scholarly journals Expressions for the second-order mixed partial derivatives of Slater-Koster matrix elements at spherical coordinate singularities

2005 ◽  
Vol 72 (16) ◽  
Author(s):  
Matthias Meister
Keyword(s):  
Second Order ◽  
Matrix Elements ◽  
Spherical Coordinate ◽  
Partial Derivatives ◽  
Derivatives Of ◽  
Coordinate Singularities

scholarly journals Simpson’s Second-Type Inequalities for Co-Ordinated Convex Functions and Applications for Cubature Formulas

2022 ◽  
Vol 6 (1) ◽  
pp. 33
Author(s):  
Sabah Iftikhar ◽  
Samet Erden ◽  
Muhammad Aamir Ali ◽  
Jamel Baili ◽  
Hijaz Ahmad
Keyword(s):  
Convex Functions ◽  
Cubature Formula ◽  
Applied Mathematics ◽  
Second Order ◽  
Absolute Value ◽  
Partial Derivatives ◽  
Cubature Formulas ◽  
Inequality Theory ◽  
Type Integral ◽  
Derivatives Of

Inequality theory has attracted considerable attention from scientists because it can be used in many fields. In particular, Hermite–Hadamard and Simpson inequalities based on convex functions have become a cornerstone in pure and applied mathematics. We deal with Simpson’s second-type inequalities based on coordinated convex functions in this work. In this paper, we first introduce Simpson’s second-type integral inequalities for two-variable functions whose second-order partial derivatives in modulus are convex on the coordinates. In addition, similar results are acquired by considering that powers of the absolute value of second-order partial derivatives of these two-variable functions are convex on the coordinates. Finally, some applications for Simpson’s 3/8 cubature formula are given.

scholarly journals The Pseudo-Laplace Filter for Vector-Based Elastic Reverse Time Migration

2021 ◽  
Vol 9 ◽  
Author(s):  
Qizhen Du ◽  
Xiaoyu Zhang ◽  
Shukui Zhang ◽  
Fuyuan Zhang ◽  
Li-Yun Fu
Keyword(s):  
Cross Correlation ◽  
Second Order ◽  
Polarity Reversal ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Imaging Condition ◽  
Partial Derivatives ◽  
Time Migration ◽  
Dot Product ◽  
Derivatives Of

The scalar images (PP and PS) can be effectively obtained in vector-based elastic reverse time migration by applying dot product–based scalar imaging conditions to the separated vector wavefields. However, the PP image suffers from polarity reversal issues when opening angles are greater than 90∘ and backscattering artifacts when opening angles are close to 180∘. To address these issues, we propose the pseudo-Laplace filter for the dot product–based scalar imaging condition. Based on the analysis of the Laplace filter in the scalar image of vector-based wavefields, the second-order parallel-oriented partial derivatives of Cartesian components cross-correlation results are selected to construct the pseudo-Laplace filter. In contrast, second-order normal-oriented partial derivatives of the Cartesian component’s cross-correlation results are omitted. The theoretical analysis with the plane wave assumption shows that the proposed pseudo-Laplace filter can solve the problems of backscattering artifacts and polarity reversal in PP images by the scalar imaging condition. Due to additional polarity correction and backscattering attenuation, numerical examples show excellent performance in PP images with a pseudo-Laplace filter. Furthermore, the application of the pseudo-Laplace filter requires trivial additional computation or storage.

scholarly journals Second Order Perturbations of Elliptic Elements with Respect to the Initial Ones

1999 ◽  
Vol 172 ◽  
pp. 453-454
Author(s):  
F.J. Marco Castillo ◽  
M.J. Martínez Usó ◽  
J.A. López Ortí
Keyword(s):  
Quadratic Form ◽  
Second Order ◽  
Initial Instant ◽  
Orbital Elements ◽  
Partial Derivatives ◽  
Orbital Parameters ◽  
The Matrix ◽  
The Individual ◽  
Derivatives Of

AbstractThe following paper is devoted to the theoretical exposition of the obtention of second order perturbations of elliptic elements and is a follow-up of previous papers (Marco et al., 1996; Marco et al., 1997) where the hypothesis was made that the matrix of the partial derivatives of the orbital elements with respect to the initial ones is the identity matrix at the initial instant only. So, we must compute them through the integration of Lagrange planetary equations and their partial derivatives.Such developments have been applied to the individual corrections of orbits together with the correction of the reference system through the minimization of a quadratic form obtained from the linearized residual. In this state two new targets emerged: 1.To be sure that the most suitable quadratic form was to be considered.2.To provide a wider vision of the behavior of the different orbital parameters in time.Both aims may be accomplished through the consideration of the second order partial derivatives of the elliptic orbital elements with respect to the initial ones.

scholarly journals A perturbation scheme for obtaining partial derivatives of Love-wave group-velocity dispersion

1975 ◽  
Vol 65 (6) ◽  
pp. 1753-1760
Author(s):  
Dan Kosloff
Keyword(s):  
Group Velocity ◽  
Love Wave ◽  
Group Velocity Dispersion ◽  
Second Order ◽  
Wave Number ◽  
Perturbation Scheme ◽  
Wave Group ◽  
Partial Derivatives ◽  
Velocity Spectra ◽  
Derivatives Of

abstract A method is derived for obtaining partial derivatives of Love-wave group-velocity spectra for a layered medium using a second-order perturbation theory. These partials are a prerequisite for systematic inversion of group-velocity spectra but they are helpful as well in trial and error methods. Mathematically the equation of motion and boundary conditions for Love waves are a singular Sturm Liouville type eigenvalue problem. In the case of a fixed wave number, the eigenvalues are the negative of the square of the frequencies. Thus, by expressing the first- and second-order perturbations of the eigenvalues in terms of partial derivatives of the frequency with respect to the wave number and material parameters of the medium, one can relate these perturbations to group-velocity partials. The scheme should be relatively economical and easy to incorporate in Love-wave dispersion codes.

Boundary Properties of Second-Order Partial Derivatives of the Poisson Integral for a Half-Space

1998 ◽  
Vol 5 (4) ◽  
pp. 385-400
Author(s):  
S. Topuria
Keyword(s):  
Half Space ◽  
Second Order ◽  
Poisson Integral ◽  
Partial Derivatives ◽  
Boundary Properties ◽  
Derivatives Of

Abstract The boundary properties of second-order partial derivatives of the Poisson integral are studied for a half-space .

scholarly journals New Hermite-Hadamard type inequalities for m and (α, m)-convex functions on the coordinates via generalized fractional integrals

2021 ◽  
Vol 40 (6) ◽  
pp. 1449-1472
Author(s):  
Seth Kermausuor
Keyword(s):  
Convex Functions ◽  
Type Inequality ◽  
Second Order ◽  
Fractional Integrals ◽  
Absolute Value ◽  
Partial Derivatives ◽  
Independent Variables ◽  
Functions Of Two Variables ◽  
Derivatives Of ◽  
Two Independent Variables

In this paper, we obtained a new Hermite-Hadamard type inequality for functions of two independent variables that are m-convex on the coordinates via some generalized Katugampola type fractional integrals. We also established a new identity involving the second order mixed partial derivatives of functions of two independent variables via the generalized Katugampola fractional integrals. Using the identity, we established some new Hermite-Hadamard type inequalities for functions whose second order mixed partial derivatives in absolute value at some powers are (α, m)-convex on the coordinates. Our results are extensions of some earlier results in the literature for functions of two variables.

scholarly journals Detection of degenerate points on the surface

2018 ◽  
Author(s):  
Marián Jenčo
Keyword(s):  
Automatic Recognition ◽  
Second Order ◽  
Ridge Line ◽  
Partial Derivatives ◽  
First Order ◽  
Convex Section ◽  
Pass Through ◽  
Derivatives Of

Landslides, bifurcations, multi-saddles and remnants of terraces are distinctive landforms. Some points on the surfaces of these objects are degenerate points. This may help us with their automatic recognition and identification. All first-order and second-order partial derivatives of analyzed function are necessary for detection of degenerate points. Terrain slope, curvatures and Hessian are required for classification of degenerate points. The paper is aimed at detection of fossil landslides. A point of landslide surface where the concave section of thalweg is turning into convex section of ridge line is a degenerate point. Two zero isolines of Hessian and zero isoline of profile, streamline and plan or tangential curvatures pass through this point. Final result of the detection procedure depends to a great extent on the quality of DEM and accuracy of derivatives.

N-Derivatives of Shannon entropy density as response functions

2020 ◽  
Vol 22 (37) ◽  
pp. 21535-21542
Author(s):  
Abdolkarim Matrodi ◽  
Siamak Noorizadeh
Keyword(s):  
Shannon Entropy ◽  
Active Sites ◽  
Entropy Density ◽  
Second Order ◽  
External Potential ◽  
Response Functions ◽  
Partial Derivatives ◽  
Derivatives Of

The exact first and second order partial derivatives of Shannon entropy density with respect to the number of electrons at constant external potential are introduced as new descriptors for prediction of the active sites of a molecule.

scholarly journals Three-Dimensional Incompressible Flow Solution of an Axial Compressor Using Pseudostream-Functions Formulation

1989 ◽  
Author(s):  
J. Z. Xu ◽  
J. C. Shi ◽  
W. Y. Ni
Keyword(s):  
Flow Field ◽  
Incompressible Flow ◽  
Three Dimensional ◽  
Axial Compressor ◽  
Second Order ◽  
Partial Derivatives ◽  
Analysis And Design ◽  
Characteristic Features ◽  
Derivatives Of ◽  
Convergent Solution

Based on considering the characteristic features of the momentum equations and meeting the requirement of the continuity, two pseudostream-functions are introduced. The principal equation of each pseudostream-function only contains the terms of its own second-order partial derivatives and does not include the second-order partial derivatives of another pseudostream-function. So the relation between the two equations are loosed considerablly and both equations may be solved separately. This property is valueful to obtain the convergent solution of flow field easily. The equations of the pseudostream-functions and the corresponding boundary conditions are given. Two incompressible flow examples show that this method may become a powerful tool in the aerothermodynamic analysis and design of a compressor.

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