Two-Dimensional Problems Involving a Single Variable

2014 ◽  
pp. 265-310
Author(s):  
J. N. Reddy
Keyword(s):  
Two Dimensional ◽  
Single Variable

Applications of Complex Numbers to Geometry

1932 ◽  
Vol 25 (4) ◽  
pp. 215-226
Author(s):  
Allen A. Shaw
Keyword(s):  
Complex Variable ◽  
Geometric Reasoning ◽  
Present Writer ◽  
Complex Numbers ◽  
Two Dimensional ◽  
Single Variable ◽  
Argand Diagram

Introduction. It was with a real pleasure that the present writer read the two excellent articles by Professors L. L. Smail and A. A. Schelkunoff on geometric applications of the complex variable.1 Both papers are important for the doctrine they expound and for the good training they give the reader in rigorous geometric reasoning on the Argand diagram. While Smail prefers the use of the complex variable in the two-dimensional form, x+iy, Schelkunoff employs and recommends the usage of the single variable z to prove the same theorems and obtains very simple and elegant demonstrations.

ATOMIC COHERENT STATES AS THE EIGENSTATES OF A TWO-DIMENSIONAL ANISOTROPIC HARMONIC OSCILLATOR IN A UNIFORM MAGNETIC FIELD

2009 ◽  
Vol 24 (38) ◽  
pp. 3129-3136 ◽  
Author(s):  
XIANG-GUO MENG ◽  
JI-SUO WANG ◽  
HONG-YI FAN
Keyword(s):  
Magnetic Field ◽  
Harmonic Oscillator ◽  
Coherent States ◽  
Uniform Magnetic Field ◽  
Hermite Polynomial ◽  
Entangled State ◽  
Two Dimensional ◽  
Entangled State Representation ◽  
Single Variable ◽  
State Representation

In the newly constructed entangled state representation embodying quantum entanglement of Einstein, Podolsky and Rosen, the usual wave function of atomic coherent state ∣τ〉 = exp (μJ+-μ*J-)∣j, -j〉 turns out to be just proportional to a single-variable ordinary Hermite polynomial of order 2j, where j is the spin value. We then prove that a two-dimensional time-independent anisotropic harmonic oscillator in a uniform magnetic field possesses energy eigenstates which can be classified as the states ∣τ〉 in terms of the spin values j.

RECURSION FILTERS FOR DIGITAL PROCESSING OF POTENTIAL FIELD DATA

Geophysics ◽  
1976 ◽  
Vol 41 (4) ◽  
pp. 712-726 ◽  
Author(s):  
B. K. Bhattacharyya
Keyword(s):  
Frequency Domain ◽  
Potential Field ◽  
Field Data ◽  
Least Squares Method ◽  
Vertical Gradient ◽  
Two Dimensional ◽  
Single Variable ◽  
Potential Field Data ◽  
Recursive Filter ◽  
Rational Expression

Zero‐phase two‐dimensional recursive filters, with a specified frequency domain response, have been designed for processing potential field data. In the case of second vertical derivative filters, it is possible to use the rational approximation of symmetrical functions of a single variable for the design of a short recursive filter. The filter so designed has an excellent response in the frequency domain. For vertical gradient and continuation filters, a method is developed for obtaining, by the least‐squares method, a rational expression for a two‐dimensional symmetrical function. In order to ensure the stability of the recursive filter, the denominator of the rational expression is approximated by a product of two factors, each being a function of a single variable. Finally, to keep the error of the filter response as small as possible, an iterative procedure is used for adjusting the zeros of the denominator and then determining the coefficients of the numerator of the rational expression.

scholarly journals MULTIVARIATE SAMPLING THEOREMS ASSOCIATED WITH MULTIPARAMETER DIFFERENTIAL OPERATORS

2005 ◽  
Vol 48 (2) ◽  
pp. 257-277 ◽  
Author(s):  
M. H. Annaby
Keyword(s):  
Lagrange Interpolation ◽  
Differential Operators ◽  
Eigenvalue Problems ◽  
Sampling Theory ◽  
Sampling Theorem ◽  
Second Order ◽  
Two Dimensional ◽  
Single Variable ◽  
Sampling Theorems ◽  
Two Parameter

AbstractWe investigate the multivariate sampling theory associated with multiparameter eigenvalue problems. A several-variable counterpart of the classical sampling theorem of Whittaker, Kotel’nikov and Shannon is given. It arose when the multiparameter system has order one. Two-dimensional sampling theorems associated with two-parameter systems of second-order differential operators will be established. The sampling formulae are of multivariate non-uniform Lagrange interpolation type. Unlike many of the known formulae, the interpolating functions are not necessarily products of single variable functions.

Spectrophotometric measurements of objective-prism spectra

1966 ◽  
Vol 24 ◽  
pp. 118-119
Author(s):  
Th. Schmidt-Kaler
Keyword(s):  
Progress Report ◽  
Two Dimensional ◽  
Objective Prism ◽  
Made In

I should like to give you a very condensed progress report on some spectrophotometric measurements of objective-prism spectra made in collaboration with H. Leicher at Bonn. The procedure used is almost completely automatic. The measurements are made with the help of a semi-automatic fully digitized registering microphotometer constructed by Hög-Hamburg. The reductions are carried out with the aid of a number of interconnected programmes written for the computer IBM 7090, beginning with the output of the photometer in the form of punched cards and ending with the printing-out of the final two-dimensional classifications.

Introductory paper

1966 ◽  
Vol 24 ◽  
pp. 3-5
Author(s):  
W. W. Morgan
Keyword(s):  
Line Intensity ◽  
Classification Scheme ◽  
Two Dimensional ◽  
Spectral Classification ◽  
Dimensional Array ◽  
Rr Lyrae ◽  
Metallic Line ◽  
Introductory Paper ◽  
Parameter Varying ◽  
Definition Of

1. The definition of “normal” stars in spectral classification changes with time; at the time of the publication of theYerkes Spectral Atlasthe term “normal” was applied to stars whose spectra could be fitted smoothly into a two-dimensional array. Thus, at that time, weak-lined spectra (RR Lyrae and HD 140283) would have been considered peculiar. At the present time we would tend to classify such spectra as “normal”—in a more complicated classification scheme which would have a parameter varying with metallic-line intensity within a specific spectral subdivision.

A one-dimensional self-gravitating stellar gas

1966 ◽  
Vol 25 ◽  
pp. 46-48 ◽  
Author(s):  
M. Lecar
Keyword(s):  
Gravitational Field ◽  
Phase Space ◽  
Motion Picture ◽  
Space Distribution ◽  
Two Dimensional ◽  
One Dimensional ◽  
Phase Space Distribution ◽  
Dimensional Phase Space ◽  
The Mean

“Dynamical mixing”, i.e. relaxation of a stellar phase space distribution through interaction with the mean gravitational field, is numerically investigated for a one-dimensional self-gravitating stellar gas. Qualitative results are presented in the form of a motion picture of the flow of phase points (representing homogeneous slabs of stars) in two-dimensional phase space.

Analysis of the 9th November 1990 flare

2000 ◽  
Vol 179 ◽  
pp. 229-232
Author(s):  
Anita Joshi ◽  
Wahab Uddin
Keyword(s):  
Image Processing ◽  
Two Dimensional ◽  
Temporal Behaviour ◽  
Contour Maps ◽  
Dimensional Measurements ◽  
Processing Software ◽  
Image Processing Software

AbstractIn this paper we present complete two-dimensional measurements of the observed brightness of the 9th November 1990Hαflare, using a PDS microdensitometer scanner and image processing software MIDAS. The resulting isophotal contour maps, were used to describe morphological-cum-temporal behaviour of the flare and also the kernels of the flare. Correlation of theHαflare with SXR and MW radiations were also studied.

High Resolution Microscopy of Multi-Layered Biological Crystals

1982 ◽  
Vol 40 ◽  
pp. 80-83
Author(s):  
H.A. Cohen ◽  
T.W. Jeng ◽  
W. Chiu
Keyword(s):  
High Resolution ◽  
Low Dose ◽  
Three Dimensional ◽  
Data Interpretation ◽  
Two Dimensional ◽  
Biological Objects ◽  
Density Maps ◽  
Density Map ◽  
Complex Crystal ◽  
High Resolution Microscopy

This tutorial will discuss the methodology of low dose electron diffraction and imaging of crystalline biological objects, the problems of data interpretation for two-dimensional projected density maps of glucose embedded protein crystals, the factors to be considered in combining tilt data from three-dimensional crystals, and finally, the prospects of achieving a high resolution three-dimensional density map of a biological crystal. This methodology will be illustrated using two proteins under investigation in our laboratory, the T4 DNA helix destabilizing protein gp32*I and the crotoxin complex crystal.

Instrumentation For Quantitative Microscopy

1977 ◽  
Vol 35 ◽  
pp. 206-209
Author(s):  
B. Ralph ◽  
A.R. Jones
Keyword(s):  
Volume Fraction ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Automatic Data ◽  
Phase Volume Fraction ◽  
Statistical Confidence ◽  
Resultant Data ◽  
Automatic Data Collection ◽  
Third Dimension ◽  
Evolution Of Microstructure

In all fields of microscopy there is an increasing interest in the quantification of microstructure. This interest may stem from a desire to establish quality control parameters or may have a more fundamental requirement involving the derivation of parameters which partially or completely define the three dimensional nature of the microstructure. This latter categorey of study may arise from an interest in the evolution of microstructure or from a desire to generate detailed property/microstructure relationships. In the more fundamental studies some convolution of two-dimensional data into the third dimension (stereological analysis) will be necessary.In some cases the two-dimensional data may be acquired relatively easily without recourse to automatic data collection and further, it may prove possible to perform the data reduction and analysis relatively easily. In such cases the only recourse to machines may well be in establishing the statistical confidence of the resultant data. Such relatively straightforward studies tend to result from acquiring data on the whole assemblage of features making up the microstructure. In this field data mode, when parameters such as phase volume fraction, mean size etc. are sought, the main case for resorting to automation is in order to perform repetitive analyses since each analysis is relatively easily performed.

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