The 2-Transitive Transplantable Isospectral Drums

Schillewaert, Jeroen; Thas, Koen

doi:10.3842/sigma.2011.080

Cited by 5 publications

(5 citation statements)

References 18 publications

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“…For each of the 7 pairs mentioned above, G is isomorphic to P SL(3, 2) × P SL(3, 2). Its permutation action on the vertices is imprimitive, in particular, it is not 2-transitive, which gives a counterexample to a conjecture expressed in [ST11]. 5.2.…”

Section: Examples Of Transplantable Pairsmentioning

confidence: 97%

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On inaudible properties of broken drums - Isospectrality with mixed Dirichlet-Neumann boundary conditions

Herbrich

2011

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We study isospectrality for manifolds with mixed Dirichlet-Neumann boundary conditions and express the well-known transplantation method in graph-and representationtheoretic terms. This leads to a characterization of transplantability in terms of monomial relations in finite groups and allows for the generating of new transplantable pairs from given ones as well as a computer-aided search for isospectral pairs. In particular, we show that the Dirichlet spectrum of a manifold does not determine whether it is connected and that an orbifold can be Dirichlet isospectral to a manifold.

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Section: Examples Of Transplantable Pairsmentioning

confidence: 97%

“…Thas et al [Tha06a,Tha06b,Tha06c,ST11] partially classified the groups appearing in (2.10) for transplantable treelike graphs with uniform loop signs. Recall that transplantable manifolds have equal heat invariants and therefore share certain geometric properties, some of which can be identified with expressions of the form (2.9):…”

Section: Okada and Shudomentioning

confidence: 99%

On inaudible properties of broken drums - Isospectrality with mixed Dirichlet-Neumann boundary conditions

Herbrich

2011

Preprint

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“…Finally, all the strong Gassmann-Sunada triples (U, V, W ) where U acts 2-transitively on U/V and U/W , and which satisfy INV, are classified in [26].…”

Section: Theorem 34 ([3]mentioning

confidence: 99%

The geometry of drums

Thas

2017

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We introduce the new concept of D-geometry (or "drum geometry"), which has been recently discovered by the author in [33] when constructing and classifying isospectral and length equivalent drums under certain constraints. We will show that any pair of length equivalent domains, and in particular any pair of isospectral domains (which makes one unable to "hear the shape of drums") which is constructed by the famous Gassmann-Sunada method, naturally defines a D-geometry, and that each D-geometry gives rise to such domains. One goal of this letter is to show that in the present theory of isospectral and length equivalent drums, many examples are controlled by finite geometrical phenomena in a very precise sense.

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“…The same principle holds for all the other examples, for both the Dirichlet and Neumann Laplacians, known from [11,16,17,23] etc. ; see [22, Section IV] and also the classifications in [33,39]. 1 There is a slight technicality here: if Ω 1 is an open set and Ω 2 differs from it by a set of capacity zero, then the Laplacians on L 2 (Ω 1 ) and L 2 (Ω 2 ) coincide without the sets being congruent; see for example [4,Section 3].…”

Section: Introductionmentioning

confidence: 99%

Disjointness-preserving operators and isospectral Laplacians

Arendt¹,

Kennedy²

2020

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All the known counterexamples to Kac' famous question "can one hear the shape of a drum", i.e., does isospectrality of two Laplacians on domains imply that the domains are congruent, consist of pairs of domains composed of copies of isometric building blocks arranged in different ways, such that the unitary operator intertwining the Laplacians acts as a sum of overlapping "local" isometries mapping the copies to each other.We prove and explore a complementary positive statement: if an operator intertwining two appropriate realisations of the Laplacian on a pair of domains preserves disjoint supports, then under additional assumptions on it generally far weaker than unitarity, the domains are congruent. We show this in particular for the Dirichlet, Neumann and Robin Laplacians on spaces of continuous functions and on L 2 -spaces.

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The 2-Transitive Transplantable Isospectral Drums

Cited by 5 publications

References 18 publications

On inaudible properties of broken drums - Isospectrality with mixed Dirichlet-Neumann boundary conditions

On inaudible properties of broken drums - Isospectrality with mixed Dirichlet-Neumann boundary conditions

The geometry of drums

Disjointness-preserving operators and isospectral Laplacians

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