By John Greenlees
This ebook involves a chain of expository articles on axiomatic, enriched and motivic homotopy concept coming up out of a NATO complicated research Institute of a similar identify on the Isaac Newton Institute for the Mathematical Sciences in Cambridge, united kingdom in September 2002.
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18 LINEAR SYSTEMS AND TRANSFORMS When fy ¼ 0, 1 ð Uðfx ; 0Þ ¼ ½h2 ðxÞ À h1 ðxÞeÀj 2p fx x dx À1 ¼ H2 ðfx Þ À H1 ðfx Þ Thus, Uðfx ; 0Þ is the difference of the 1-D FT of the functions h2 ðxÞ and h1 ðxÞ. 6 REAL FOURIER TRANSFORM Sometimes it is more convenient to represent the Fourier transform with real sine and cosine basis functions. Then, it is referred to as the real Fourier transform (RFT). What was discussed as the Fourier transform before would then be the complex Fourier transform (CFT) [Ersoy, 1994].
3-12) can be written as h pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃi 2 2 Uðx; y; zÞ ¼ F À1 F½Uðx; y; 0Þejkz 1Àax Àay ð4:3-15Þ Aðfx ; fy ; 0Þ ¼ F½Uðx; y; 0Þ for particular values of fx and fy is the complex amplitude of a plane wave traveling in the direction specified by the direction 1 cosines ax ¼ 2pfx , ay ¼ 2pfy , and az ¼ 2pfz , where fz ¼ 2p ½1 À a2x À a2y 1=2 . The effect of propagation is to modify the relative phases of the various plane waves by pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ 2 2 e jkz 1Àax Àay without changing their amplitudes.
Then, we can write E ¼ E0 ejðkzþwtÞ ex E0 H ¼ À ejðkzþwtÞ ey ; ð3:7-18Þ where Real  is assumed from the context. Intensity or irradiance I is defined as the time-averaged power given by w I¼ 2p 2p=w ð jSjdt ¼ em E02 ; 2 ð3:7-19Þ 0 where S is the Poynting vector. It is observed that the intensity is proportional to the square of the field magnitude. This will be assumed to be true in general unless otherwise specified. 1 INTRODUCTION When the wavelength of a wave field is larger than the ‘‘aperture’’ sizes of the diffraction device used to control the wave, the scalar diffraction theory can be used.
Axiomatic, Enriched and Motivic Homolopy Theory by John Greenlees