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.

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.

scholarly journals A remark on the derivative of the one-dimensional Hardy-Littlewood maximal function

2002 ◽  
Vol 65 (2) ◽  
pp. 253-258 ◽  
Author(s):  
Hitoshi Tanaka
Keyword(s):  
Sobolev Space ◽  
Maximal Function ◽  
60Th Birthday ◽  
One Dimensional ◽  
Partial Derivatives ◽  
First Order ◽  
Littlewood Maximal Function ◽  
The One ◽  
Derivatives Of

Dedicated to Professor Kôzô Yabuta on the occasion of his 60th birthdayJ. Kinnunen proved that of P > 1, d ≤ 1 and f is a function in the Sobolev space W1,P(Rd), then the first order weak partial derivatives of the Hardy-Littlewood maximal function ℳf belong to LP(Rd). We shall show that, when d = 1, Kinnunen's result can be extended to the case where P = 1.

Potential fields and their partial derivatives produced by a 2D homogeneous polygonal source: A summary with some revisions

Geophysics ◽  
2011 ◽  
Vol 76 (4) ◽  
pp. L29-L34 ◽  
Author(s):  
Zhen Jia ◽  
Shiguo Wu
Keyword(s):  
Magnetic Field ◽  
Cross Section ◽  
Observation Point ◽  
Potential Fields ◽  
Data Sets ◽  
Partial Derivatives ◽  
First Order ◽  
Gravity And Magnetic ◽  
Derivatives Of ◽  
The Magnetic Field

We summarized and revised the present forward modeling methods for calculating the gravity- and magnetic-field components and their partial derivatives of a 2D homogeneous source with a polygonal cross section. The responses of interest include the gravity-field components and their first- and second-order partial derivatives and the magnetic-field components and their first-order partial derivatives. The revised formulas consist of several basic quantities that are common in all the formulations. A singularity appears when the observation point coincides with a polygon vertex. This singularity is removable for the gravity formulas but not for the others. The compact forms of the revised formulas make them easy to implement. We compare the gravity- and magnetic-field components and their partial derivatives produced by a 2D prism whose polygonal cross section approximates a cylinder with the corresponding analytical fields and partial derivatives of the cylinder. The perfect fittings presented by both data sets confirm the reliability of the updated formulas.

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.

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