Permittivity profile inversion of a one-dimensional medium using the fractional linear transformation and the phase compensation constant

2001 ◽  
Vol 31 (1) ◽  
pp. 10-13
Author(s):  
Choongho Song ◽  
Sangseol Lee
Keyword(s):  
Linear Transformation ◽  
Phase Compensation ◽  
Fractional Linear Transformation ◽  
One Dimensional ◽  
Profile Inversion

scholarly journals Notes on projective structures on complex manifolds

1989 ◽  
Vol 116 ◽  
pp. 63-88 ◽  
Author(s):  
Björn Gustafsson ◽  
Jaak Peetre
Keyword(s):  
Riemann Surface ◽  
Linear Transformation ◽  
Projective Structure ◽  
Complex Manifolds ◽  
Projective Structures ◽  
Fractional Linear Transformation ◽  
Local Coordinates

Consider a Riemann surface X equipped with a projective structure, that is, a covering of X with coordinate neighborhoods U and corresponding (holomorphic) local coordinates {t} such that in the intersection U ∩ U′ of any two such coordinate neighborhoods U and U′ change of local coordinates is mediated by a fractional linear transformation

scholarly journals Sums of Linear Transformations in Higher Dimensions

2019 ◽  
Vol 70 (3) ◽  
pp. 965-984 ◽  
Author(s):  
Akshat Mudgal
Keyword(s):  
Natural Number ◽  
Linear Transformation ◽  
Finite Subset ◽  
Absolute Constant ◽  
Higher Dimensions ◽  
Dimensional Subspace ◽  
Main Term ◽  
Linear Transformations ◽  
One Dimensional

AbstractIn this paper, we prove the following two results. Let d be a natural number and q, s be co-prime integers such that 1<qs<|q|. Then there exists a constant δ>0 depending only on q, s and d such that for any finite subset A of ℝd that is not contained in a translate of a hyperplane, we have |q⋅A+s⋅A|≥(|q|+|s|+2d−2)|A|−Oq,s,d(|A|1−δ).The main term in this bound is sharp and improves upon an earlier result of Balog and Shakan. Secondly, let L∈GL2(ℝ) be a linear transformation such that L does not have any invariant one-dimensional subspace of ℝ2. Then, for all finite subsets A of ℝ2, we have |A+L(A)|≥4|A|−O(|A|1−δ), for some absolute constant δ>0. The main term in this result is sharp as well.

XX. On the classification of Loci

1878 ◽  
Vol 169 ◽  
pp. 663-681 ◽  
Keyword(s):  
Finite Number ◽  
Linear Transformation ◽  
Infinite System ◽  
Flat Space ◽  
One Dimensional

By a curve we mean a continuous one-dimensional aggregate of any sort of elements, and therefore not merely a curve in the ordinary geometrical sense, but also a singly infinite system of curves, surfaces, complexes, &c., such that one condition is sufficient to determine a finite number of them. The elements may be regarded as determined by k coordinates; and then, if these be connected by k —1 equations of any order, the curve is either the whole aggregate of common solutions of these equations, or, when this breaks up into algebraically distinct parts, the curve is one of these parts. It is thus convenient to employ still further the language of geometry, and to speak of such a curve as the complete or partial intersection of k —1 loci in flat space of k dimensions, or, as we shall sometimes say, in a k -flat. If a certain number, say h , of the equations are linear, it is evidently possible by a linear transformation to make these equations equate h of the coordinates to zero ; it is then convenient to leave these coordinates out of consideration altogether, and only to regard the remaining k — h —1 equations between k — h coordinates. In this case the curve will, therefore, be regarded as a curve in flat space of k — h dimensions. And, in general, when we speak of a curve as in flat space of k dimensions, we mean that it cannot exist in flat space of k —1 dimensions.

Linear Groups Generated by Reflection Tori

1999 ◽  
Vol 51 (6) ◽  
pp. 1149-1174 ◽  
Author(s):  
A. M. Cohen ◽  
H. Cuypers ◽  
H. Sterk
Keyword(s):  
Vector Space ◽  
Linear Transformation ◽  
Arbitrary Dimension ◽  
General Linear ◽  
Linear Groups ◽  
One Dimensional ◽  
General Linear Groups ◽  
Dimensional Group ◽  
Fixed Axis

AbstractA reflection is an invertible linear transformation of a vector space fixing a given hyperplane, its axis, vectorwise and a given complement to this hyperplane, its center, setwise. A reflection torus is a one-dimensional group generated by all reflections with fixed axis and center.In this paper we classify subgroups of general linear groups (in arbitrary dimension and defined over arbitrary fields) generated by reflection tori.

Three‐dimensional velocity profile inversion from finite‐offset scattering data

Geophysics ◽  
1981 ◽  
Vol 46 (6) ◽  
pp. 837-842 ◽  
Author(s):  
S. Raz
Keyword(s):  
Velocity Profile ◽  
Three Dimensional ◽  
Scattering Data ◽  
One Dimensional ◽  
Velocity Variations ◽  
Profile Inversion ◽  
Relationship Of ◽  
The Relationship

The reconstruction of three‐dimensional (3-D) velocity variations from finite‐offset scattering data is formulated. Reduction to the limiting cases of zero and small offset distances as well as the case of one‐dimensional (1-D) stratification is given. An inherent increase in complexity is cited and interpreted. The relationship of the proposed inversion to the F-K migration is discussed.

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