Fourier Expansions of Doubly Periodic Functions of the Third Kind

1930 ◽  
Vol 31 (4) ◽  
pp. 641
Author(s):  
John D. Elder
Keyword(s):  
Periodic Functions ◽  
The Third ◽  
Fourier Expansions

Periodic Models of Seasonality

2013 ◽  
Author(s):  
Lars Peter Hansen ◽  
Thomas J. Sargent
Keyword(s):  
Spectral Density ◽  
Competitive Equilibrium ◽  
Information Flows ◽  
Periodic Functions ◽  
Third Way ◽  
Monthly Data ◽  
The Third ◽  
Periodic Models ◽  
Time Invariant ◽  
Over Time

Until now, each of the matrices defining preferences, technologies, and information flows has been specified to be constant over time. This chapter relaxes this assumption and lets the matrices be strictly periodic functions of time. The aim is to apply and extend an idea of Denise Osborn (1988) and Richard Todd (1983, 1990) to arrive at a model of seasonality as a hidden periodicity. Seasonality can be characterized in terms of a spectral density. A variable is said to “have a seasonal” if its spectral density displays peaks at or in the vicinity of the frequencies commonly associated with the seasons of the year, for example, every 12 months for monthly data, every four quarters for quarterly data. Within a competitive equilibrium, it is possible to think of three ways of modeling seasonality. The first two ways can be represented within the time-invariant setup of our previous chapters, while the third way departs from the assumption that the matrices that define our economies are time invariant. The chapter focuses on a third model of seasonality following Todd. It specifies an economy in terms of matrices whose elements are periodic functions of time.

scholarly journals Fourier-Based Automatic Transformation between Mapping Shapes—Cadastral and Land Registry Applications

2020 ◽  
Vol 9 (8) ◽  
pp. 482
Author(s):  
Juan Francisco Reinoso-Gordo ◽  
Rocío Romero-Zaliz ◽  
Carlos León-Robles ◽  
Jesús Mataix-SanJuan ◽  
Marcelo Antonio Nero
Keyword(s):  
Computer Vision ◽  
Information System ◽  
Geographic Information System ◽  
Least Squares ◽  
Reference Object ◽  
Fourier Descriptors ◽  
Periodic Functions ◽  
Fourier Expansions ◽  
Elliptic Fourier Descriptors ◽  
Least Squares Approach

Sometimes it is necessary to know the transformation to apply to a mapping shape in order to locate its true place. Such an operation can be computed if a corresponding reference object exists and we can identify corresponding points in both shapes. Nevertheless our approach does not need to match any corresponding point beforehand. The method proposed defines a polygon in the frequency domain—two periodic functions are derived from a polygonal or polygon. According to the theory of elliptic Fourier descriptors those two periodic functions can be expressed by Fourier expansions. The transformation can be computed using the coefficients of the harmonics from the corresponding shapes without taking into account where each polygon vertex is placed in the spatial domain. The transformation parameters will be derived by a least squares approach. The geomatics and geosciences applications of this method go from photogrammetry, geographic information system, computer vision, to cadaster and real estates.

scholarly journals A Korovkin Type Approximation Theorem and Its Applications

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Malik Saad Al-Muhja
Keyword(s):  
Approximation Theorem ◽  
Modulus Of Smoothness ◽  
Linear Operators ◽  
Stieltjes Integral ◽  
Periodic Functions ◽  
Third Order ◽  
Korovkin Type Approximation Theorem ◽  
The Third ◽  
Type Approximation ◽  
Statistical Limit

We present a Korovkin type approximation theorem for a sequence of positive linear operators defined on the space of all real valued continuous and periodic functions viaA-statistical approximation, for the rate of the third order Ditzian-Totik modulus of smoothness. Finally, we obtain an interleave between Riesz's representation theory and Lebesgue-Stieltjes integral-i, for Riesz's functional supremum formula via statistical limit.

Progress report on 21-cm observations at low latitude between longitudes (l 11) 0° and 66°

1967 ◽  
Vol 31 ◽  
pp. 177-179
Author(s):  
W. W. Shane
Keyword(s):  
Preliminary Report ◽  
Progress Report ◽  
Galactic Centre ◽  
Low Latitude ◽  
The Third ◽  
Introductory Lecture ◽  
Centre Region

In the course of several 21-cm observing programmes being carried out by the Leiden Observatory with the 25-meter telescope at Dwingeloo, a fairly complete, though inhomogeneous, survey of the regionl11= 0° to 66° at low galactic latitudes is becoming available. The essential data on this survey are presented in Table 1. Oort (1967) has given a preliminary report on the first and third investigations. The third is discussed briefly by Kerr in his introductory lecture on the galactic centre region (Paper 42). Burton (1966) has published provisional results of the fifth investigation, and I have discussed the sixth in Paper 19. All of the observations listed in the table have been completed, but we plan to extend investigation 3 to a much finer grid of positions.

The orbit of Pluto over a long interval of time

1966 ◽  
Vol 25 ◽  
pp. 227-229 ◽  
Author(s):  
D. Brouwer
Keyword(s):  
Periodic Orbit ◽  
Numerical Integration ◽  
Restricted Problem ◽  
Three Dimensional ◽  
The Third ◽  
Circular Restricted Problem ◽  
Resonance Relation

The paper presents a summary of the results obtained by C. J. Cohen and E. C. Hubbard, who established by numerical integration that a resonance relation exists between the orbits of Neptune and Pluto. The problem may be explored further by approximating the motion of Pluto by that of a particle with negligible mass in the three-dimensional (circular) restricted problem. The mass of Pluto and the eccentricity of Neptune's orbit are ignored in this approximation. Significant features of the problem appear to be the presence of two critical arguments and the possibility that the orbit may be related to a periodic orbit of the third kind.

Monte Carlo Algorithms for Moments of Transition Arrays

1988 ◽  
Vol 102 ◽  
pp. 79-81
Author(s):  
A. Goldberg ◽  
S.D. Bloom
Keyword(s):  
Monte Carlo ◽  
State Vector ◽  
Basis Vector ◽  
Collective State ◽  
Dispersion Characteristics ◽  
Dipole Strength ◽  
The Third ◽  
Monte Carlo Algorithms ◽  
Third Moment ◽  
Very High

AbstractClosed expressions for the first, second, and (in some cases) the third moment of atomic transition arrays now exist. Recently a method has been developed for getting to very high moments (up to the 12th and beyond) in cases where a “collective” state-vector (i.e. a state-vector containing the entire electric dipole strength) can be created from each eigenstate in the parent configuration. Both of these approaches give exact results. Herein we describe astatistical(or Monte Carlo) approach which requires onlyonerepresentative state-vector |RV> for the entire parent manifold to get estimates of transition moments of high order. The representation is achieved through the random amplitudes associated with each basis vector making up |RV>. This also gives rise to the dispersion characterizing the method, which has been applied to a system (in the M shell) with≈250,000 lines where we have calculated up to the 5th moment. It turns out that the dispersion in the moments decreases with the size of the manifold, making its application to very big systems statistically advantageous. A discussion of the method and these dispersion characteristics will be presented.

Probe size and 5th-order aberration

1987 ◽  
Vol 45 ◽  
pp. 134-135 ◽  
Author(s):  
Zhifeng Shao
Keyword(s):  
Electron Probe ◽  
Spherical Aberration ◽  
Wave Length ◽  
Cylindrical Symmetry ◽  
Electron Wave ◽  
Third Order ◽  
The Third ◽  
Probe Radius ◽  
Small Probe ◽  
Probe Size

A small electron probe has many applications in many fields and in the case of the STEM, the probe size essentially determines the ultimate resolution. However, there are many difficulties in obtaining a very small probe.Spherical aberration is one of them and all existing probe forming systems have non-zero spherical aberration. The ultimate probe radius is given byδ = 0.43Csl/4ƛ3/4where ƛ is the electron wave length and it is apparent that δ decreases only slowly with decreasing Cs. Scherzer pointed out that the third order aberration coefficient always has the same sign regardless of the field distribution, provided only that the fields have cylindrical symmetry, are independent of time and no space charge is present. To overcome this problem, he proposed a corrector consisting of octupoles and quadrupoles.

Sign in / Sign up

Export Citation Format

Share Document