown part Γ0 of the boundary of a bounded domain, Ω⊂ℝN. We consider variations in the eigenvalues and propose a conformal mapping tool to reconstruct a part of the boundary curve of the two-dimensional bounded domain based on the Cauchy data of a holomorphic function that maps the unit disk onto the unknown domain. The boundary values of this holomorphic function are obtained by solving a nonlocal differential Bessel equation. Then, the unknown boundary is obtained as the image of the boundary of the unit disk by solving an ill-posed Cauchy problem for holomorphic functions via a regularized power expansion. The Cauchy data were restricted to a nonvanishing function and to the normal derivatives without zeros. We prove the existence and uniqueness of the holomorphic function being considered and use the fixed-point method to numerically analyze the results of convergence. We’ll calculate the eigenvalues and compare the result with the shape obtained via minimization functional method, as developed in a previous study. Further, we’ll observe via simulations the shape of Γ and if it preserves its properties with varying the eigenvalues.

Existence and multiplicity of solutions for some Styklov problem involving p(x)-Laplacian operator

In this paper, we consider a class of -Laplacian problems of the form: where , is a bounded domain with Lipschitz boundary is outer unit normal derivative. The functions a, b, p, q, g and f are assumed to satisfy suitable assymptions. The existence and the multiplicity of solutions is obtained by using variational methods, and mountain pass lemma combined with Ekeland variational principle.

A Numerical Method for Boundary Value Problems for a Homogeneous Equation with the Squared Laplace Operator with the Use of Interpolating Wavelets

We present an effective numerical method for the recovery of biharmonic functions in a disk from continuous boundary values of these functions and of their normal derivatives using wavelets that are harmonic in the disk and interpolating on its boundary on dyadic rational grids. The expansions of solutions of boundary value problems into cumbersome interpolation series in the wavelet basis are replaced by sequences of their partial sums that are compactly presentable in the subspace bases of the corresponding multiresolution analysis (MRA) of Hardy spaces $$h_{\infty}(K)$$ of functions harmonic in the disk. Effective estimates are obtained for the approximation of solutions by partial sums of any order in terms of the best approximation of the boundary functions by trigonometric polynomials of a slightly smaller order. As a result, to provide the required accuracy of the representation of the unknown biharmonic functions, one can choose in advance the scaling parameter of the corresponding MRA subspace such that the interpolation projection to this space defines a simple analytic representation of the corresponding partial sums of interpolation series in terms of appropriate compressions and shifts of the scaling functions, skipping complicated iterative procedures for the numerical construction of the coefficients of expansion of the boundary functions into series in interpolating wavelets. We write solutions using interpolating and interpolating-orthogonal wavelets based on modified Meyer wavelets; the latter are convenient to apply if the boundary values of the boundary value problem are given approximately, for example, are found experimentally. In this case, one can employ the usual, well-known procedures of discrete orthogonal wavelet transformations for the analysis and refinement (correction) of the boundary values.

Volumetric discretization element with invariant displacements approximation in the SSS numerical analysis problems for the building structures

The article describes the sought quantities invariant approximation method in relation to the volumetric discretization element in the form of a parallelepiped with the main nodes located at its vertices. The displacement vector components and their partial derivatives of the first order were selected as the sought-for unknowns of the volume discretization element. An approximating expression when implementing the invariant approximation technique is written for the displacement vector of an element inner zone point and not for components of this vector isolated from each other. The approximating expressions obtained on the sought quantities’ invariant approximation basis contain the curvilinear coordinates system parameters used. The discretization volume element compatibility is improved due to the use of Lagrange multipliers in additional nodes located in the middle of the parallelepiped bases’ edges. The Lagrange correction factors ensure the displacement vectors normal components derivatives’ equality along the normals to the element base edges’ midpoints. The expanded stiffness matrix and the column of the volumetric discretization element nodal forces are formed by minimizing the conditional Lagrange functional.

Selective Turn‐On Aluminium Ions Detection of NBD(+) Appended Schiff‐Base Fluorophore

AbstractThe Al3+ sensor, 7‐nitrobenz‐2‐oxa‐1, 3‐diazole (NBD) derived fluorescent probe 2‐((E)‐(4‐(phenylamino)benzo[c][1,2,5]oxadiazol‐7‐ylimino)methyl) phenol (3) was synthesized and characterized by various spectral methods and crystallography. The compound (3) exhibits dual channel (fluorometric and colorimetric) sensor behaviour. It has high selectivity for Al3+ detection, produced significant fluorescence enhancement in the presence of Al3+ ion. It shows ‘‘turn off–on’’ fluorescent responses toward Al3+ in semi aqueous medium. Here, the correlative the properties of a photo sensor (NBD based schiff‘s base 3 with normal NBD derivatives 1, 2) towards Al3+ was studied. The solvent effect and cation sensing properties were investigated from ultraviolet spectroscopy and fluorescence spectroscopic methods. The facile synthetic probe 3 will alternative and open new opportunities in bio‐imaging of Al3+ in various cell lines. This will certainly be helpful in understanding the fundamental mechanism of aluminium‐induced human diseases.

Ambient residual penalty approximation of partial differential equations on embedded submanifolds

In this paper, we present a novel approach to the approximate solution of elliptic partial differential equations on compact submanifolds of $\mathbb {R}^{d}$ , particularly compact surfaces and the surface equation ${\Delta }_{\mathbb {M}} u - \lambda u=f$ . In the course of this, we reconsider differential operators on such submanifolds to deduce suitable penalty based functionals. These functionals are based on the residual of the equation in an integral representation, extended by a penalty on the first-order normal derivative. The general framework we develop is accompanied by error analysis and exemplified by numerical examples employing tensor product B-splines.

An Open Boundary Condition for High-order Solutions of Magnetohydrodynamics on Unstructured Grids

In this paper a characteristics-based open boundary condition (CBC) is proposed for the magnetohydrodynamic (MHD) system of equations. The algorithm is carefully designed and implemented in the context of a high-order flux reconstruction (FR) scheme under the Generalised Lagrange Multiplier (GLM)-MHD system of equations. It is implemented by adding the contribution of the characteristic equation directly to the corrected flux term in the FR scheme dispensing with solving time-dependent characteristic equations along boundary faces. The CBC method is shown to be more accurate and robust than commonly used zero normal derivative (ZND) and approximate Riemann solver boundary conditions (ARBC) in solving 1D, 2D, and 3D test problems. The CBC method is successfully applied to simulate challenging problems of magnetic reconnection for which other options failed to get stable results over long-period time integration.

Analogs of the Liouville Property for Harmonic Functions on Unbounded Domains

We obtain some analogs of the Liouville property for the function that is harmonic on the exterior of a Jordan domain $ G\subset{\mathbb{C}} $ and has constant boundary values of the function itself and its normal derivative. We show that these conditions cannot be relaxed in general.

Sparse Data Interpolation and Smoothing on Embedded Submanifolds

Energy minimization is one of the properties that make univariate splines so favorable in many problems of approximation and estimation; interpolation in and extrapolation from sparse data sites and smoothing of noisy data in particular. In this paper, we present a novel approach to approximate energy minimization on certain classes of submanifolds that gives rise to new methods for extrapolation and smoothing on submanifolds. To accomplish this, we minimize intrinsic functionals approximately by minimising a suitable extrinsic formulation of the functional augmented by a penalty on the first order normal derivative. The general framework we develop is accompanied by error analysis and exemplified by tensor product B-splines.

The Dirichlet problem for rectangle and new identities for elliptic integrals and functions

In the paper, results of comparison of two different methods of exact solution of the Dirichlet problem for rectangle are presented, namely, method of conformal mapping and method of variables’ separation. By means of this procedure normal derivative of Green’s function for rectangular domain was expressed via Jacobian elliptic functions. Under approaching to rectangle’s boundaries these formulas give new representations of the Dirac delta function. Moreover in the framework of suggested ideology a number of identities for the complete elliptic integral of the first kind were obtained. These formulas may be applied to summation of both numerical and functional series; also they may be useful for analytic number theory.

A high‐order harmonic polynomial method for solving the Laplace equation with complex boundaries and its application to free‐surface flows. Part I: Two‐dimensional cases

SummaryA high‐order harmonic polynomial method (HPM) is developed for solving the Laplace equation with complex boundaries. The “irregular cell” is proposed for the accurate discretization of the Laplace equation, where it is difficult to construct a high‐quality stencil. An advanced discretization scheme is also developed for the accurate evaluation of the normal derivative of potential functions on complex boundaries. Thanks to the irregular cell and the discretization scheme for the normal derivative of the potential functions, the present method can avoid the drawback of distorted stencils, that is, the possible numerical inaccuracy/instability. Furthermore, it can involve stationary or moving bodies on the Cartesian grid in an accurate and simple way. With the proper free‐surface tracking methods, the HPM has been successfully applied to the accurate and stable modeling of highly nonlinear free‐surface potential flows with and without moving bodies, that is, sloshing, water entry, and plunging breaker.

Rapid detection and quantification of sucrose adulteration in cow milk using Attenuated total reflectance-Fourier transform infrared spectroscopy coupled with multivariate analysis.

Adulteration of milk to gain economic benefit has become a common practice in recent years. Sucrose is illegally added in milk to reconstitute its compositional requirement by improving the total solid contents. The present study is aimed to use FTIR spectroscopy in combination with multivariate chemometric modelling for the differentiation and quantification of sucrose in cow milk. Pure milk and adulterated milk spectra (0.5-7.5% w/v) were observed in the spectral region 4000-400cm-1. Principal component analysis (PCA) was used for the discrimination of pure milk and adulterated milk. Soft independent modelling of class analogy (SIMCA) was able to classify test samples with a classification efficiency of 100%. Partial least square regression (PLS-R) and principle component regression (PCR) models were established for normal spectra, 1st derivative and 2nd derivative for the quantification of sucrose in milk. PLS-R model (normal spectra) in the combined wavenumber range of 1070-980cm-1 showed the best prediction based on parameters like coefficient of determination (R2) (Cal: 0.996; Val: 0.993), RMSE (Cal: 0.15% w/v; Val: 0.20% w/v), RE% (Cal: 4.9% w/v; Val: 5.1% w/v) and RPD (13.40). This method has a detection level of 0.5% w/v sucrose adulteration.

RETRACTED ARTICLE: Segmentation of spinal cord from computed tomography images based on level set method with Gaussian kernel

In the human body, organs segmentation is the most imperative issues in therapeutic applications. The challenges are connected with medicinal image segmentation and low complexity between required organ and incorporating tissues. There exist a wide range of methodologies for how a segmentation problem can be comprehended. These methods want to have a spot specific region of individual bones. The particular part remains a test for spinal cord segmentation. As a result of the beforehand expressed downsides of the current spinal cord segmentation procedures, this paper proposes a modified spatial fuzzy C clustering with level set segmentation method to incorporate Neumann Boundary Condition, a third function, called by the level set evolution. Neumann Boundary Condition is utilized to specify the normal derivative of the function present on any surface. The proposed method gives better results of segmentation of the spinal cord organs. The execution of the proposed method proves its superiority in term of accuracy as compared with the other methods.

On an inverse Robin spectral problem

We consider the problem of the recovery of a Robin coefficient on a part γ ⊂ ∂Ω of the boundary of a bounded domain Ω from the principal eigenvalue and the boundary values of the normal derivative of the principal eigenfunction of the Laplace operator with Dirichlet boundary condition on ∂Ω\\γ. We prove the uniqueness, as well as local Lipschitz stability of the inverse problem. Moreover, we present an iterative reconstruction algorithm with numerical computations in two dimensions showing the accuracy of the method.

Complete Asymptotics for Solution of Singularly Perturbed Dynamical Systems with Single Well Potential

We consider a singularly perturbed boundary value problem ( − ε 2 ∆ + ∇ V · ∇ ) u ε = 0 in Ω , u ε = f on ∂ Ω , f ∈ C ∞ ( ∂ Ω ) . The function V is supposed to be sufficiently smooth and to have the only minimum in the domain Ω . This minimum can degenerate. The potential V has no other stationary points in Ω and its normal derivative at the boundary is non-zero. Such a problem arises in studying Brownian motion governed by overdamped Langevin dynamics in the presence of a single attracting point. It describes the distribution of the points at the boundary ∂ Ω , at which the trajectories of the Brownian particle hit the boundary for the first time. Our main result is a complete asymptotic expansion for u ε as ε → + 0 . This asymptotic is a sum of a term K ε Ψ ε and a boundary layer, where Ψ ε is the eigenfunction associated with the lowest eigenvalue of the considered problem and K ε is some constant. We provide complete asymptotic expansions for both K ε and Ψ ε ; the boundary layer is also an infinite asymptotic series power in ε . The error term in the asymptotics for u ε is estimated in various norms.

Nonlinear stability and existence of compressible vortex sheets in 2D elastodynamics

The nonlinear stability and local existence of compressible vortex sheets for the two-dimensional isentropic elastic fluid are established in the usual Sobolev spaces. The problem has a characteristic free boundary, and the Kreiss–Lopatinskiĭ condition is satisfied only in a weak form. This paper completes the previous works [6,7] of the first three authors where the weakly linear stability of the rectilinear vortex sheets is proved by means of an upper triangularization technique. Our proof is based on certain higher-order energy estimates and an appropriate modification of the Nash–Moser iteration. In particular, the estimate for the normal derivatives of the characteristic variables can be recovered from that for the linearized divergences and vorticities.