HOW THIN IS A THIN BED?

Geophysics ◽  
1973 ◽  
Vol 38 (6) ◽  
pp. 1176-1180 ◽  
Author(s):  
M. B. Widess
Keyword(s):  
Order Of Approximation ◽  
First Order ◽  
Reflective Power ◽  
Typical Frequency

Based on reflective properties, a thin bed may be conveniently defined as one whose thickness is less than about [Formula: see text] where [Formula: see text] is the (predominant) wavelength computed using the velocity of the bed. The amplitude of a reflection from a thin bed is to the first order of approximation equal to [Formula: see text] where b is the thickness of the bed and A is the amplitude of the reflection if the bed were to be very thick. The equation shows that a bed as thin as 10 ft has, for typical frequency and velocity, considerably more reflective power than is usually attributed to it.

Role of Single Word Variables in Recall of Statistical Approximations to English

1966 ◽  
Vol 19 (2) ◽  
pp. 627-634 ◽  
Author(s):  
Mari J. K. Brown
Keyword(s):  
Free Recall ◽  
Higher Order ◽  
Order Of Approximation ◽  
Single Word ◽  
First Order ◽  
Original Order ◽  
Sequential Organization ◽  
The Individual

Free recall of lists at different orders of approximation to English was compared to the recall of the same lists when the order of the words had been scrambled to destroy their sequential organization. Recall of the organized lists showed the typical improvement with increasing order of approximation. Recall of the scrambled lists was unrelated to the original order of approximation. The results indicate that increased recall with increasing order of approximation to English is not produced by systematic differences in the characteristics of the individual words comprising the approximations. When recall of the organized lists was scored in terms of the number of longer sequences present in recall, the number of recalled sequences of any given length increased as order of approximation to English increased, with the first order list showing proportionally less organization in recall than the second and higher order lists.

scholarly journals Restructured class of estimators for population mean using an auxiliary variable under simple random sampling scheme

2020 ◽  
Vol 16 (1) ◽  
pp. 61-75
Author(s):  
S. Baghel ◽  
S. K. Yadav
Keyword(s):  
Random Sampling ◽  
Simple Random Sampling ◽  
Auxiliary Variable ◽  
Sampling Scheme ◽  
Order Of Approximation ◽  
Study Variable ◽  
Population Mean ◽  
First Order ◽  
New Class ◽  
Class Of Estimators

AbstractThe present paper provides a remedy for improved estimation of population mean of a study variable, using the information related to an auxiliary variable in the situations under Simple Random Sampling Scheme. We suggest a new class of estimators of population mean and the Bias and MSE of the class are derived upto the first order of approximation. The least value of the MSE for the suggested class of estimators is also obtained for the optimum value of the characterizing scaler. The MSE has also been compared with the considered existing competing estimators both theoretically and empirically. The theoretical conditions for the increased efficiency of the proposed class, compared to the competing estimators, is verified using a natural population.

scholarly journals Some Median Type Estimators to Estimate the Finite Population Mean

2020 ◽  
pp. 48-58
Author(s):  
Waqar Hafeez ◽  
Javid Shabbir ◽  
Muhammad Taqi Shah ◽  
Shakeel Ahmed
Keyword(s):  
Linear Regression ◽  
Finite Population ◽  
Auxiliary Variable ◽  
Maximum Efficiency ◽  
Order Of Approximation ◽  
Study Variable ◽  
Sample Mean ◽  
First Order ◽  
Regression Estimators ◽  
Class Of Estimators

Researchers always appreciates estimators of finite population quantities, especially mean, with maximum efficiency for reaching to valid statistical inference.  Apart from ratio, product and regression estimators, exponential estimators are widely considered by survey statisticians. Motivated from the idea of exponential type estimators, in this article, we propose some new estimators utilizing known median of the study variable with mean of auxiliary variable. Theoretical properties of the suggested estimators are studied up to first order of approximation. In addition, an empirical and simulation study the comparison of median based proposed class of estimators with sample mean, ratio and linear regression estimators  are discussed. The results expose that the proposed estimators are more efficient than the existing estimators.

scholarly journals Connection Between Non-Rotating Local Reference Frames by Means of the World Function

1991 ◽  
Vol 127 ◽  
pp. 262-265
Author(s):  
J.M. Gambi ◽  
P. Romero ◽  
A.San Miguel ◽  
F. Vicente
Keyword(s):  
Reference Frames ◽  
Riemann Tensor ◽  
Space Time ◽  
General Character ◽  
Order Of Approximation ◽  
First Order ◽  
The World ◽  
Local Reference ◽  
General Relativistic ◽  
World Function

AbstractBy means of the world function an approximate transformation showing the Riemann tensor between the Fermi coordinates associated to two non-rotating local reference frames is derived in a General Relativistic space-time. One of the observer’s world lines is resticted to be a time-like geodesic of the space-time, and the other is a time-like curve of a general character. The space-time where the transformation is evaluated is supposed to be of small curvature, and the calculations are carried out in a first order of approximation with respect to the Riemann tensor.

scholarly journals The classical geometrization of the electromagnetism

2015 ◽  
Vol 12 (09) ◽  
pp. 1560022 ◽  
Author(s):  
Celso de Araujo Duarte
Keyword(s):  
General Relativity ◽  
Magnetic Monopoles ◽  
Electromagnetic Theory ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Unified Theory ◽  
Quantum Field ◽  
Order Of Approximation ◽  
First Order ◽  
The 19Th Century

Following the line of the history, if by one side the electromagnetic theory was consolidated on the 19th century, the emergence of the special and the general relativity theories on the 20th century opened possibilities of further developments, with the search for the unification of the gravitation and the electromagnetism on a single unified theory. Some attempts to the geometrization of the electromagnetism emerged in this context, where these first models resided strictly on a classical basis. Posteriorly, they were followed by more complete and embracing quantum field theories. The present work reconsiders the classical viewpoint, with the purpose of showing that at first-order of approximation the electromagnetism constitutes a geometric structure aside other phenomena as gravitation, and that magnetic monopoles do not exist at least up to this order of approximation. Even though being limited, the model is consistent and offers the possibility of an experimental test of validity.

scholarly journals Acoustic–vorticity coupling in linearly varying shear flows using the WKB method

2013 ◽  
Vol 469 (2153) ◽  
pp. 20120708 ◽  
Author(s):  
Gael Favraud ◽  
Vincent Pagneux
Keyword(s):  
Acoustic Mode ◽  
Rigid Rotation ◽  
Wkb Method ◽  
Plane Couette Flow ◽  
First Order ◽  
Acoustic Modes ◽  
Hyperbolic Flow ◽  
Linear Flows ◽  
Flow Plane ◽  
Typical Frequency

The evolution of acoustic and vorticity perturbations in a two-dimensional incompressible linear flow is investigated. A weighted decomposition of the flow into a hyperbolic part and a rotation part allows continuous spanning of all linear flows such as hyperbolic flow, plane Couette flow and rigid rotation for instance. Using the Kelvin non-modal approach, the equations governing the time evolution of plane wave perturbations are reduced into a system of three first-order ordinary differential equations. This system is analysed using a WKB method where the small parameter ε is the ratio of the shear rate of the flow over the typical frequency of the perturbations. With this method, a basis of three modes naturally appears: two acoustic modes and one vorticity mode. At finite but small ε , couplings between the modes appear when the length of the wavenumber is minimal. For hyperbolic flow, incident vorticity mode generates the two acoustic modes, and an incident acoustic mode generates the other acoustic mode. More generally, for all flows, the hyperbolic part of the flow is responsible of the coupling between acoustic and vorticity modes, but also of the coupling between the two acoustic modes. These phenomena are illustrated by displaying wavepacket evolutions.

Stability of General Hill’s Equation With Three Independent Parameters

1972 ◽  
Vol 39 (1) ◽  
pp. 276-278 ◽  
Author(s):  
K. Hamer ◽  
M. R. Smith
Keyword(s):  
Perturbation Method ◽  
New Method ◽  
Order Of Approximation ◽  
Hill’S Equation ◽  
Stability Boundaries ◽  
First Order ◽  
Hill's Equation ◽  
The Stability ◽  
Independent Parameters

The stability of Hill’s equation with three independent parameters, two of which are small, is analyzed using a perturbation method. It is shown that, except for periodic terms of a special type, existing methods of determining stability boundaries fail. A new method, which works successfully to the first order of approximation, is described.

scholarly journals TELEPARALLEL LAGRANGE GEOMETRY AND A UNIFIED FIELD THEORY: LINEARIZATION OF THE FIELD EQUATIONS

2013 ◽  
Vol 10 (03) ◽  
pp. 1250092 ◽  
Author(s):  
M. I. WANAS ◽  
NABIL L. YOUSSEF ◽  
A. M. SID-AHMED
Keyword(s):  
Field Theory ◽  
Approximate Solution ◽  
Order Of Approximation ◽  
Unified Field Theory ◽  
Field Equations ◽  
First Order ◽  
Unified Field ◽  
Vertical Field ◽  
Geometric Objects ◽  
Linearized Theory

This paper is a natural continuation of our previous paper: "Teleparallel Lagrange geometry and a unified field theory, Class. Quantum Grav.27 (2010) 045005 (29 pp)". In this paper, we apply a linearization scheme on the field equations obtained in the above-mentioned paper. Three important results under the linearization assumption are accomplished. First, the vertical fundamental geometric objects of the EAP-space lose their dependence on the positional argument x. Secondly, our linearized theory in the Cartan-type case coincides with the GFT in the first-order of approximation. Finally, an approximate solution of the vertical field equations is obtained.

scholarly journals A Note on Generalized Exponential Type Estimator for Population Variance in Survey Sampling

2015 ◽  
Vol 38 (2) ◽  
pp. 385-397 ◽  
Author(s):  
Javid Shabbir ◽  
Sat Gupta
Keyword(s):  
Exponential Type ◽  
Auxiliary Variable ◽  
Survey Sampling ◽  
Data Sets ◽  
Numerical Comparison ◽  
Order Of Approximation ◽  
Population Variance ◽  
First Order ◽  
Better Than

<p>Recently a new generalized estimator for population variance using information on the auxiliary variable has been introduced by Asghar, Sanaullah &amp; Hanif (2014). In that paper there was some inaccuracy in the bias and MSE expressions. In this paper, we provide the correct expressions for bias and MSE of the Asghar et al. (2014) estimator, up to the first order of approximation. We also propose a new generalized exponential type estimator for population variance which performs better than the existing estimators. Four data sets are used for numerical comparison of efficiencies.</p>

scholarly journals ON THE DEVELOPMENT OF RATIO TYPE ESTIMATORS USING AUXILIARY INFORMATION

2021 ◽  
Vol 21 (1) ◽  
pp. 163-170
Author(s):  
MUHAMMAD IJAZ ◽  
ATTA ULLAH ◽  
TOLGA ZAMAN
Keyword(s):  
Mean Squared Error ◽  
Auxiliary Information ◽  
Ratio Estimator ◽  
Order Of Approximation ◽  
Large Sample ◽  
First Order ◽  
Squared Error ◽  
Ratio Estimators ◽  
Modified Forms

The paper produces some new modified forms of the ratio estimators using the auxiliary information. The large sample properties, that is, the bias and mean squared error up to the first order of approximation are determined. The comparison is made with other existing estimators by using an applied data. It has been observed that the proposed estimators have a fewer mean squared error and leads to the efficient results as compared to the classical ratio estimator, Sisodia and Dwivedi, Singh and Kakran, Upadhyaya and Singh estimators.

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