By Vasili M. Babic, Vladimir S. Buldyrev, Edward F. Kuester

ISBN-10: 0387191895

ISBN-13: 9780387191898

ISBN-10: 3540191895

ISBN-13: 9783540191896

Within the examine of short-wave diffraction difficulties, asymptotic tools - the ray strategy, the parabolic equation process, and its additional improvement because the "etalon" (model) challenge strategy - play an incredible position. those are the how you can be handled during this publication. The functions of asymptotic equipment within the thought of wave phenomena are nonetheless faraway from being exhausted, and we are hoping that the ideas set forth the following might help in fixing a few difficulties of curiosity in acoustics, geophysics, the physics of electromagnetic waves, and maybe in quantum mechanics. moreover, the e-book will be of use to the mathematician attracted to modern difficulties of mathematical physics. every one bankruptcy has been annotated. those notes supply a short historical past of the matter and cite references facing the content material of that exact bankruptcy. the most textual content mentions in simple terms these courses that designate a given argument or a selected calculation. to be able to store paintings for the reader who's attracted to just some of the issues thought of during this ebook, we've got incorporated a move chart indicating the interdependence of chapters and sections. The authors reflect on it their friendly responsibility to thank M.M. Popov, who, on the authors' request, wrote Sects. 10.1-7, and in addition los angeles. Molotkov, who was once focused on all phases of getting ready this ebook. l. a.. Molotkov wrote Sect. 7.5 and such a lot of Chap. eleven, and was once our coauthor in writing Chap. 6. we're indebted to contributors on the seminar of the Leningrad element of the V.A. Steklov Institute of arithmetic and Leningrad kingdom collage at the mathematical idea of diffraction, yet specially to V.F. Lazutkin, for his or her confident feedback. We owe greatly to the editors of the ebook. Their cautious paintings, now and then going some distance past the bounds of direct editorial tasks, enabled us to do away with a couple of shortcomings within the manuscript

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**Additional resources for Short-wavelength diffraction theory: asymptotic methods**

**Example text**

F , the evaluation algorithm, is a deterministic algorithm that takes as input parameters I and an element x ∈ DI , and outputs an element y ∈ RI . We write this as y := fI (x). ) Π is a permutation family if the following additionally hold: 1. For all I output by Gen, the distribution defined by the output of Samp(I) is (statistically close to) the uniform distribution on DI . 2. For all I output by Gen it holds that DI = RI and the function fI is a bijection. If Π is a permutation family and there exists a polynomial p such that DI = {0, 1} p(k) for all I output by Gen(1k ), then we say that Π is a permutation family over bit-strings.

Samp, the sampling algorithm, is a probabilistic algorithm that takes as input parameters I and outputs an element x ∈ DI whose distribution is statistically close to the uniform distribution over DI . We usually leave Samp implicit, and just write x ← DI . • f0 and f1 , the evaluation algorithms, are deterministic algorithms that take as input parameters I and an element x ∈ DI , and output an element y ∈ DI . We write this as y := fI,0 (x) or y := fI,1 (x). 42 2 Cryptographic Hardness Assumptions • The following is negligible for all PPT algorithms A: Pr (I, td) ← Gen(1k ); (x0 , x1 ) ← A(I) : fI,0 (x0 ) = fI,1 (x1 ) .

The distribution on r is statistically close to uniform.

### Short-wavelength diffraction theory: asymptotic methods by Vasili M. Babic, Vladimir S. Buldyrev, Edward F. Kuester

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