Second Derivative Research Articles

Estimating two groups of fracture weaknesses using azimuthal differences in partially incidence-angle-stacked seismic amplitudes

Geophysical and geologic data, e.g., results obtained using rock cores, outcrops, and borehole images, reveal the presence of multiple sets of fractures in rock. To analyze how fracture interactions affect seismic anisotropy and dispersion, we consider an effective model that contains two sets of orthogonal fractures. In seismic amplitude analysis, we focus on the case of subsurface target zones containing a fracture network composed of primary and secondary fracture sets. Focusing on gas-bearing rocks with small fracture densities, we formulate simplified stiffness parameters in terms of two groups of fracture weaknesses, which are related to the primary and secondary fractures, respectively. Using the simplified stiffness parameters, we derive the PP-wave reflection coefficient and the azimuthal elastic impedance (AEI) as functions of the two groups of normal and tangential fracture weaknesses. Based on the derived PP-wave reflection coefficient and AEI, we propose a method and workflow in which the azimuthal variations of the partial incidence-angle-stacked seismic data are used to estimate the AEI and differences in AEI (which we refer to as DEI) is input to a nonlinear inversion for two groups of fracture weaknesses. In the nonlinear inversion, the initial values of fracture weaknesses are obtained based on a two-term approximation of the PP-wave reflection coefficient. First- and second-order derivatives of DEI with respect to fracture weakness parameters are calculated to generate the update in the unknown parameter vector. Synthetic seismic gathers with varying signal-to-noise ratios are used to validate the robustness of the estimation. Applying the inversion method to a real data set, we obtain what we interpret to be reliable fracture weakness estimates that produce azimuthal amplitude differences matching those extracted from real seismic data. The results provide a potential tool for determining if two sets of fractures have developed in a reservoir.

Precise diagnosis of lung cancer enabled by improved FTIR-based machine learning

Micro-morphologic lesions of lung cancers (LC) are usually related to the developed tumors, lack of sensitivity to early lung cancer and specificity to tissue lesions of different origins due to similarity detected by medical imaging techniques. Attenuated total Reflection-Fourier transform infrared spectroscopy (ATR-FTIR) using accessible blood samples has identified many biomolecular markers and characteristic spectral changes, which is related to early cancer diagnosis. In this study, for precise diagnosis of LCs, the in-situ serum-based ATR-FTIR spectroscopy experiments were carried out with 156 serum samples divided into 5 different types of LC groups and controls. The peak-to-peak area ratios between different functional groups contributed to the discovery of potential FTIR markers of LCs. Moreover, we also constructed a two-dimensional second-order derivative spectral (2D-SD-IR) feature dataset based on infrared molecular fingerprints (IMF) of LCs, including absorbance and wave number shifts of FTIR vibrational peaks for LC early diagnosis with low time cost-effectiveness and high classification accuracy. By comparing, the diagnosis models of partial least square discriminant analysis (PLS-DA) and back propagation (BP) neural network based on 2D-SD-IR are suitable to differentiate LCs and pathological stages accurately, and the high sensitivity and specificity of BP reached up to 98.7%. The proposed 2D-SD-IR machine learning protocol for LCs’ classification demonstrates the great advantage of aid clinical blood biochemical testing.

Qualitative and Quantitative Assessments of Apple Quality Using Vis Spectroscopy Combined with Improved Particle-Swarm-Optimized Neural Networks.

Exploring a cost-effective and high-accuracy optical detection method is of great significance in promoting fruit quality evaluation and grading sales. Apples are one of the most widely economic fruits, and a qualitative and quantitative assessment of apple quality based on soluble solid content (SSC) was investigated via visible (Vis) spectroscopy in this study. Six pretreatment methods and principal component analysis (PCA) were utilized to enhance the collected spectra. The qualitative assessment of apple SSC was performed using a back-propagation neural network (BPNN) combined with second-order derivative (SD) and Savitzky-Golay (SG) smoothing. The SD-SG-PCA-BPNN model's classification accuracy was 87.88%. To improve accuracy and convergence speed, a dynamic learning rate nonlinear decay (DLRND) strategy was coupled with the model. After that, particle swarm optimization (PSO) was employed to optimize the model. The classification accuracy was 100% for testing apples via the SD-SG-PCA-PSO-BPNN model combined with a Gaussian DLRND strategy. Then, quantitative assessments of apple SSC values were performed. The correlation coefficient (r) and root-square-mean error for prediction (RMSEP) in testing apples were 0.998 and 0.112 °Brix, surpassing a commercial fructose meter. The results demonstrate that Vis spectroscopy combined with the proposed synthetic model has significant value in qualitative and quantitative assessments of apple quality.

Global existence for a class of non-equilibrium reaction–diffusion systems with flux limitation

For a class of general reaction–diffusion systems with flux limitation in one dimension, when homogeneous Neumann boundary conditions are considered, we will prove global upper and lower boundness estimates for the positive classical solutions and boundness for their first-order spatial derivatives. In particular, the non-equilibrium radiation flux-limited diffusion coupled with material heat conduction model is included as a special case of the general reaction–diffusion systems. The key point of our method is to at first prove that the second-order spatial derivative of the solution of the second diffusion equation can be linearly controlled by the first-order spatial derivative of the solution of the flux-limited diffusion equation. Then, with this control, we can apply the Bernstein method to the flux-limited diffusion equation, while a stratifying time technique is introduced to get a complete first order spatial derivative estimate. With these global prior estimates, the global existence and uniqueness of positive classical solutions for the general flux-limited diffusion systems are proved.

Organic Matter Retrieval in Black Soil Based on Oblique Extremum Signatures

How to extract the indicative signatures from the spectral data is an important issue for further retrieval based on remote sensing technique. This study provides new insight into extracting indicative signatures by identifying oblique extremum points, rather than local extremum points traditionally known as absorption points. A case study on retrieving soil organic matter (SOM) contents from the black soil region in Northeast China using spectral data revealed that the oblique extremum method can effectively identify weak absorption signatures hidden in the spectral data. Moreover, the comparison of retrieval outcomes using various indicative signature extraction methods reveals that the oblique extremum method outperforms the correlation analysis and traditional extremum methods. The experimental findings demonstrate that the radial basis function (RBF) neural network retrieval model exposes the nonlinear relationship between reflectance (or reflectance transformation results) and the SOM contents. Additionally, an improved oblique extremum method based on the second-order derivative is provided. Overall, this research presents a novel perspective on indicative signature extraction, which could potentially offer better retrieval performance than traditional methods.

Massive Quantum Vortices in Superfluids

We consider the dynamical properties of quantum vortices with filled massive cores, hence the term “massive vortices”. While the motion of massless vortices is described by first-order motion equations, the inclusion of core mass introduces a second-order time derivative in the motion equations and thus doubles the number of independent dynamical variables needed to describe the vortex. The simplest possible system where this physics is present, i.e. a single massive vortex in a circular domain, is thoroughly discussed. We point out that a massive vortex can exhibit various dynamical regimes, as opposed to its massless counterpart, which can only precess at a constant rate. The predictions of our analytical model are validated by means of numerical simulations of coupled Gross-Pitaevskii equations, which indeed display the signature of the core inertial mass. Eventually, we discuss a nice formal analogy between the motion of massive vortices and that of massive charges in two-dimensional domains pierced by magnetic fields.

3D static bending analysis of functionally graded piezoelectric microplates resting on an elastic medium subjected to electro-mechanical loads using a size-dependent Hermitian C 2 finite layer method based on the consistent couple stress theory

Based on the consistent couple stress theory (CCST), we develop a size-dependent Hermitian C 2 finite layer method (FLM) for carrying out the three-dimensional (3D) static bending analysis of a simply-supported, functionally graded (FG) piezoelectric microplate which is placed under closed-circuit surface conditions. The microplate of interest is assumed to be resting on a Winkler-Pasternak foundation and subjected to either sinusoidal or uniformly distributed electro-mechanical loads. By setting the material length scale parameter at zero and ignoring the piezoelectric and flexoelectric effects, we reduce the formulation of the Hermitian C 2 FLM for analyzing FG piezoelectric microplates to that for analyzing FG piezoelectric macroplates and FG elastic microplates, respectively. The accuracy and the convergence rate of the Hermitian C 2 FLM are assessed by comparing the solutions it produces with the exact and approximate 3D solutions of FG piezoelectric macroplates and FG elastic microplates reported in the literature. Because the Hermitian C 2 FLM requires that the first-order and second-order derivatives of the primary variables must be continuous at each nodal plane, which in turn leads to their solutions converging rapidly and being able to obtain the accurate results of the electric and elastic variables induced in the microplates, especially for the transverse shear and normal stresses and the electric displacements. The effects of piezoelectricity, flexoelectricity, and the material length scale parameter on the deformations and stresses induced in the FG piezoelectric microplates are significant. Highlights A CCST-based Hermitian C 2 finite layer method (FLM) is developed for analyzing functionally graded (FG) piezoelectric microplates. The current FLM can be reduced to that for analyzing FG elastic microplates by ignoring the piezoelectric and flexoelectric effects. The current FLM can be reduced to that for analyzing FG piezoelectric macroplates by setting the material length scale parameter at zero. Implementing the current FLM reveals that its solutions converge rapidly and are in excellent agreement with those obtained using the exact 3D models. The effects of piezoelectricity, flexoelectricity, and the material length scale parameter on the static bending behavior of FG piezoelectric microplates are significant.

Noise Estimation for MRI Images with Revised Theory on Histograms of Second-order Derivatives

Previous research by the author in the use of histograms of second-order derivatives showed that the differences between pixels in MRI images can be determined without referring to the ground truth for the purpose of noise reduction. Yet, the results of the previous research also showed that the methodologies used could not prevent false positives and negatives. To address this problem, a technique has been developed to involve multiple conditions that utilize the statistics of the histograms and the circumstances of neighbours in the vicinity of a pixel. The confusion matrix for this method shows that the technique has marginal but consistent improvement across the noise levels that are tested, compared to prior methods

On Hermite–Hadamard-Type Inequalities for Functions Satisfying Second-Order Differential Inequalities

Hermite–Hadamard inequality is a double inequality that provides an upper and lower bounds of the mean (integral) of a convex function over a certain interval. Moreover, the convexity of a function can be characterized by each of the two sides of this inequality. On the other hand, it is well known that a twice differentiable function is convex, if and only if it admits a nonnegative second-order derivative. In this paper, we obtain a characterization of a class of twice differentiable functions (including the class of convex functions) satisfying second-order differential inequalities. Some special cases are also discussed.

Unified discontinuous Galerkin finite-element framework for transient conjugated radiation-conduction heat transfer.

Research on conjugated radiation-conduction (CRC) heat transfer in participating media is of vital scientific and engineering significance due to its extensive applications. Appropriate and practical numerical methods are essential to forecast the temperature distributions during the CRC heat-transfer processes. Here, we established a unified discontinuous Galerkin finite-element (DGFE) framework for solving transient CRC heat-transfer problems in participating media. To overcome the mismatch between the second-order derivative in the energy balance equation (EBE) and the DGFE solution domain, we rewrite the second-order EBE as two first-order equations and then solve both the radiative transfer equation (RTE) and the EBE in the same solution domain, resulting in the unified framework. Comparisons between the DGFE solutions with published data confirm the accuracy of the present framework for transient CRC heat transfer in one- and two-dimensional media. The proposed framework is further extended to CRC heat transfer in two-dimensional anisotropic scattering media. Results indicate that the present DGFE can precisely capture the temperature distribution at high computational efficiency, paving the way for a benchmark numerical tool for CRC heat-transfer problems.

Two crucial suggestions on tool paths in slow-tool-servo diamond cutting of micro-structured functional surfaces

In ultra-precision machining (UPM) of micro-structured functional surfaces (μ-SFS), tool servo dynamic errors (tool servo tracking error and tool servo induced vibration) would deteriorate surface quality as two crucial issues. At the machined surface, the tool servo tracking error manifests as surface fluctuation due to its phase shift, and the tool servo induced vibration presents with periodical patterns. To eliminate or reduce the tool servo dynamic errors, theoretical and experimental results innovatively reveal that: (1) for the mathematical prerequisite the tool path should be at least fourth-order differentiable (snap), and (2) for the physical mechanism the second-order derivative of tool path (required acceleration) should be less than the rated acceleration provided by the tool servo. It is of significance that the study draws up two crucial suggestions on the tool path in UPM of μ-SFS to effectively improve surface quality.

Costa’s concavity inequality for dependent variables based on the multivariate Gaussian copula

AbstractAn extension of Shannon’s entropy power inequality when one of the summands is Gaussian was provided by Costa in 1985, known as Costa’s concavity inequality. We consider the additive Gaussian noise channel with a more realistic assumption, i.e. the input and noise components are not independent and their dependence structure follows the well-known multivariate Gaussian copula. Two generalizations for the first- and second-order derivatives of the differential entropy of the output signal for dependent multivariate random variables are derived. It is shown that some previous results in the literature are particular versions of our results. Using these derivatives, concavity of the entropy power, under certain mild conditions, is proved. Finally, special one-dimensional versions of our general results are described which indeed reveal an extension of the one-dimensional case of Costa’s concavity inequality to the dependent case. An illustrative example is also presented.

Second-Order Derivative Spectrophotometric Method for Simultaneous Estimation of Azilsartan Medoxomil and Cilnidipine in Combined Dosage Form

Second-Order Derivative Spectrophotometric Method for Simultaneous Estimation of Azilsartan Medoxomil and Cilnidipine in Combined Dosage Form

An effective high-order five-point stencil, based on integrated-RBF approximations, for the first biharmonic equation and its applications in fluid dynamics

PurposeThe purpose of this paper is to present a new discretisation scheme, based on equation-coupled approach and high-order five-point integrated radial basis function (IRBF) approximations, for solving the first biharmonic equation, and its applications in fluid dynamics.Design/methodology/approachThe first biharmonic equation, which can be defined in a rectangular or non-rectangular domain, is replaced by two Poisson equations. The field variables are approximated on overlapping local regions of only five grid points, where the IRBF approximations are constructed to include nodal values of not only the field variables but also their second-order derivatives and higher-order ones along the grid lines. In computing the Dirichlet boundary condition for an intermediate variable, the integration constants are used to incorporate the boundary values of the first-order derivative into the boundary IRBF approximation.FindingsThese proposed IRBF approximations on the stencil and on the boundary enable the boundary values of the derivative to be exactly imposed, and the IRBF solution to be much more accurate and not influenced much by the RBF width. The error is reduced at a rate that is much greater than four. In fluid dynamics applications, the method is able to capture well the structure of steady highly non-linear fluid flows using relatively coarse grids.Originality/valueThe main contribution of this study lies in the development of an effective high-order five-point stencil based on IRBFs for solving the first biharmonic equation in a coupled set of two Poisson equations. A fast rate of convergence (up to 11) is achieved.

Структурный анализ функции управления динамической системой в частных производных разного порядка

For a dynamic system described by a partial differential equation of different order, the problem of constructing software control in an analytical form is solved. The primary method of research is the cascade decomposition method, the improved algorithm of which includes three main stages: the forward move, the central stage and the reverse move. The method is based on the properties of the matrix coefficient with a second-order partial derivative of the control function. The noetherianness of the coefficient causes the splitting of the original space into direct sums of subspaces. A scheme for structuring subspaces in accordance with the properties of the matrix coefficient is given. The direct move of decomposition is realized, which consists in a step-by-step transition to equivalent hierarchically structured systems of two levels in subspaces. The step-by-step structuring of the components of the state function into functions from subspaces while using matrix coefficients - projectors is performed. The resulting functions from subspaces are called pseudo-state and pseudo-control functions. Graphical visualization of the hierarchical structure of the source space in the form of a diagram is performed. This scheme reflects the essential connections between the components of the subspaces of each decomposition level. Finite-dimensional spaces are considered, which causes the complete completion of the first stage of the algorithm in a finite number of steps not exceeding the dimension of the original space. During the decomposition, the conditions at the start and end points are reduced, so that at the end of each step of the forward stroke, one additional condition appears at each point for each second-level system. In the process of implementing the first stage of the algorithm, the properties of matrix coefficients entailing full controllability or unmanageability of the initial system are established; the properties of functions in the initial conditions necessary for the implementation of the controlled process are also revealed. The criterion of complete controllability of the initial system is deduced. For a fully controlled system, a transition is made to the central stage of the algorithm – the construction of a defining basis function that satisfies all additional conditions for partial derivatives in time at each point resulting from the reduction of the original ones. It is the presence of this defining basic function that lays the prerequisites for constructing the state and control functions of the initial system at the final stage of the algorithm. A diagram visualizing the procedure of step-by-step restoration of the components of the state function in the process of implementing the reverse course is given. The reverse is completed by explicitly constructing first the state function, then the control function. A qualitative analysis of the management structure of the system under study is carried out.

Analytical Formulation of the Second-Order Derivative of Energy for the Orbital-Optimized Variational Quantum Eigensolver: Application to Polarizability.

We develop a quantum-classical hybrid algorithm to calculate the analytical second-order derivative of the energy for the orbital-optimized variational quantum eigensolver (OO-VQE), which is a method to calculate eigenenergies of a given molecular Hamiltonian by utilizing near-term quantum computers and classical computers. We show that all quantities required in the algorithm to calculate the derivative can be evaluated on quantum computers as standard quantum expectation values without using any ancillary qubits. We validate our formula by numerical simulations of quantum circuits for computing the polarizability of the water molecule, which is the second-order derivative of the energy, with respect to the electric field. Moreover, the polarizabilities and refractive indices of thiophene and furan molecules are calculated as a test bed for possible industrial applications. We finally analyze the error scaling of the estimated polarizabilities obtained by the proposed analytical derivative versus the numerical derivative obtained by the finite difference. Numerical calculations suggest that our analytical derivative requires fewer measurements (runs) on quantum computers than the numerical derivative to achieve the same fixed accuracy.

Noncollinear and Spin-Flip TDDFT in Multicollinear Approach

Time-dependent density functional theory (TDDFT) is one of the most important tools for investigating the excited states of electrons. The TDDFT calculation for spin-conserving excitation, where collinear functionals are sufficient, has obtained great success and has become routine. However, TDDFT for noncollinear and spin-flip excitations, where noncollinear functionals are needed, is less widespread and still a challenge nowadays. This challenge lies in the severe numerical instabilities that root in the second-order derivatives of commonly used noncollinear functionals. To be free from this problem radically, noncollinear functionals with numerical stable derivatives are desired, and our recently developed approach, called the multicollinear approach, provides an option. In this work, the multicollinear approach is implemented in noncollinear and spin-flip TDDFT, and prototypical tests are given.

Evaluation of Hyperspectral Monitoring Model for Aboveground Dry Biomass of Winter Wheat by Using Multiple Factors

The aboveground dry biomass (AGDB) of winter wheat can reflect the growth and development of winter wheat. The rapid monitoring of AGDB by using hyperspectral technology is of great significance for obtaining the growth and development status of winter wheat in real time and promoting yield increase. This study analyzed the changes of AGDB based on a winter wheat irrigation experiment. At the same time, the AGDB and canopy hyperspectral reflectance of winter wheat were obtained. The effect of spectral preprocessing algorithms such as reciprocal logarithm (Lg), multiple scattering correction (MSC), standardized normal variate (SNV), first derivative (FD), and second derivative (SD); sample division methods such as the concentration gradient method (CG), the Kennard–Stone method (KS), and the sample subset partition based on the joint X–Y distances method (SPXY); sample division ratios such as 1:1 (Ratio1), 3:2 (Ratio2), 2:1 (Ratio3), 5:2 (Ratio4), and 3:1 (Ratio5); dimension reduction algorithms such as uninformative variable elimination (UVE); and modeling algorithms such as partial least-squares regression (PLSR), stepwise multiple linear regression (SMLR), artificial neural network (ANN), and support vector machine (SVM) on the hyperspectral monitoring model of winter wheat AGDB was studied. The results showed that irrigation can improve the AGDB and canopy spectral reflectance of winter wheat. The spectral preprocessing algorithm can change the original spectral curve and improve the correlation between the original spectrum and the AGDB of winter wheat and screen out the bands of 1400 nm, 1479 nm, 1083 nm, 741 nm, 797 nm, and 486 nm, which have a high correlation with AGDB. The calibration sets and validation sets divided by different sample division methods and sample division ratios have different data-distribution characteristics. The UVE method can obviously eliminate some bands in the full-spectrum band. SVM is the best modeling algorithm. According to the universality of data, the better sample division method, sample division ratio, and modeling algorithm are SPXY, Ratio4, and SVM, respectively. Combined with the original spectrum and by using UVE to screen bands, a model with stable performance and high accuracy can be obtained. According to the particularity of data, the best model in this study is FD-CG-Ratio4-Full-SVM, for which the R2c, RMSEc, R2v, RMSEv, and RPD are 0.9487, 0.1663 kg·m−2, 0.7335, 0.3600 kg·m−2, and 1.9226, respectively, which can realize hyperspectral monitoring of winter wheat AGDB. This study can provide a reference for the rational irrigation of winter wheat in the field and provide a theoretical basis for monitoring the AGDB of winter wheat by using hyperspectral remote sensing technology.

Partition functions on squashed seven-spheres and holography

Our paper presents two main results. First, we study the renormalized free energies of Euclidean Einstein gravity in asymptotically AdS8 and various field theories on a squashed seven sphere. In the gravity theory, we demonstrate the absence of the Hawking-Page transition, while in the field theory, we focus on the O(N) vector model and the massless free fermion model. The conformal symmetry governs the universal behaviors of the free energies for small and large squashings, which we confirm numerically and analytically. Second, we evaluate the second-order derivative of CFT free energy with respect to the squashing parameter, finding universal results that hold for generic conformal field theories. We examine two different squashings, one with an SU(2) bundle, which is the primary focus of our paper, and another with a U(1) bundle, where our results align with the conjectured formula from the gravity side in the literature.

Study on Hyperspectral Monitoring Model of Total Flavonoids and Total Phenols in Tartary Buckwheat Grains.

Tartary buckwheat is a common functional food. Its grains are rich in flavonoids and phenols. The rapid measurement of flavonoids and phenols in buckwheat grains is of great significance in promoting the development of the buckwheat industry. This study, based on multiple scattering correction (MSC), standardized normal variate (SNV), reciprocal logarithm (Lg), first-order derivative (FD), second-order derivative (SD), and fractional-order derivative (FOD) preprocessing spectra, constructed hyperspectral monitoring models of total flavonoids content and total phenols content in tartary buckwheat grains. The results showed that SNV, Lg, FD, SD, and FOD preprocessing had different effects on the original spectral reflectance and that FOD can also reflect the change process from the original spectrum to the integer-order derivative spectrum. Compared with the original spectrum, MSC, SNV, Lg, FD, and SD transformation spectra can improve the correlation between spectral data and total flavonoids and total phenols in varying degrees, while the correlation between FOD spectra of different orders and total flavonoids and total phenols in grains was different. The monitoring models of total flavonoids and total phenols in grains based on MSC, SNV, Lg, FD, and SD transformation spectra achieved the best accuracy under SD and FD transformation, respectively. Therefore, this study further constructed monitoring models of total flavonoids and total phenols content in grains based on the FOD spectrum and achieved the best accuracy under 1.6 and 0.6 order derivative preprocessing, respectively. The R2c, RMSEc, R2v, RMSEv, and RPD were 0.8731, 0.1332, 0.8384, 0.1448, and 2.4475 for the total flavonoids model, and 0.8296, 0.2025, 0.6535, 0.1740, and 1.6713 for the total phenols model. The model can realize the rapid measurement of total flavonoids content and total phenols content in tartary buckwheat grains, respectively.