5. To infinity …

2020 ◽  
pp. 84-108
Author(s):  
Glen Van Brummelen
Keyword(s):  
Fourier Analysis ◽  
Trigonometric Series ◽  
Gibbs Phenomenon ◽  
Cartesian Coordinates ◽  
Mathematical Objects ◽  
Major Shift

‘To infinity … ’ looks at how infinite trigonometric series are used to compute π. It shows how Machin’s formula used the inverse tangent series to compute π to a hundred places. Lord Kelvin’s use of Fourier analysis in studying tide behaviour is also explained along with the Gibbs phenomenon. The invention of Cartesian coordinates and calculus in the 17th and 18th centuries reflects a major shift in mathematics. Geometry gradually changed from being synthetic (in the style of Euclid) to being analytic. The basic mathematical objects―originally points, lines, and shapes as well as numbers―became functions that accept input quantities and produce output quantities.

THE USE OF LINEAR FILTERING IN GRAVITY PROBLEMS

Geophysics ◽  
1971 ◽  
Vol 36 (6) ◽  
pp. 1174-1203 ◽  
Author(s):  
C. C. Ku ◽  
W. M. Telford ◽  
S. H. Lim
Keyword(s):  
Fourier Transform ◽  
Fourier Analysis ◽  
Frequency Domain ◽  
Gibbs Phenomenon ◽  
Downward Continuation ◽  
Main Lobe ◽  
Linear Filtering ◽  
Space Domain ◽  
Actual Field ◽  
High Pass

The technique of Fourier analysis is reviewed and the equivalence and relative advantages of convolution filtering in the space domain and multiplication filtering in the frequency domain are demonstrated with actual field examples. We discuss the design of ideal filters in terms of the relationships between the main lobe and the side lobes. Cut‐and‐try methods appear to favor the hanning window or the hamming window, since these windows minimize the Gibbs phenomenon associated with the downward continuation or high‐pass filtering operation. New sets of coefficients for convolution filtering, based upon Fourier transform theory and the sampling theory, are derived.

The LDE-Type Flare Occurrence (1969-1993)

1994 ◽  
Vol 144 ◽  
pp. 279-282
Author(s):  
A. Antalová
Keyword(s):  
Time Series ◽  
Fourier Analysis ◽  
Sunspot Cycle ◽  
List Type ◽  
Type Number ◽  
Solar Cycles ◽  
Type Iv ◽  
Long Duration ◽  
Cycle Maximum ◽  
Flare Index

AbstractThe occurrence of LDE-type flares in the last three cycles has been investigated. The Fourier analysis spectrum was calculated for the time series of the LDE-type flare occurrence during the 20-th, the 21-st and the rising part of the 22-nd cycle. LDE-type flares (Long Duration Events in SXR) are associated with the interplanetary protons (SEP and STIP as well), energized coronal archs and radio type IV emission. Generally, in all the cycles considered, LDE-type flares mainly originated during a 6-year interval of the respective cycle (2 years before and 4 years after the sunspot cycle maximum). The following significant periodicities were found:• in the 20-th cycle: 1.4, 2.1, 2.9, 4.0, 10.7 and 54.2 of month,• in the 21-st cycle: 1.2, 1.6, 2.8, 4.9, 7.8 and 44.5 of month,• in the 22-nd cycle, till March 1992: 1.4, 1.8, 2.4, 7.2, 8.7, 11.8 and 29.1 of month,• in all interval (1969-1992):a)the longer periodicities: 232.1, 121.1 (the dominant at 10.1 of year), 80.7, 61.9 and 25.6 of month,b)the shorter periodicities: 4.7, 5.0, 6.8, 7.9, 9.1, 15.8 and 20.4 of month.Fourier analysis of the LDE-type flare index (FI) yields significant peaks at 2.3 - 2.9 months and 4.2 - 4.9 months. These short periodicities correspond remarkably in the all three last solar cycles. The larger periodicities are different in respective cycles.

FOURIER ANALYSIS OF HYSTERETIC DAMPING MECHANISMS

1985 ◽  
Vol 46 (C10) ◽  
pp. C10-171-C10-173 ◽  
Author(s):  
D. N. BESHERS ◽  
V. F. CORONEL
Keyword(s):  
Fourier Analysis ◽  
Hysteretic Damping
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