scholarly journals An estimation of p-adic sizes of common zeros of partial derivative polynomials associated with a quartic form

2014 ◽  
Author(s):  
Yap Hong Keat ◽  
Kamel Ariffin Mohd Atan ◽  
Siti Hasana Sapar ◽  
Mohamad Rushdan Md Said
Keyword(s):  
Partial Derivative ◽  
Quartic Form ◽  
Common Zeros

scholarly journals An explicit estimation of p-adic sizes of common zeros of partial derivative polynomials associated with a complete quartic form

2015 ◽  
Author(s):  
Yap Hong Keat ◽  
Kamel Ariffin Mohd Atan ◽  
Siti Hasana Sapar ◽  
Mohamad Rushdan Md Said
Keyword(s):  
Partial Derivative ◽  
Quartic Form ◽  
Common Zeros

A METHOD OF ESTIMATING THE p-ADIC SIZES OF COMMON ZEROS OF PARTIAL DERIVATIVE POLYNOMIALS ASSOCIATED WITH A QUINTIC FORM

2009 ◽  
Vol 05 (03) ◽  
pp. 541-554 ◽  
Author(s):  
S. H. SAPAR ◽  
K. A. MOHD. ATAN
Keyword(s):  
Partial Derivative ◽  
Newton Polyhedron ◽  
Exponential Sum ◽  
Common Root ◽  
Partial Derivatives ◽  
Sum Formula ◽  
Derivatives Of ◽  
Common Zeros

It is known that the value of the exponential sum [Formula: see text] can be derived from the estimate of the cardinality |V|, the number of elements contained in the set [Formula: see text] where [Formula: see text] is the partial derivatives of [Formula: see text] with respect to [Formula: see text]. The cardinality of V in turn can be derived from the p-adic sizes of common zeros of the partial derivatives [Formula: see text]. This paper presents a method of determining the p-adic sizes of the components of (ξ,η) a common root of partial derivative polynomials of f(x,y) in Zp[x,y] of degree five based on the p-adic Newton polyhedron technique associated with the polynomial. The degree five polynomial is of the form f(x,y) = ax5 + bx4y + cx3y2 + sx + ty + k. The estimate obtained is in terms of the p-adic sizes of the coefficients of the dominant terms in f.

Estimation of p–Adic Sizes of Common Zeroes of Partial Derivative Polynomials Associated with a Quintic Form

2012 ◽  
Author(s):  
S.H. Sapar ◽  
K. A. Mohd. Atan
Keyword(s):  
Partial Derivative ◽  
Exponential Sums ◽  
Newton Polyhedron ◽  
Exponential Sum ◽  
Partial Derivatives ◽  
Quintic Polynomial ◽  
Z Ring ◽  
Complete Set ◽  
Derivatives Of ◽  
Common Zeros

Katakan x = {xi, x2,...,xn} vektor dalam ruang Zn dengan Z menandakan gelanggang integer dan q integer positif, f polinomial dalam x dengan pekali dalam Z. Hasil tambah eksponen yang disekutukan dengan f ditakrifkan sebagai S (f;q) = exp (2πif (x)/ q) yang dinilaikan bagi semua nilai x di dalam reja lengkap modulo q. Nilai S(f;q) adalah bersandar kepada penganggaran bilangan unsur |V|, yang terdapat dalam set V = {x mod q | fx = 0 mod q} dengan fx menandakan polinomial-polinomial terbitan separa f terhadap x. Untuk menentukan kekardinalan bagi V, maklumat mengenai saiz p-adic pensifar sepunya perlu diperolehi. Makalah ini membincangkan suatu kaedah penentuan saiz p-adic bagi komponen (ξ,η) pensifar sepunya pembezaan separa f(x,y) dalam Zp[x, y] berdarjah lima berasaskan teknik polihedron Newton yang disekutukan dengan polinomial terbabit. Polinomial berdarjah lima yang dipertimbangkan berbentuk f(x,y) = ax5 + bx4y + cx3y2 + dx2y3 + exy4 + my5 + nx + ty + k. Kata kunci: Hasil tambah eksponen, kekardinalan, saiz p–adic, polihedron Newton Let x = {xi, x2,...,xn} be a vector in a space Zn with Z ring of integers and let q be a positive integer, f a polynomial in x with coefficients in Z. The exponential sum associated with f is defined as S (f;q) = exp (2πif (x)/ q) where the sum is taken over a complete set of residues modulo q. The value of S (f;q) has been shown to depend on the estimate of the cardinality | V |, the number of elements contained in the set V = {x mod q | fx = 0 mod q} where fx is the partial derivatives of f with respect to x. To determine the cardinality of V, the information on the p-adic sizes of common zeros of the partial derivatives polynomials need to be obtained. This paper discusses a method of determining the p-adic sizes of the components of (ξ,η) a common root of partial derivative polynomials of f(x, y) in Zp[x, y] of degree five based on the p-adic Newton polyhedron technique associated with the polynomial. The quintic polynomial is of the form f(x,y) = ax5 + bx4y + cx3y2 + dx2y3 + exy4 + my5 + nx + ty + k. Key words: Exponential sums, cardinality, p–adic sizes, Newton polyhedron

scholarly journals A plane quartic curve with twelve undulations

1945 ◽  
Vol 35 ◽  
pp. 10-13 ◽  
Author(s):  
W. L. Edge
Keyword(s):  
Homogeneous Coordinates ◽  
Quartic Curve ◽  
Base Points ◽  
Octahedral Group ◽  
Quartic Form ◽  
Image Position ◽  
Quartic Curves ◽  
Set Of Points

The pencil of quartic curveswhere x, y, z are homogeneous coordinates in a plane, was encountered by Ciani [Palermo Rendiconli, Vol. 13, 1899] in his search for plane quartic curves that were invariant under harmonic inversions. If x, y, z undergo any permutation the ternary quartic form on the left of (1) is not altered; nor is it altered if any, or all, of x, y, z be multiplied by −1. There thus arises an octahedral group G of ternary collineations for which every curve of the pencil is invariant.Since (1) may also be writtenthe four linesare, as Ciani pointed out, bitangents, at their intersections with the conic C whose equation is x2 + y2 + z2 = 0, to every quartic of the pencil. The 16 base points of the pencil are thus all accounted for—they consist of these eight contacts counted twice—and this set of points must of course be invariant under G. Indeed the 4! collineations of G are precisely those which give rise to the different permutations of the four lines (2), a collineation in a plane being determined when any four non-concurrent lines and the four lines which are to correspond to them are given. The quadrilateral formed by the lines (2) will be called q.

scholarly journals Depth-Extrapolation-Based True-Amplitude Full-Wave-Equation Migration from Topography

2021 ◽  
Vol 11 (7) ◽  
pp. 3010
Author(s):  
Hao Liu ◽  
Xuewei Liu
Keyword(s):  
Wave Equation ◽  
Partial Derivative ◽  
Model Experiment ◽  
Surface Velocity ◽  
Seismic Exploration ◽  
Initial Condition ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Full Wave ◽  
Time Migration

The lack of an initial condition is one of the major challenges in full-wave-equation depth extrapolation. This initial condition is the vertical partial derivative of the surface wavefield and cannot be provided by the conventional seismic acquisition system. The traditional solution is to use the wavefield value of the surface to calculate the vertical partial derivative by assuming that the surface velocity is constant. However, for seismic exploration on land, the surface velocity is often not uniform. To solve this problem, we propose a new method for calculating the vertical partial derivative from the surface wavefield without making any assumptions about the surface conditions. Based on the calculated derivative, we implemented a depth-extrapolation-based full-wave-equation migration from topography using the direct downward continuation. We tested the imaging performance of our proposed method with several experiments. The results of the Marmousi model experiment show that our proposed method is superior to the conventional reverse time migration (RTM) algorithm in terms of imaging accuracy and amplitude-preserving performance at medium and deep depths. In the Canadian Foothills model experiment, we proved that our method can still accurately image complex structures and maintain amplitude under topographic scenario.

scholarly journals A Partial Derivative Approach to the Change of Scale Formula for the Function Space Integral

Entropy ◽  
2020 ◽  
Vol 23 (1) ◽  
pp. 26
Author(s):  
Young Sik Kim
Keyword(s):  
Fourier Transform ◽  
Function Space ◽  
Partial Derivative ◽  
Directional Derivative ◽  
Feynman Integral ◽  
Wiener Integral ◽  
Vector Calculus ◽  
Change Of Scale

We investigate the partial derivative approach to the change of scale formula for the functon space integral and we investigate the vector calculus approach to the directional derivative on the function space and prove relationships among the Wiener integral and the Feynman integral about the directional derivative of a Fourier transform.

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