Dirichlet spectrum and heat content

McDonald, Patrick; Meyers, Robert A.

doi:10.1016/s0022-1236(02)00076-9

Cited by 30 publications

(35 citation statements)

References 7 publications

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“…Equation should be interpreted as connecting the

L^{1}

‐moment spectrum to the Mellin transform of the heat content. As noted in , the Mellin transform of the heat content takes the form of a Dirichlet series

ζ_{normalΩ} false(s false) = \sum_{λ \in {spec}^{*} false(normalΩ false)} a_{λ}^{2} {((), \frac{1}{λ})}^{s}

and the

L^{1}

‐moment spectrum is related to the Dirichlet spectrum via the equality

Γ false(k + 1 false) ζ_{normalΩ} false(k false) = T_{k} false(normalΩ false),

where

Γ (k)

is the gamma function. There is a great deal of information contained in the relationship .…”

Section: Background and Notationmentioning

confidence: 99%

“…We call the sequence defined by the

L^{1}

‐ moment spectrum associated to the domain

normalΩ

. It is known that the

L^{1}

‐moment spectrum determines the heat content asymptotics and, generically, the Dirichlet spectrum . Recent work suggests the moment sequence determines a number of geometric invariants of the associated domain and is of value in establishing comparison results (cf.…”

Section: Introductionmentioning

confidence: 99%

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Exit time moments and eigenvalue estimates

Dryden

Langford

McDonald

2017

Bull. London Math. Soc.

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We give upper bounds on the principal Dirichlet eigenvalue associated to a smoothly bounded domain in a complete Riemannian manifold; the bounds involve L1‐norms of exit time moments of Brownian motion. Our results generalize a classical inequality of Pólya. We also prove lower bounds for Dirichlet eigenvalues using invariants that arise during the examination of the relationship between the heat content and exit time moments.

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“…Equation should be interpreted as connecting the

L^{1}

‐moment spectrum to the Mellin transform of the heat content. As noted in , the Mellin transform of the heat content takes the form of a Dirichlet series

ζ_{normalΩ} false(s false) = \sum_{λ \in {spec}^{*} false(normalΩ false)} a_{λ}^{2} {((), \frac{1}{λ})}^{s}

and the

L^{1}

‐moment spectrum is related to the Dirichlet spectrum via the equality

Γ false(k + 1 false) ζ_{normalΩ} false(k false) = T_{k} false(normalΩ false),

where

Γ (k)

is the gamma function. There is a great deal of information contained in the relationship .…”

Section: Background and Notationmentioning

confidence: 99%

“…We call the sequence defined by the

L^{1}

‐ moment spectrum associated to the domain

normalΩ

. It is known that the

L^{1}

Section: Introductionmentioning

confidence: 99%

Exit time moments and eigenvalue estimates

Dryden

Langford

McDonald

2017

Bull. London Math. Soc.

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“…We conclude, using the theory of classical moment problems (as in [MM1], [MM2]), Proposition 2.6. Let D, D be domains in G with nonempty boundary.…”

Section: Background and Notationmentioning

confidence: 73%

“…Let D, D be domains in G with nonempty boundary. Then The moment spectrum contains a great deal of information related to the spectral (and, in the smooth case, Riemannian) geometry of D. For example, in the smooth case it is a theorem that the moment spectrum determines the heat content, and for generic domains in Euclidean space, the moment spectrum determines the Dirichlet spectrum [MM2].…”

Section: Background and Notationmentioning

confidence: 99%

“…This provides good reasons to believe that the same might be true in the continuous case. Preliminary results which clarify the relationship of the heat content to Dirichlet spectrum in the continuous case can be found in [MM2]. This paper is organized as follows: In the second section we recall the necessary facts from graph theory, discrete potential theory and the theory of Markov chains.…”

Section: Introductionmentioning

confidence: 99%

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Isospectral polygons, planar graphs and heat content

McDonald

Meyers

2003

Proc. Amer. Math. Soc.

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Abstract. Given a pair of planar isospectral, nonisometric polygons constructed as a quotient of the plane by a finite group, we construct an associated pair of planar isospectral, nonisometric weighted graphs. Using the natural heat operators on the weighted graphs, we associate to each graph a heat content. We prove that the coefficients in the small time asymptotic expansion of the heat content distinguish our isospectral pairs. As a corollary, we prove that the sequence of exit time moments for the natural Markov chains associated to each graph, averaged over starting points in the interior of the graph, provides a collection of invariants that distinguish isospectral pairs in general.

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Logarithmic expected packet delivery delay in mobile ad hoc wireless networks

Rio

Sarkar

2004

Wireless Communications

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It has been shown that in a mobile ad hoc wireless network per node throughput remains constant as the number of nodes approaches infinity, when the nodes lie in a disk of unit area. However, this requires that some packets are delivered indirectly by a relay node. In this paper, we prove, in the same setup, that D(k) 2 Â(log k), where D(k) is the expected delay time of a mobile ad hoc wireless network with k nodes. As consequence, we have concluded that mobile ad hoc wireless networks are not scalable for real-time applications. z Observe that the rôle played by the unitary disk in Reference [6] is not of importance for their results. For example, the estimate lim z!0 FðzÞ=z 2= ¼ , obtained in Reference [6], does not depend on the fact that the nodes lie on the unitary disk. This estimate only depends on the fact that the measure used on the unitary disk is the restriction of the Lebesgue measure. The same estimate can be obtained for the unitary square, see Reference [11].

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Dirichlet spectrum and heat content

Cited by 30 publications

References 7 publications

Exit time moments and eigenvalue estimates

Exit time moments and eigenvalue estimates

Isospectral polygons, planar graphs and heat content

Logarithmic expected packet delivery delay in mobile ad hoc wireless networks

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