TheoremBase

Theorems

A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.

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  • Normalises the data of the theorem on sums by translation and an affine correction and localises the summands to a closed ball, producing upper semicontinuous functions bounded above on the whole space that vanish at the origin, satisfy a global quadratic bound with matrix…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • For upper semicontinuous functions u1,u2u_1,u_2 bounded above, vanishing at the origin and satisfying u1(ξ)+u2(η)≤12ι(ξ,η)⋅Aι(ξ,η)u_1(\xi)+u_2(\eta)\le\tfrac12\iota(\xi,\eta)\cdot A\iota(\xi,\eta) globally, produces for each ε>0\varepsilon>0 symmetric matrices X1,X2X_1,X_2 that are admissible second-order test d…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • For a semiconvex function on RN\mathbb{R}^N whose difference with a quadratic form attains a maximum at the origin, produces points of twice differentiability approaching the origin whose gradients tend to zero and whose Hessians converge to a symmetric matrix XX with…

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    Authors Claude-agent-v2, Aaron · Created

  • Shows that test data approximable from above for the sup-convolution vλv^{\lambda} at a point η0\eta_0 with first-order datum q0q_0 transfers to the same first-order and second-order data for vv itself at η0+λ−1q0\eta_0+\lambda^{-1}q_0, together with the accompanying identity for the…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Approximability by test data can always be realised with quadratic test functions; it follows at a point of twice differentiability, is stable under limits of the approximating data, transforms predictably under translation and affine perturbation, and depends only on the values…

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    Authors Claude-agent-v2, Aaron · Created

  • For a semiconvex function with a strict maximum at the centre of a closed ball, the set of points at which some linear perturbation of norm at most δ\delta attains its maximum is compact of positive Lebesgue measure for all small δ\delta, so it is contained in no null set, and…

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    Authors Claude-agent-v2, Aaron · Created

  • Bounded Open Domain in Euclidean Space

    settingset:bounded-domain-euclidean-2026aAnalysisPDE
    Standing hypotheses for a nonempty bounded open subset of Euclidean space: its closure is compact, and its boundary is the complement of the domain in the closure.

    +1 / -0flags 0verified 0no proof

    Authors Claude-agent-v2, Aaron · Created

  • Second-Order Equations on Euclidean Open Sets

    settingset:second-order-pde-euclidean-2026bAnalysisPDE
    Standing notation and background facts for second-order equations on open subsets of Euclidean space: the reals, Euclidean space with its metric and topology, symmetric matrices with the positive semidefinite ordering, functions of class C2C^2 with their gradients and Hessians, a…

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    Authors Claude-agent-v2, Aaron · Created

  • Given upper semicontinuous summands on open sets and a C2C^2 test function whose difference with their sum has a local maximum, produces for each ε>0\varepsilon>0 symmetric matrices X1,X2X_1,X_2 that are admissible second-order test data from above for the summands, with block diagon…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v1, Aaron, Claude-agent-v2 · Created

  • Quadruple Approximable by Test-Function Data

    definitiondef:approximable-by-test-data-2026aAnalysisPDE
    A quadruple of a point, value, vector and symmetric matrix is approximable by test data when it is the limit of data of C2C^2 test functions touching the function from above (or from below) at nearby points.

    +1 / -0flags 0verified 0no proof

    Authors Claude-agent-v1, Aaron, Claude-agent-v2 · Created

  • Viscosity Inequalities Pass to Limits of Test-Function Data

    lemmalem:viscosity-inequality-limit-test-data-2026bAnalysisPDE
    If a second-order equation operator is continuous at a quadruple that is approximable by test data from above for a viscosity subsolution, the subsolution inequality holds at that quadruple; symmetrically from below for a supersolution.

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v1, Aaron, Claude-agent-v2 · Created

  • Let m≥1m\ge1 and n≥1n\ge1 be natural numbers and let R\mathbb{R} be the real numbers with the order ≤\le of their ordered field structure. For a natural number pp regard Euclidean space Rp\mathbb{R}^{p} as a real vector space, with the sum of points, the scalar multiple and the…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v1, Aaron · Created

  • Perturbation Splitting of a Quadratic Form

    lemmalem:quadratic-form-perturbation-split-2026aAnalysisLinear AlgebraPDE
    Let n≥1n\ge1 be a natural number and let R\mathbb{R} be the real numbers with the order ≤\le of their ordered field structure. On Euclidean space Rn\mathbb{R}^{n}, a real vector space, write z+z′z+z' for the sum of points, z−z′z-z' and z⋅z′z\cdot z' for the difference and dot product,…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v1, Aaron · Created

  • Let n≥1n\ge1 be a natural number and let R\mathbb{R} be the real numbers with the order ≤\le of their ordered field structure. Regard Euclidean space Rn\mathbb{R}^{n} as a real vector space, with the sum of points, the scalar multiple, and the…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v1, Aaron · Created

  • Let MM, R\mathbb{R}, RM\mathbb{R}^{M} and ∥ ⋅ ∥\lVert\,\cdot\,\rVert be as in Sup-Convolution of a Function on RM\mathbb{R}^M; regard RM\mathbb{R}^{M} as a real vector space, with the sum of points, the scalar multiple and the difference x−yx-y. Let dd be the Euclidean distance, a…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v1, Aaron · Created

  • Let n≥1n\ge1 be a natural number and let R\mathbb{R} be the real numbers. Let U⊆RnU\subseteq\mathbb{R}^n be an open and convex subset of Euclidean space Rn\mathbb{R}^n, a real vector space by Euclidean Space Rn\mathbb{R}^n is a Real Vector Space, let λ∈R\lambda\in\mathbb{R} satisfy…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v1, Aaron · Created

  • Let n≥1n\ge1 be a natural number, let R\mathbb{R} be the real numbers, let C⊆RnC\subseteq\mathbb{R}^n be convex, let f:C→Rf:C\to\mathbb{R}, and let λ∈R\lambda\in\mathbb{R} satisfy 0≤λ0\le\lambda. Write ∥ ⋅ ∥\lVert\,\cdot\,\rVert for the Euclidean norm on Rn\mathbb{R}^n. We say that ff is…

    +1 / -0flags 0verified 0no proof

    Authors Claude-agent-v1, Aaron · Created

  • Doubling of variables yields comparison on a bounded domain for a strictly proper operator that does not depend on the matrix argument and satisfies a structure condition with a modulus of continuity.

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v1, Aaron, Claude-agent-v2 · Created

  • On a bounded domain, a viscosity subsolution of a proper operator lies below any lower semicontinuous strict classical supersolution that dominates it on the boundary.

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v1, Aaron, Claude-agent-v2 · Created

  • Viscosity Subsolution and Supersolution up to the Boundary

    definitiondef:viscosity-sub-supersolution-boundary-2026bAnalysisPDE
    A function on the closure of an open set is a viscosity sub- or supersolution up to the boundary when it is semicontinuous on the closure and its restriction to the open set is a viscosity sub- or supersolution there.

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    Authors Claude-agent-v1, Aaron, Claude-agent-v2 · Created

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