Partial derivatives of the one-dimensional apparent resistivity

1980 ◽  
Vol 24 (1) ◽  
pp. 39-50
Author(s):  
Josef Pek ◽  
M. Hvoždara
Keyword(s):  
Apparent Resistivity ◽  
One Dimensional ◽  
Partial Derivatives ◽  
The One ◽  
Derivatives Of

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.

Estimation of the integral of the square of derivatives of symmetric probability densities of one-dimensional random variables

Metrologiya ◽  
2020 ◽  
pp. 15-27
Author(s):  
Aleksandr V. Lapko ◽  
Vasiliy A. Lapko
Keyword(s):  
Standard Deviation ◽  
Probability Density ◽  
Random Variables ◽  
Random Variable ◽  
Density Estimates ◽  
One Dimensional ◽  
Probability Densities ◽  
The One ◽  
Derivatives Of ◽  
Selection Of

When substantiating the method of fast selection of the bandwidth of kernel probability density estimates, a constant was found that is a functional of the second density derivative. In this paper, the obtained result is generalized to derivatives of symmetric probability densities of different orders. The functional dependences of the constants under study on the coeffi cient of antikurtosis of a random variable are established. The regularities peculiar to them are investigated. Based on the results obtained, a method for estimating functionals from derived probability densities has been developed, which involves the following actions. In the original sample estimated standard deviation of the one-dimensional random variables and the coeffi cient of antikurtosis. Using the reconstructed functional dependences on the antikurtosis coeffi cient, the constants are estimated, which are functionals of the derivatives of the probability density. With known estimates of the standard deviation of the investigated random variable and the considered constant, the values of the functional from the derivative of the probability density of the selected order are calculated. The obtained results are confi rmed by the analysis of the data of computational experiments. It is established that with increasing order of the derivative, the values of the estimates of the studied functionals increase. This fact is explained by the complication of the integrand function in the considered functionals. The proposed method provides objective results for the fi rst three derivatives of the probability density of a random variable. The obtained conclusions are confi rmed by the results of the confi dence estimation of the investigated functionals.

scholarly journals From a Mechanical Lagrangian to the Schrödinger Equation: A Modified Version of the Quantum Newton Law

2003 ◽  
Vol 18 (19) ◽  
pp. 3347-3368 ◽  
Author(s):  
A. Bouda
Keyword(s):  
Kinetic Term ◽  
Third Order ◽  
One Dimensional ◽  
The Third ◽  
Classical Potential ◽  
The One ◽  
Quantum Motion ◽  
Derivatives Of ◽  
Hamilton Jacobi Equation

In the one-dimensional stationary case, we construct a mechanical Lagrangian describing the quantum motion of a nonrelativistic spinless system. This Lagrangian is written as a difference between a function T, which represents the quantum generalization of the kinetic energy and which depends on the coordinate x and the temporal derivatives of x up the third order, and the classical potential V(x). The Hamiltonian is then constructed and the corresponding canonical equations are deduced. The function T is first assumed to be arbitrary. The development of T in a power series together with the dimensional analysis allow us to fix univocally the series coefficients by requiring that the well-known quantum stationary Hamilton–Jacobi equation be reproduced. As a consequence of this approach, we formulate the law of the quantum motion representing a new version of the quantum Newton law. We also analytically establish the famous Bohm relation [Formula: see text] outside the framework of the hydrodynamical approach and show that the well-known quantum potential, although it is a part of the kinetic term, plays really the role of an additional potential as assumed by Bohm.

An Efficient Calculation of Photonic Crystal Band Structures Using Taylor Expansions

2014 ◽  
Vol 16 (5) ◽  
pp. 1355-1388 ◽  
Author(s):  
Dirk Klindworth ◽  
Kersten Schmidt
Keyword(s):  
Photonic Crystal ◽  
Eigenvalue Problem ◽  
Brillouin Zone ◽  
Taylor Expansion ◽  
Dispersion Curves ◽  
Band Structures ◽  
One Dimensional ◽  
The One ◽  
Derivatives Of ◽  
Selection Of

AbstractIn this paper we present an efficient algorithm for the calculation of photonic crystal band structures and band structures of photonic crystal waveguides. Our method relies on the fact that the dispersion curves of the band structure are smooth functions of the quasi-momentum in the one-dimensional Brillouin zone. We show the derivation and computation of the group velocity, the group velocity dispersion, and any higher derivative of the dispersion curves. These derivatives are then employed in a Taylor expansion of the dispersion curves. We control the error of the Taylor expansion with the help of a residual estimate and introduce an adaptive scheme for the selection of nodes in the one-dimensional Brillouin zone at which we solve the underlying eigenvalue problem and compute the derivatives of the dispersion curves. These derivatives are then employed in a Taylor expansion of the dispersion curves. We control the error of the Taylor expansion with the help of a residual estimate and introduce an adaptive scheme for the selection of nodes in the one-dimensional Brillouin zone at which we solve the underlying eigenvalue problem and compute the derivatives of the dispersion curves. The proposed algorithm is not only advantageous as it decreases the computational effort to compute the band structure but also because it allows for the identification of crossings and anti-crossings of dispersion curves, respectively. This identification is not possible with the standard approach of solving the underlying eigenvalue problem at a discrete set of values of the quasi-momentum without taking the mode parity into account.

A Third-Order Accurate Direct Eulerian GRP Scheme for One-Dimensional Relativistic Hydrodynamics

2014 ◽  
Vol 4 (2) ◽  
pp. 95-131 ◽  
Author(s):  
Kailiang Wu ◽  
Zhicheng Yang ◽  
Huazhong Tang
Keyword(s):  
Second Order ◽  
Third Order ◽  
Sonic Point ◽  
One Dimensional ◽  
Eulerian Formulation ◽  
Numerical Fluxes ◽  
The One ◽  
Derivatives Of ◽  
Limiting Values ◽  
Generalised Riemann Problem

AbstractA third-order accurate direct Eulerian generalised Riemann problem (GRP) scheme is derived for the one-dimensional special relativistic hydrodynamical equations. In our GRP scheme, the higher-order WENO initial reconstruction is employed, and the local GRPs in the Eulerian formulation are directly and analytically resolved to third-order accuracy via the Riemann invariants and Rankine-Hugoniot jump conditions, to get the approximate states in numerical fluxes. Unlike a previous second-order accurate GRP scheme, for the non-sonic case the limiting values of the second-order time derivatives of the fluid variables at the singular point are also needed for the calculation of the approximate states; while for the sonic case, special attention is paid because the calculation of the second-order time derivatives at the sonic point is difficult. Several numerical examples are given to demonstrate the accuracy and effectiveness of our GRP scheme.

scholarly journals Koszul Complexes and Pole Order Filtrations

2014 ◽  
Vol 58 (2) ◽  
pp. 333-354 ◽  
Author(s):  
Alexandru Dimca ◽  
Gabriel Sticlaru
Keyword(s):  
Linear Systems ◽  
Homogeneous Polynomial ◽  
The Other ◽  
Koszul Complex ◽  
Partial Derivatives ◽  
Nodal Surface ◽  
Pole Order ◽  
Open Set ◽  
The One ◽  
Derivatives Of

AbstractWe study the interplay between the cohomology of the Koszul complex of the partial derivatives of a homogeneous polynomial f and the pole order filtration P on the cohomology of the open set U = ℙn \ D, with D the hypersurface defined by f = 0. The relation is expressed by some spectral sequences. These sequences may, on the one hand, in many cases be used to determine the filtration P for curves and surfaces and, on the other hand, to obtain information about the syzygies involving the partial derivatives of the polynomial f. The case of a nodal hypersurface D is treated in terms of the defects of linear systems of hypersurfaces of various degrees passing through the nodes of D. When D is a nodal surface in ℙ3, we show that F2H3(U) ≠ P2H3(U) as soon as the degree of D is at least 4.

scholarly journals A Class of Bridges of Iterated Integrals of Brownian Motion Related to Various Boundary Value Problems Involving the One-Dimensional Polyharmonic Operator

2011 ◽  
Vol 2011 ◽  
pp. 1-32
Author(s):  
Aimé Lachal
Keyword(s):  
Brownian Motion ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Classical Problem ◽  
Polyharmonic Operator ◽  
One Dimensional ◽  
The One ◽  
Interpolation Polynomials ◽  
Derivatives Of ◽  
Hermite Interpolation Polynomials

Let be the linear Brownian motion and the -fold integral of Brownian motion, with being a positive integer: for any In this paper we construct several bridges between times and of the process involving conditions on the successive derivatives of at times and . For this family of bridges, we make a correspondence with certain boundary value problems related to the one-dimensional polyharmonic operator. We also study the classical problem of prediction. Our results involve various Hermite interpolation polynomials.

scholarly journals Identification of Unentangled–Entangled Border in the Luttinger Liquid Phase

Crystals ◽  
2019 ◽  
Vol 9 (2) ◽  
pp. 105 ◽  
Author(s):  
Somayyeh Nemati ◽  
Fatemeh Khastehdel Fumani ◽  
Saeed Mahdavifar
Keyword(s):  
Liquid Phase ◽  
Quantum Discord ◽  
Luttinger Liquid ◽  
Quantum Correlations ◽  
Transverse Magnetic ◽  
One Dimensional ◽  
Spin Pairs ◽  
The One ◽  
Derivatives Of ◽  
Xx Model

Quantum discord and entanglement are both criteria for distinguishing quantum correlations in a quantum system. We studied the effect of the transverse magnetic field on the quantum discord of the one-dimensional spin-1/2 XX model. This study focused on the pair of spins at different distances. We show that quantum discord is finite for all studied spin pairs in the Luttinger liquid phase. In addition, relying on our calculations, we show that the derivatives of quantum discord can be used to identify the border between entangled and separable regions in the Luttinger liquid phase.

scholarly journals Partial Derivatives for the One-Dimensional Magnetotelluric Problem

1976 ◽  
Vol 44 (2) ◽  
pp. 359-378 ◽  
Author(s):  
F. Abramovici ◽  
M. Landisman ◽  
Y. Shoham
Keyword(s):  
One Dimensional ◽  
Partial Derivatives ◽  
The One

THE INVERSE PROBLEM OF MAGNETOTELLURIC SOUNDING

Geophysics ◽  
1968 ◽  
Vol 33 (6) ◽  
pp. 972-979 ◽  
Author(s):  
Francis T. Wu
Keyword(s):  
Inverse Problem ◽  
Apparent Resistivity ◽  
Least Squares Method ◽  
Nonlinear Least Squares ◽  
Initial Guess ◽  
Magnetotelluric Sounding ◽  
Partial Derivatives ◽  
Linearly Independent ◽  
Nonlinear Least Squares Method ◽  
Derivatives Of

Based on the model of a flat layered earth, a nonlinear, least‐squares method is used to invert magnetotelluric sounding curves to obtain the layer resistivities and thicknesses. Partial derivatives of the apparent resistivity with respect to layer parameters show the manner in which the layer parameters are contrbuting to the apparent resistivities. Uniqueness of the inversion is not guaranteed, but when the partial derivatives are linearly independent and the relative magnitudes of the layer resistivities of the initial guess are not too far from the correct ones, the convergence of the method to the correct values seems to be ensured.

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