Gaussian Formula Research Articles

Kinetic study of dry magnetic separation based on Gauss-Maxwell magnetic stress tensor: A 3D finite element method (FEM)

In this paper, two artificial magnetic particles (1#, 2#) with known magnetic parameters were taken as the objects, a finite element model based on Gauss’s law was established for the calculation of the transient Maxwell magnetic stress tensor on the surfaces of particles and kinetic study, a high-speed camera was used to obtain the motion behaviors of magnetic particles in comparison with the simulation. According to the results of multiple simulation-experiment comparisons, the motion behaviors of magnetic particles in the finite element simulation were consistent with experimental phenomena under the identical conditions, indicated that the accuracy of the model is reliable. In the comparison of two kinds of magnetic force calculations, the magnetic force FM based on the Gaussian formula had a similar tendency to FD based on the kinetic calculations, and since the FM was obtained by converting the surface tension of particles, it more accurately reflected the overall magnetic force and magnetic torque acting on the particles. Kinetic analysis showed that the magnetic force acting on a particle was strictly dependent on its magnetization, dynamic and non-uniform magnetization caused the magnetic particle to be subjected to magnetic force and magnetic torque in the non-uniform magnetic field, resulting in displacements and flips. In addition, compared to the particle release attitude, the influence of the distribution of magnetic substance and the particle’s release position on the displacement was particularly significant. The 3D finite element model established for dry magnetic separation can be further used for the study of dynamic, non-uniform magnetization and the force of magnetic particle or grain groups, which is of certain significance for the kinetic study of magnetic separation and improving research of magnetic separation equipment.

A Gaussian interval type-2 fuzzy characterization method based on heterogeneous big data and its application in forest ecological assessment

The under-forest economy is a new agricultural and forestry production mode. It can improve the utilization efficiency of forest land and make full use of forest resources, forest space, and forest ecological environment through different forms of industries such as understory planting and breeding. The under-forest economy has important ecological and economic benefits, significant to rural revitalization and industrial structure adjustment. It has been widely discussed by people in recent years. The forest ecological security level is a core factor to be considered in selecting sites for the under-forest economy. In practice, some forest ecological security evaluation indexes usually show heterogeneous forms. To comprehensively evaluate the level of forest ecological security, this study constructs a new multi-indicator fuzzy evaluation method based on heterogeneous big data. Firstly, a new linguistic scale function is defined to realize the consistent conversion of multi-granularity linguistic information. A Gaussian interval type-2 fuzzy aggregation formula for heterogeneous large-scale data is proposed. This formula is used to describe the differences in evaluation information between individuals caused by different knowledge backgrounds of evaluators. Then, the Gaussian interval type-2 fuzzy cross-entropy is defined, and it is verified to satisfy the excellent properties. This cross-entropy is applied to determine the objective weights of evaluation indexes. Considering the risk-averse psychology of experts and combining the prospect theory, the MULTIMOORA method in the Gaussian interval type-2 fuzzy environment is proposed. Finally, the applicability of the constructed Gaussian interval type-2 fuzzy comprehensive evaluation method (GIT2FCE) is verified through a case study conducted on the site selection of the under-forest economic industry.

Molecular-Scale Liquid Density Fluctuations and Cavity Thermodynamics.

Equilibrium density fluctuations at the molecular level produce cavities in a liquid and can be analyzed to shed light on the statistics of the number of molecules occupying observation volumes of increasing radius. An information theory approach led to the conclusion that these probabilities should follow a Gaussian distribution. Computer simulations confirmed this prediction across various liquid models if the size of the observation volume is not large. The reversible work required to create a cavity and the chance of finding no molecules in a fixed observation volume are directly correlated. The Gaussian formula for the latter probability is scrutinized to derive the changes in enthalpy and entropy, which arise from the cavity creation. The reversible work of cavity creation has a purely entropic origin as a consequence of the solvent-excluded volume effect produced by the inaccessibility of a region of the configurational space. The consequent structural reorganization leads to a perfect compensation of enthalpy and entropy changes. Such results are coherent with those obtained from Lee in his direct statistical mechanical study.

Accuracy of Ten Intraocular Lens Formulas in Spherical Equivalent of Toric Intraocular Lens Power Calculation.

The aim of this work is to evaluate the accuracy of the Barrett Universal II (BU II), Emmetropia verifying optical (EVO) 2.0, Haigis, Hoffer Q, Hoffer QST (Savini/Taroni) (HQST), Holladay 1, Kane, Ladas Super, Sanders-Retzlaff-Kraff/theoretical (SRK/T), and T2 intraocular lens (IOL) power formulas for calculating spherical equivalent (SE) of toric IOL. This study enrolled consecutive patients who underwent phacoemulsification and toric IOL implantation at the Eye Hospital of Wenzhou Medical University in Hangzhou from 2015 to 2022. We compared the new-generation formulas with Gaussian optics-based standard formulas, and calculated the mean absolute error (MAE), median absolute error (MedAE), and percentage of eyes within ± 0.25 diopter (D), ± 0.50D, ± 0.75D and ± 1.00D of the target refraction. Subgroup analyses were conducted based on the anterior chamber depth (ACD), keratometry (K), and toricity (T). A total of 207 eyes of 207 patients were included in this study. Overall, the Kane and EVO2.0 formulas demonstrated the lowest MedAEs. The EVO2.0 formula exhibited the highest percentage of eyes within ± 0.50D, ± 0.75D, ± 1.00D. Moreover, the EVO2.0 formula showed the lowest MedAE for flat K subgroup, the highest percentage of eyes within ± 0.50D, ± 1.00D for shallow ACD subgroup, the highest percentage of eyes within ± 0.75D for regular ACD, flat K, T2-T3, T4-T5 subgroups. The Kane and formula performed the lowest MedAE in the T4-T5 subgroup. Application of the Kane and EVO2.0 formulas significantly improved the prediction of postoperative SE outcome for toric IOL compared to the other formulas.

Extended homogeneous bivariate orthogonal polynomials: symbolic and numerical Gaussian cubature formula

Extended homogeneous bivariate orthogonal polynomials: symbolic and numerical Gaussian cubature formula

Gauss, Mean and Total Curvature Formulae of Rational Bezier Curves in Minkowski 4-Space

In this paper Gaussian, Mean and total curvature formulae of Rational Bezier Curves with asymptotic frame field are calculated by using its curvatures in 3- dimensional lightlike cone in Minkowski 4- Space. Our main intention is to introduce and investigate some differential geometric properties of the the Rational Bezier Curves with asymptotic frame field in 3- dimensional lightlike cone in Minkowski 4- Space by using its curvatures.

Minkowski functionals of CMB polarization intensity with pynkowski: theory and application to Planck and future data

ABSTRACT The angular power spectrum of the cosmic microwave background (CMB) anisotropies is a key tool to study the Universe. However, it is blind to the presence of non-Gaussianities and deviations from statistical isotropy, which can be detected with other statistics such as Minkowski functionals (MFs). These tools have been applied to CMB temperature and E-mode anisotropies with no detection of deviations from Gaussianity and isotropy. In this work, we extend the MF formalism to the CMB polarization intensity, P2 = Q2 + U2. We use the Gaussian kinematic formula to derive the theoretical predictions of MFs for Gaussian isotropic fields. We develop a software that computes MFs on P2healpix maps and apply it to simulations to verify the robustness of both theory and methodology. We then estimate MFs of P2 maps from Planck, both in pixel space and needlet domain, comparing them with realistic simulations that include CMB and instrumental noise residuals. We find no significant deviations from Gaussianity or isotropy in Planck CMB polarization intensity. However, MFs could play an important role in the analysis of CMB polarization measurements from upcoming experiments with improved sensitivity. Therefore, we forecast the ability of MFs applied to P2 maps to detect much fainter non-Gaussian anisotropic signals than with Planck data for two future complementary experiments: the LiteBIRD satellite and the ground-based Simons Observatory. We publicly release the software to compute MFs in arbitrary scalar healpix maps as a fully documented python package called pynkowski.

Confidence tubes for curves on SO(3) and identification of subject-specific gait change after kneeling

Abstract In order to identify changes of gait patterns, e.g. due to prolonged occupational kneeling, which might be a major risk factor for the development of knee osteoarthritis, we develop confidence tubes for curves following a perturbation model on SO(3) using the Gaussian kinematic formula which are equivariant under gait similarities and have precise coverage even for small sample sizes. Applying them to gait curves from eight volunteers undergoing kneeling tasks and adjusting for different walking speeds and marker replacement at different visits, allows us to identify at which phases of the gait cycle the gait pattern changed due to kneeling.

Error estimates for a Gaussian rule involving Bessel functions

This paper deals with the estimation of the quadrature error of a Gaussian formula for weight functions involving powers, exponentials and Bessel functions of the first kind. For this purpose, in this work the averaged and generalized averaged Gaussian rules are employed, together with a tentative a priori approximation of the error. The numerical examples confirm the reliability of these approaches.

Prediction of total corneal power in keratoconus using anterior surface data

ABSTRACT Clinical relevance Keratoconus results in an increase in anterior and posterior curvatures and a reduction in corneal thickness. Anterior corneal ectasia is partially compensated by remodelling the corneal epithelium. Therefore, there is an alteration in the relationship between corneal surfaces and variation in corneal power. The variation in corneal power is one of the sources that induces errors in IOL power calculation. Background This study aimed to assess a method for predicting total corneal power in keratoconus using several anterior surface parameters at 3 mm and 4 mm. Methods Tomographic data obtained using Pentacam (Oculus, Germany) were analysed from 280 eyes of 140 patients with keratoconus using anterior and posterior keratometry, anterior Q-value at 8 mm, central corneal thickness, Kmax location and value, and true net power at 4 mm (TNP). Calculated total corneal power (TCPc) at 3 mm was obtained using the Gauss formula. Predicted total corneal power at 3 mm (TCPp3) and 4 mm (TCPp4) was obtained from univariate (TCPp3u and TCPp4u) and multivariate linear regression formulae (TCPp3m and TCPp4m). SimK, anterior Q-value, vertical location, and Kmax value were used in the multivariate formulae. Mean absolute error (MAE) and median absolute error (MedAE) were also calculated. Absolute frequencies within dioptric ranges of all formulas divided for keratoconus grading were evaluated. Results TCPc and TNP exhibited a good correlation (R2 = 0.58, p < 0.05) with a higher dispersion above 50 D of corneal power. Highly significant correlations were observed between TCPp3u and TCPc (R2 = 0.978, p < 0.05) and TCPp3m and TCPc (R2 = 0.989, p < 0.05). Lower but significant correlations were observed between TCPp4u and TNP (R2 = 0.692, p < 0.05) and between TCPp4m and TNP (R2 = 0.887, p < 0.05). The best results for TCP prediction at 3 and 4 mm were obtained with TCPp3m and TCPp4m as follows: MAE of TCPp3m was 0.24 ± 0.20 (SD) D with MedAE of 0.20 D, while MAE of TCPp4m was 0.96 ± 0.77 D with MedAE of 0.80 D. The 3 mm multivariate regression formula results in higher absolute frequencies of prediction errors in the total eyes within 0.5 D (93%) than the univariate formula (81%). At 4mm, the multivariate regression formula has a lower percentage within 0.5 D (32%) than the univariate formula (41%), but the percentage of the multivariate formula is higher within 1 D (63%) than the univariate formula (56%). Conclusion All formulas show a decrease in accuracy with increasing grades of keratoconus. Multivariate linear regression formulae using only anterior surface data can predict TCP with good approximation in eyes with keratoconus in cases where posterior surface parameters are unavailable. The vertical location of Kmax and the anterior asphericity could play a relevant role in the prediction of total corneal power in keratoconus.

A new characterization of Aminov surface with regards to its Gauss map in E^4

In this work, we focus on Aminov surface with regard to its Gauss map in E^4. Firstly, we write the covariant derivatives according to linear combinations of orthonormal vectors and separate the equalities using Gauss and Weingarten formulas. Then, we get the laplace of the Gauss map. After giving some conditions, we yield as main results: Aminov surfaces can not have harmonic Gauss map and can not have pointwise one-type Gauss map of I. kind in E^4. Further, we give an example of helical cylinder which is also congruent to an Aminov surface. Lastly, we obtain the conditions of having pointwise one-type Gauss map of II. kind.

On the Maximum Principle for Optimal Control Problems of Stochastic Volterra Integral Equations with Delay

In this paper, we prove both necessary and sufficient maximum principles for infinite horizon discounted control problems of stochastic Volterra integral equations with finite delay and a convex control domain. The corresponding adjoint equation is a novel class of infinite horizon anticipated backward stochastic Volterra integral equations. Our results can be applied to discounted control problems of stochastic delay differential equations and fractional stochastic delay differential equations. As an example, we consider a stochastic linear-quadratic regulator problem for a delayed fractional system. Based on the maximum principle, we prove the existence and uniqueness of the optimal control for this concrete example and obtain a new type of explicit Gaussian state-feedback representation formula for the optimal control.

On trigonometric interpolation in an even number of points

In contrast to odd-length trigonometric interpolants, even-length trigonometric interpolants need not be unique; this is apparent from the representation of the interpolant in the (real or complex) Fourier basis, which possesses an extra degree of freedom in the choice of the highest-order basis function in the even case. One can eliminate this degree of freedom by imposing a constraint, but then the interpolant may cease to exist for certain choices of the interpolation points. On the other hand, the Lagrange representation developed by Gauss always produces an interpolant despite having no free parameters. We discuss the choice Gauss's formula makes for the extra degree of freedom and show that, when the points are equispaced, its choice is optimal in the sense that it minimizes both the standard and 2-norm Lebesgue constants for the interpolation problem. For non-equispaced points, we give numerical evidence that Gauss's formula is no longer optimal and consider interpolants of minimal 2-norm instead. We show how to modify Gauss's formula to produce a minimal-norm interpolant and that, if the points are equispaced, no modification is necessary.

Barycentric rational interpolation method for solving time-dependent fractional convection-diffusion equation

&lt;abstract&gt;&lt;p&gt;The time-dependent fractional convection-diffusion (TFCD) equation is solved by the barycentric rational interpolation method (BRIM). Since the fractional derivative is the nonlocal operator, we develop a spectral method to solve the TFCD equation to get the coefficient matrix as a full matrix. First, the fractional derivative of the TFCD equation is changed to a nonsingular integral from the singular kernel to a density function. Second, efficient quadrature of the new Gauss formula are constructed to simply compute it. Third, matrix equation of discrete the TFCD equation is obtained by the unknown function replaced by a barycentric rational interpolation basis function. Then, the convergence rate of BRIM is proved. Finally, a numerical example is given to illustrate our result.&lt;/p&gt;&lt;/abstract&gt;

Barycentric rational interpolation method for solving fractional cable equation

&lt;abstract&gt;&lt;p&gt;A fractional cable (FC) equation is solved by the barycentric rational interpolation method (BRIM). As the fractional derivative is a nonlocal operator, we develop a spectral method to solve the FC equation to get the coefficient matrix as the full matrix. First, the fractional derivative of the FC equation is changed to a nonsingular integral from the singular kernel to the density function. Second, an efficient quadrature of a new Gauss formula is constructed to compute it simply. Third, a matrix equation of the discrete FC equation is obtained by the unknown function replaced by a barycentric rational interpolation basis function. Then, convergence rate for FC equation of the BRIM is derived. At last, a numerical example is given to illustrate our results.&lt;/p&gt;&lt;/abstract&gt;

Rational symbolic cubature rules over the first quadrant in a Cartesian plane

In this paper we introduce a new symbolic Gaussian formula for theevaluation of an integral over the first quadrant in a Cartesianplane, in particular with respect to the weight function$w(x)=\exp(-x^T x-1/x^T x)$, where $x=(x_1,x_2)^T\in \mathbb{R}^2_+$. Itintegrates exactly a class of homogeneous Laurent polynomials withcoefficients in the commutative field of rational functions in twovariables. It is derived using the connection between orthogonalpolynomials, two-point Pad&eacute; approximants, and Gaussian cubatures.We also discuss the connection to two-point Pad&eacute;-typeapproximants in order to establish symbolic cubature formulas ofinterpolatory type. Numerical examples are presented toillustrate the different formulas developed in the paper.

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Biquaternion generalizations of Maxwell's and Dirac's equations and properties of their solutions

The differential algebra of biquaternions is used to construct generalized solutions of the biquaternion wave (biquaternion) equation, special cases of which are the systems of equations of Maxwell and Dirac. The method of generalized functions is used to construct solutions and study their properties. Fundamental and generalized solutions are constructed, an analogue of the Kirchhoff formula for the wave equation, which gives a solution to the Cauchy problem. A dynamic analog of the Gauss formula is obtained, which gives a solution to the biwave equation in a limited region under known boundary and initial conditions for the desired biquaternion. Using the constructed analogues of the Kirchhoff and Green's formula, one can obtain analogues of the Green's formula under nonzero initial conditions by decomposing the solution of Eq. (1) into two biquaternions, one of which satisfies the initial conditions. Conditions on the boundary for the second biquaternion will be obtained using the boundary values of the original biquaternion and the first constructed one. After that, formulas similar to (30) give an integral representation of the second biquaternion. Since the equations of Maxwell and Dirac are a special case of biwave equations, the constructed solutions can be used for problems of electrodynamics and field theory. They can be used in experiments, so the field characteristics of EM fields at the boundary can be measured experimentally without solving the SHIE. The constructed solutions can be used in field theory, elementary particles and electro-gravimagnetic interactions.

Maximum ground level concentration for Gaussian plume formula using standard and Briggs methods

AbstractThis work aims to determine the maximum concentration and downwind distance at the earth's surface in two methods Briggs and standard for two conditions are slightly unstable and neutral by using the Gaussian equation at different effective heights and wind seeped at (8.9, 8.5 and 8 m/s) under lateral and vertical dispersion coefficient measurement.

Surface Reconstruction from Point Clouds without Normals by Parametrizing the Gauss Formula

We propose Parametric Gauss Reconstruction (PGR) for surface reconstruction from point clouds without normals. Our insight builds on the Gauss formula in potential theory, which represents the indicator function of a region as an integral over its boundary. By viewing surface normals and surface element areas as unknown parameters, the Gauss formula interprets the indicator as a member of some parametric function space. We can solve for the unknown parameters using the Gauss formula and simultaneously obtain the indicator function. Our method bypasses the need for accurate input normals as required by most existing non-data-driven methods, while also exhibiting superiority over data-driven methods, since no training is needed. Moreover, by modifying the Gauss formula and employing regularization, PGR also adapts to difficult cases such as noisy inputs, thin structures, sparse or nonuniform points, for which accurate normal estimation becomes quite difficult. Our code is publicly available at https://github.com/jsnln/ParametricGaussRecon .