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For later publicationv0.9.0 local

Boundary conditions and compact embeddings

A. Author — draft

1Boundary cases

We consider a σ-finite measure space (X, μ) and a densely defined, self-adjoint operator L on L²(X, μ).

Theorem 2.1 Let (X, μ) be a finite measure space and L have compact resolvent. Let (Vₙ) be a sequence of finite-dimensional subspaces with dense union. The claimed convergence then holds for every admissible sequence.

2Auxiliary results

Lemma 2.2 The a priori estimate gives C > 0 such that the stated norm bound exists.
Proof: standard.

\displaystyle S_n\rightharpoonup S\;\text{in}\;H^1(\Omega)

the limit is weak and remains bound to the boundary assumptions.

For later publicationv0.8.1 local

Stability of discrete operator families

A. Author — draft

1Stability

We consider a σ-finite measure space (X, μ) and a densely defined, self-adjoint operator L on L²(X, μ).

2Auxiliary results

Lemma 2.2 The a priori estimate gives C > 0 such that the stated norm bound exists.
Proof: standard.

\displaystyle \lambda_k(A_n)\to\lambda_k(A)

for every isolated eigenvalue sequence under the stated assumptions.

For later publicationv0.7.3 local

On the convergence of adaptive spectral methods

A. Author — draft

1Introduction

We consider a σ-finite measure space (X, μ) and a densely defined, self-adjoint operator L on L²(X, μ).

2Auxiliary results

Lemma 2.2 The a priori estimate gives C > 0 such that the stated norm bound exists.
Proof: standard.

\displaystyle \lim_{n\to\infty}\left\lVert (I-P_n)f\right\rVert_{L^2}=0,

where Pₙ denotes the L²-orthogonal projection onto Vₙ.

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$\text{[math]}$

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Please state the boundary assumption before Theorem 2.1.

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Emmy Noether

1882–1935

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Which invariance carries this statement?

A question suggested by MathPaperAI

$\text{[math]}$

Co-authorMaryam Mirzakhani

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Does the convergence still hold without compactness?

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v0.7.3 just nowv0.7.2 18 min. ago

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Emmy NoetherWhich invariance carries this statement?
Co-author: Maryam MirzakhaniDoes the convergence still hold without compactness?
Supervisor: Leonhard EulerPlease state the boundary assumption before Theorem 2.1.

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One paragraph.Four waysto think further.

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Trace and continue

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Everything a manuscript needs, nothing it doesn’t

Proofs a kernel has checked

Conjectures put under fire

Every claim knows where it came from

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A machine that says no

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A Millennium Problem as a walkable map

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