The dominant asymptotic term of solution to the Hamer equation

2013 ◽  
Vol 81 (2) ◽  
pp. 121-134
Author(s):  
A.A. Soloviev
Keyword(s):  
Asymptotic Term

Emendations to a proof in the general three-dimensional theory of oscillating sources of waves

1990 ◽  
Vol 427 (1872) ◽  
pp. 31-42 ◽  
Keyword(s):  
Source Region ◽  
Three Dimensional ◽  
Radiation Condition ◽  
Careful Analysis ◽  
Asymptotic Result ◽  
Dimensional Theory ◽  
Asymptotic Term ◽  
Anisotropic System ◽  
Cauchy's Theorem ◽  
Asymptotic Forms

Asymptotic forms far from the source region of waves generated by oscillating sources in a linear homogeneous anisotropic system were derived by a method of proof that requires emendation although the final conclusions remain unchanged. An intermediate asymptotic result, in the form of an integral over that part S + of the whole real wavenumber surface S on which a certain inequality (related to the radiation condition) is satisfied, needs modification as described in §2; but, as shown in §3, it is the modified form that is correctly estimated as in the final conclusions. Thus the proof is given two necessary emendations that cancel out. A careful analysis in §4 of why they cancel shows that the original intermediate result regains validity if S + , besides including that part of the real wavenumber surface S on which the inequality ∂ ω /∂ k 1 > 0 is satisfied (where ω is frequency and k 1 the component of wavenumber in the direction chosen for wave estimation), is considered as being continued on the complex wavenumber surface S, beyond the curve C on which ∂ ω /∂ k 1 = 0, in the negative pure-imaginary k 1 -direction. This change is required to ensure the proper application of Cauchy’s theorem. Furthermore, the removal of any discontinuity at C prevents the appearance of an additional asymptotic term that would be unavoidably associated with such a singularity. I am grateful to Professor V. A. Borovikov for stimulating me to make these necessary clarifications.

How close is the nearest node in a wireless network?

2020 ◽  
Author(s):  
Amy L L Middleton ◽  
Antal A Járai ◽  
Jonathan H P Dawes ◽  
Keith Briggs
Keyword(s):  
Communication Networks ◽  
Model Problem ◽  
Path Loss ◽  
Small Cell ◽  
Nearest Neighbour ◽  
Distributed Network ◽  
Asymptotic Term ◽  
Spatially Distributed ◽  
And Performance ◽  
Theoretical Results

Abstract The ability of small-cell wireless networks to self-organize is crucial for improving capacity and performance in modern communication networks. This paper considers one of the most basic questions: what is the expected distance to a cell’s nearest neighbour in a spatially distributed network? We analyse a model problem in the asymptotic limit of large total received signal and compare the accuracy of different heuristics. We also analytically consider the effects of fading. Our analysis shows that the most naive heuristic systematically underestimates the distance to the nearest node; this is substantially corrected in cases of interest by inclusion of the next-order asymptotic term. We illustrate our theoretical results explicitly or several combinations of signal and path loss parameters and show that our theory is well supported by numerical simulations.

scholarly journals Quantum random walks in one dimension via generating functions

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Andrew Bressler ◽  
Robin Pemantle
Keyword(s):  
Random Walks ◽  
Generating Functions ◽  
Nearest Neighbor ◽  
Function Analysis ◽  
One Dimension ◽  
Quantum Random Walks ◽  
Linear Speed ◽  
One Dimensional ◽  
Asymptotic Term ◽  
International Audience

International audience We analyze nearest neighbor one-dimensional quantum random walks with arbitrary unitary coin-flip matrices. Using a multivariate generating function analysis we give a simplified proof of a known phenomenon, namely that the walk has linear speed rather than the diffusive behavior observed in classical random walks. We also obtain exact formulae for the leading asymptotic term of the wave function and the location probabilities.

Illustration of the anisotropic Porod law

2002 ◽  
Vol 35 (3) ◽  
pp. 304-313 ◽  
Author(s):  
S. Ciccariello ◽  
J.-M. Schneider ◽  
B. Schönfeld ◽  
G. Kostorz
Keyword(s):  
Small Angle ◽  
Small Angle Scattering ◽  
Two Dimensional ◽  
Simple Derivation ◽  
Ellipsoidal Particles ◽  
Asymptotic Term ◽  
Angle Scattering ◽  
Porod Law

A simple derivation of the leading asymptotic term of small-angle scattering intensities relevant to anisotropic particulate samples is reported. The result is illustrated for the case of ellipsoidal particles. It is discussed under what circumstances the anisotropic Porod law can be applied to scattering intensities collected by a two-dimensional detector.

Leading asymptotic term of the solution to the hamer equation

2011 ◽  
Vol 84 (1) ◽  
pp. 555-557
Author(s):  
A. A. Soloviev
Keyword(s):  
Asymptotic Term

scholarly journals On the exterior magnetic field and silent sources in magnetoencephalography

2004 ◽  
Vol 2004 (4) ◽  
pp. 307-314 ◽  
Author(s):  
George Dassios ◽  
Fotini Kariotou
Keyword(s):  
Magnetic Field ◽  
Dipole Moment ◽  
Physical Reality ◽  
Dipole Source ◽  
Position Vector ◽  
Asymptotic Term ◽  
Mathematical Approximation ◽  
Quadrupole Term ◽  
The Brain ◽  
The Magnetic Field

Two main results are included in this paper. The first one deals with the leading asymptotic term of the magnetic field outside any conductive medium. In accord with physical reality, it is proved mathematically that the leading approximation is a quadrupole term which means that the conductive brain tissue weakens the intensity of the magnetic field outside the head. The second one concerns the orientation of the silent sources when the geometry of the brain model is not a sphere but an ellipsoid which provides the best possible mathematical approximation of the human brain. It is shown that what characterizes a dipole source as “silent” is not the collinearity of the dipole moment with its position vector, but the fact that the dipole moment lives in the Gaussian image space at the point where the position vector meets the surface of the ellipsoid. The appropriate representation for the spheroidal case is also included.

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