Modulary groups of t×t matrices

Newman, Morris; Smart, John

doi:10.1215/s0012-7094-63-03028-x

Cited by 47 publications

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“…This Lorentz transformation freedom may be viewed as providing a choice of "inertial frame" normalisation at each radius, and may be used to normalise certain of the remaining metric coefficients, as described below. This interpretation is supported by a comparison [9] between the Robinson-Trautman metrics and the NU form [10] of the Minkowski metric in coordinates using null cones with base point describing a timelike curve. This comparison may also be made in the NQS coordinates, and supports both the interpretation of Robinson-Trautman spacetimes as describing an accelerated black hole rapidly settling down to a Schwarzschild black hole in uniform motion, and the interpretation of the NQS freedom as representing a choice of reference frame at each radius and time.…”

Section: Introductionmentioning

confidence: 88%

Shear-free null quasi-spherical space–times

Bartnik

1997

Journal of Mathematical Physics

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We study the residual gauge freedom within the null quasi-spherical (NQS) gauge for spacetimes admitting an expanding shear-free null foliation. By constructing the most general NQS coordinates subordinate to such a foliation, we obtain both a clear picture of the geometric nature of the residual coordinate freedom, and an explicit construction of nontrivial NQS metrics representing some well-known spacetimes, such as Schwarzschild, accelerated Minkowski, and Robinson-Trautman. These examples will be useful in testing numerical evolution codes. The geometric gauge freedom consists of an arbitrary boost and rotation at each coordinate sphere -and this freedom may be used to normalise the coordinate to an "inertial" frame.

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Section: Introductionmentioning

confidence: 88%

Shear-free null quasi-spherical space–times

Bartnik

1997

Journal of Mathematical Physics

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“…different ways. To prove the last part of the theorem we observe that, if 2 and these are equal, by (3). If C is a common transversal, we have…”

Section: A Well-known Theorem In Group Theorymentioning

confidence: 93%

Common Transversals

Rankin¹

1966

Proceedings of the Edinburgh Mathematical Society

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A well-known theorem in group theory [(8), p. 11, Satz 3] asserts that, when H is a subgroup of finite index in a group G, there exists a system of common representatives of the right cosets and the left cosets of H in G. Various proofs and generalisations, mainly involving combinatorial rather than grouptheoretical ideas, are known, and an excellent account of the subject is to be found in Chapter 5 of Ryser's book (6), where references to the literature are given. The purpose of the present paper is to use group-theoretical ideas to prove theorems of a similar nature. The motivation for this work comes from the theory of Hecke operators, and one of the main objects is to provide a simple proof of a result given by Petersson (4), which is needed in order to prove the normality of these operators.If a set S is partitioned in r ways as a union of disjoint non-void subsets, a subset C of S is called a system of common representatives, or a common transversal, for the r partitions when C contains exactly one element in common with each of the subsets. Usually we shall take r = 2, but we also give some results for r>2 in § 3.We shall find the following notation useful. Let 5 be a set on whose elements a binary operation (denoted by juxtaposition) is defined. Let A and B be subsets of S and put C = {c: c = ab, a e A, b e B). Then we write, as usual, C = AB. If, however, each c e C can be expressed uniquely in the form c = ab, where ae A and beB, we write C = A . B.

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“…In particular, M. Newman [5] and D. JVlcQuillan [4] have shown that if Γ(n) C G CΓ (1), (n, 6) = 1, then G = Γ(d) or Γ(d) where d \ n. The only additional result that we need is (2.1) f(n)Γ(k) = Γ(l), (n,fc)=l, which was proved by M. Newman and J. R. Smart in [6].…”

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confidence: 84%

“…is an element of G(ra (1/2) )' but is certainly not an element of Γ 2 (8) when m = 2 or Γ 3 (6) when m = 3.…”

Section: Classification Theorems For Normal Congruence Subgroupsmentioning

confidence: 99%

Normal congruence subgroups of the Hecke groupsG(2^(1∕2)) andG(3^(1∕2))

Parson¹

1977

Pacific J. Math.

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Modulary groups of t×t matrices

Cited by 47 publications

References 0 publications

Shear-free null quasi-spherical space–times

Shear-free null quasi-spherical space–times

Common Transversals

Normal congruence subgroups of the Hecke groupsG(2^(1∕2)) andG(3^(1∕2))

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Modulary groups of t×t matrices

Cited by 47 publications

References 0 publications

Shear-free null quasi-spherical space–times

Shear-free null quasi-spherical space–times

Common Transversals

Normal congruence subgroups of the Hecke groupsG(2(1∕2)) andG(3(1∕2))

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Normal congruence subgroups of the Hecke groupsG(2^(1∕2)) andG(3^(1∕2))