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"Theoretical mathematics”: Toward a cultural synthesis of mathematics and theoretical physics
 BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY
, 1993
"... Is speculative mathematics dangerous? Recent interactions between physics and mathematics pose the question with some force: traditional mathematical norms discourage speculation, but it is the fabric of theoretical physics. In practice there can be benefits, but there can also be unpleasant and de ..."
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Cited by 24 (1 self)
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Is speculative mathematics dangerous? Recent interactions between physics and mathematics pose the question with some force: traditional mathematical norms discourage speculation, but it is the fabric of theoretical physics. In practice there can be benefits, but there can also be unpleasant and destructive consequences. Serious caution is required, and the issue should be considered before, rather than after, obvious damage occurs. With the hazards carefully in mind, we propose a framework that should allow a healthy and positive role for speculation.
Spectral theory for perturbed Krein Laplacians in nonsmooth domains
, 2010
"... We study spectral properties for HK,Ω, the Krein–von Neumann extension of the perturbed Laplacian − ∆ + V defined on C ∞ 0 (Ω), where V is measurable, bounded and nonnegative, in a bounded open set Ω ⊂ Rn belonging to a class of nonsmooth domains which contains all convex domains, along with all do ..."
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Cited by 7 (7 self)
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We study spectral properties for HK,Ω, the Krein–von Neumann extension of the perturbed Laplacian − ∆ + V defined on C ∞ 0 (Ω), where V is measurable, bounded and nonnegative, in a bounded open set Ω ⊂ Rn belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class C1,r, r> 1/2. In particular, in the aforementioned context we establish the Weyl asymptotic formula #{j ∈ N  λK,Ω,j ≤ λ} = (2π) −n vnΩ  λ n/2 + O ` λ (n−(1/2))/2 ´ as λ → ∞, where vn = πn/2 /Γ((n/2)+1) denotes the volume of the unit ball in Rn, and λK,Ω,j, j ∈ N, are the nonzero eigenvalues of HK,Ω, listed in increasing order according to their multiplicities. We prove this formula by showing that the perturbed Krein Laplacian (i.e., the Krein–von Neumann extension of − ∆ + V defined on C ∞ 0 (Ω)) is spectrally equivalent to the buckling of a clamped plate problem, and using an abstract result of Kozlov from the mid 1980’s. Our work builds on that of Grubb in the early 1980’s, who has considered similar issues for elliptic operators in smooth domains, and shows that the question posed by Alonso and Simon in 1980 pertaining to the validity of the above Weyl asymptotic formula continues to have an affirmative answer in this nonsmooth setting. We also study certain exteriortype domains Ω = Rn \K, n ≥ 3, with K ⊂ Rn compact and vanishing
Potential Theory Of The FarthestPoint Distance Function
"... We study the farthest{point distance function, which measures the distance from z 2 C to the farthest point or points of a given compact set E in the plane. The logarithm of this distance is subharmonic as a function of z, and equals the logarithmic potential of a unique probability measure with unb ..."
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Cited by 6 (4 self)
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We study the farthest{point distance function, which measures the distance from z 2 C to the farthest point or points of a given compact set E in the plane. The logarithm of this distance is subharmonic as a function of z, and equals the logarithmic potential of a unique probability measure with unbounded support. This measure E has many interesting properties that reect the topology and geometry of the compact set E. We prove E (E) 2 for polygons inscribed in a circle, with equality if and only if E is a regular ngon for some odd n. Also we show E (E) = 2 for smooth convex sets of constant width. We conjecture E (E) 2 for all E. 1.
Pseudoconvexity, Analytic Discs, and Invariant Metrics
"... Abstract: We begin by studying characterizations of pseudoconvexity, and also of finite type, using analytic discs. The results presented are analogous to wellknown ideas from the real variable setting in which “pseudoconvex ” is replaced by “convex ” and “analytic disc ” is replaced by “line segme ..."
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Cited by 1 (0 self)
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Abstract: We begin by studying characterizations of pseudoconvexity, and also of finite type, using analytic discs. The results presented are analogous to wellknown ideas from the real variable setting in which “pseudoconvex ” is replaced by “convex ” and “analytic disc ” is replaced by “line segment”. The second part of the paper concerns regularity results for the Kobayashi metric. Of course this metric is defined using analytic discs, so the discussion is a natural extension of that in the first part of the paper. We also comment on the Carathéodory metric. 0
Spectral Theory for . . . IN NONSMOOTH DOMAINS
, 2009
"... We study spectral properties for HK,Ω, the Krein–von Neumann extension of the perturbed Laplacian − ∆ + V defined on C ∞ 0 (Ω), where V is measurable, bounded and nonnegative, in a bounded open set Ω ⊂ Rn belonging to a class of nonsmooth domains which contains all convex domains, along with all do ..."
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We study spectral properties for HK,Ω, the Krein–von Neumann extension of the perturbed Laplacian − ∆ + V defined on C ∞ 0 (Ω), where V is measurable, bounded and nonnegative, in a bounded open set Ω ⊂ Rn belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class C1,r, r> 1/2. In particular, in the aforementioned context we establish the Weyl asymptotic formula #{j ∈ N  λK,Ω,j ≤ λ} = (2π) −n vnΩ  λ n/2 + O ` λ (n−(1/2))/2 ´ as λ → ∞, where vn = πn/2 /Γ((n/2)+1) denotes the volume of the unit ball in Rn, and λK,Ω,j, j ∈ N, are the nonzero eigenvalues of HK,Ω, listed in increasing order according to their multiplicities. We prove this formula by showing that the perturbed Krein Laplacian (i.e., the Krein–von Neumann extension of − ∆ + V defined on C ∞ 0 (Ω)) is spectrally equivalent to the buckling of a clamped plate problem, and using an abstract result of Kozlov from the mid 1980’s. Our work builds on that of Grubb in the early 1980’s, who has considered similar issues for elliptic operators in smooth domains, and shows that the question posed by Alonso and Simon in 1980 pertaining to the validity of the above Weyl asymptotic formula continues to have an affirmative answer in this nonsmooth setting. We also study certain exteriortype domains Ω = Rn \K, n ≥ 3, with K ⊂ Rn compact and vanishing
CORRESPONDING ABSTRACT BUCKLING PROBLEM, AND WEYLTYPE SPECTRAL ASYMPTOTICS FOR PERTURBED KREIN LAPLACIANS IN NONSMOOTH DOMAINS
, 2012
"... Dedicated with great pleasure to Michael Demuth on the occasion of his 65th birthday. Abstract. In the first (and abstract) part of this survey we prove the unitary equivalence of the inverse of the Krein–von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed ..."
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Dedicated with great pleasure to Michael Demuth on the occasion of his 65th birthday. Abstract. In the first (and abstract) part of this survey we prove the unitary equivalence of the inverse of the Krein–von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed, strictly positive operator, S ≥ εIH for some ε> 0 in a Hilbert space H to an abstract buckling problem operator. In the concrete case where S = −∆  C ∞ 0 (Ω) in L 2 (Ω;d n x) for Ω ⊂ R n an open, bounded (and sufficiently regular) domain, this recovers, as a particular case of a general result due to G. Grubb, that the eigenvalue problem for the Krein Laplacian SK (i.e., the Krein–von Neumann extension of S), SKv = λv, λ ̸ = 0, is in onetoone correspondence with the problem of the buckling of a clamped plate, where u and v are related via the pair of formulas (−∆) 2 u = λ(−∆)u in Ω, λ ̸ = 0, u ∈ H 2 0(Ω), u = S −1 F (−∆)v, v = λ−1 (−∆)u,
A SURVEY ON THE KREIN–VON NEUMANN EXTENSION, THE CORRESPONDING ABSTRACT BUCKLING PROBLEM, AND WEYLTYPE SPECTRAL ASYMPTOTICS FOR PERTURBED KREIN LAPLACIANS IN NONSMOOTH DOMAINS
"... Dedicated with great pleasure to Michael Demuth on the occasion of his 65th birthday. Abstract. In the first (and abstract) part of this survey we prove the unitary equivalence of the inverse of the Krein–von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed ..."
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Dedicated with great pleasure to Michael Demuth on the occasion of his 65th birthday. Abstract. In the first (and abstract) part of this survey we prove the unitary equivalence of the inverse of the Krein–von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed, strictly positive operator, S ≥ εIH for some ε> 0 in a Hilbert space H to an abstract buckling problem operator. In the concrete case where S = −∆  C ∞ 0 (Ω) in L2 (Ω; dnx) for Ω ⊂ Rn an open, bounded (and sufficiently regular) domain, this recovers, as a particular case of a general result due to G. Grubb, that the eigenvalue problem for the Krein Laplacian SK (i.e., the Krein–von Neumann extension of S), SKv = λv, λ ̸ = 0, is in onetoone correspondence with the problem of the buckling of a clamped plate, where u and v are related via the pair of formulas (−∆) 2 u = λ(−∆)u in Ω, λ ̸ = 0, u ∈ H 2 0 (Ω), u = S −1 F (−∆)v, v = λ−1 (−∆)u, with SF the Friedrichs extension of S.