Optimal 𝐶² two-dimensional interpolatory ternary subdivision schemes with two-ring stencils
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Bibliographic record
Abstract
For any interpolatory ternary subdivision scheme with two-ring stencils for a regular triangular or quadrilateral mesh, we show that the critical Hölder smoothness exponent of its basis function cannot exceed <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="log Subscript 3 Baseline 11 left-parenthesis almost-equals 2.18266 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>log</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mo> </mml:mo> <mml:mn>11</mml:mn> <mml:mo stretchy="false">(</mml:mo> <mml:mo> ≈ </mml:mo> <mml:mn>2.18266</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\log _3 11 (\approx 2.18266)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where the critical Hölder smoothness exponent of a function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f colon double-struck upper R squared right-arrow from bar double-struck upper R"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false"> ↦ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">f : \mathbb {R}^2\mapsto \mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is defined to be <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="nu Subscript normal infinity Baseline left-parenthesis f right-parenthesis colon-equal sup left-brace right-brace colon nu colon element-of element-of f of LipLip nu period"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi> ν </mml:mi> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-REL"> <mml:mo>≔</mml:mo> </mml:mrow> <mml:mo movablelimits="true" form="prefix">sup</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi> ν </mml:mi> <mml:mo>:</mml:mo> <mml:mi>f</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mi>Lip</mml:mi> <mml:mo> </mml:mo> <mml:mi> ν </mml:mi> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\nu _\infty (f) \coloneq \sup \{ \nu : f\in \operatorname {Lip} \nu \}.</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> On the other hand, for both regular triangular and quadrilateral meshes, we present several examples of interpolatory ternary subdivision schemes with two-ring stencils such that the critical Hölder smoothness exponents of their basis functions do achieve the optimal smoothness upper bound <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="log Subscript 3 Baseline 11"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>log</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mo> </mml:mo> <mml:mn>11</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\log _3 11</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Consequently, we obtain optimal smoothest <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C squared"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">C^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> interpolatory ternary subdivision schemes with two-ring stencils for the regular triangular and quadrilateral meshes. Our computation and analysis of optimal multidimensional subdivision schemes are based on the projection method and the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l Subscript p"> <mml:semantics> <mml:msub> <mml:mi> ℓ </mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\ell _p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -norm joint spectral radius.
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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