The Eigenvalues of an Equilateral Triangle

Pinsky, Mark A.

doi:10.1137/0511073

Cited by 106 publications

(66 citation statements)

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“…For proofs, see the recent exposition (including completeness) by McCartin [26], building on work of Práger [33]. A different approach is due to Pinsky [30].…”

Section: Eigenfunctions Of the Equilateral Trianglementioning

confidence: 99%

Maximizing Neumann fundamental tones of triangles

Laugesen

Siudeja

2009

Journal of Mathematical Physics

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“…For proofs, see the recent exposition (including completeness) by McCartin [26], building on work of Práger [33]. A different approach is due to Pinsky [30].…”

Section: Eigenfunctions Of the Equilateral Trianglementioning

confidence: 99%

Maximizing Neumann fundamental tones of triangles

Laugesen

Siudeja

2009

Journal of Mathematical Physics

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“…The title of the article refers to the problem which was solved in [20] for an equilateral triangle, which is the fundamental domain of the affine Weyl group W aff , corresponding to the simple Lie algebra A 2 and to the compact group SU (3). Through the use of an entirely different method [21], orbit functions provide a solution to this problem for any compact simple Lie group.…”

Section: Introductionmentioning

confidence: 99%

Orbit Functions

Klimyk¹

2006

SIGMA

192

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Abstract. In the paper, properties of orbit functions are reviewed and further developed. Orbit functions on the Euclidean space E n are symmetrized exponential functions. The symmetrization is fulfilled by a Weyl group corresponding to a Coxeter-Dynkin diagram. Properties of such functions will be described. An orbit function is the contribution to an irreducible character of a compact semisimple Lie group G of rank n from one of its Weyl group orbits. It is shown that values of orbit functions are repeated on copies of the fundamental domain F of the affine Weyl group (determined by the initial Weyl group) in the entire Euclidean space E n . Orbit functions are solutions of the corresponding Laplace equation in E n , satisfying the Neumann condition on the boundary of F . Orbit functions determine a symmetrized Fourier transform and a transform on a finite set of points.

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“…The eigenvalues and the eigenfunctions in (3.1) and (3.3) are the direct consequences of Pinsky [11]. Then for the 3-dimensional case, we obtain (3.2) and (3.4) by applying the separation of variable method.…”

Section: Then the Eigenvalues And The Eigenfunctions Of-a For Diricmentioning

confidence: 99%

“…https://doi.org/10.1017/S1446788700036314 [11] Spatial averaging for inertial manifolds 135 DEFINITION. For a given (bounded Lipschitz) domain ficS",n<3, and choice of boundary conditions for the Laplacian, we say the weaker principle of spatial averaging holds if there exists a quantity £ > 0 such that for every e > 0, K > 0 and any bounded subset SB C H 2 , there exists arbitrarily large k = k(S8) > K, such that The main difference between the weaker PSA and PSA is the choice of k and the upper bound of the estimate (4.10).…”

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confidence: 99%

An extension of the principle of spatial averaging for inertial manifolds

Kwean¹

1999

J Aust Math Soc A

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In this paper we extend a theorem of Mallet-Paret and Sell for the existence of an inertial manifold for a scalar-valued reaction diffusion equation to new physical domains Q n C R", n = 2, 3. For their result the Principle of Spatial Averaging (PSA), which certain domains may possess, plays a key role for the existence of an inertial manifold. Instead of the PSA, we define a weaker PSA and prove that the domains Q n with appropriate boundary conditions for the Laplace operator, A, satisfy a weaker PSA. This weaker PSA is enough to ensure the existence of an inertial manifold for a specific class of scalar-valued reaction diffusion equations on each domain €!" under suitable conditions. 1991 Mathematics subject classification (Amer. Math. Soc): primary 35P20, 34C29, 34C30, 35K57.

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The Eigenvalues of an Equilateral Triangle

Cited by 106 publications

References 1 publication

Maximizing Neumann fundamental tones of triangles

Maximizing Neumann fundamental tones of triangles

Orbit Functions

An extension of the principle of spatial averaging for inertial manifolds

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