Continuous wavelet transform of Schwartz distributions in 𝒟′(ℝ<sup>𝑛</sup>), 𝑛 ≤ 1
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Notice bibliographique
Résumé
Abstract In this paper, we extend the continuous wavelet transform to Schwartz distributions in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:msup><m:mi mathvariant="script">D</m:mi><m:mo>′</m:mo></m:msup><m:mo></m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:msup><m:mi mathvariant="double-struck">R</m:mi><m:mi>n</m:mi></m:msup><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> \mathcal{D}^{\prime}(\mathbb{R}^{n}) , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>n</m:mi><m:mo>≥</m:mo><m:mn>1</m:mn></m:mrow></m:math> n\geq 1 , and derive the corresponding wavelet inversion formula (valid modulo a constant distribution) interpreting convergence in the weak distributional sense. The kernel of our wavelet transform is an element <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>ψ</m:mi><m:mo></m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mi>x</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> \psi(x) of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi mathvariant="script">D</m:mi><m:mo></m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:msup><m:mi mathvariant="double-struck">R</m:mi><m:mi>n</m:mi></m:msup><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> \mathcal{D}(\mathbb{R}^{n}) , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>n</m:mi><m:mo>≥</m:mo><m:mn>1</m:mn></m:mrow></m:math> n\geq 1 , which, when integrated along each of the real axes <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:msub><m:mi>X</m:mi><m:mn>1</m:mn></m:msub><m:mo>,</m:mo><m:msub><m:mi>X</m:mi><m:mn>2</m:mn></m:msub><m:mo>,</m:mo><m:msub><m:mi>X</m:mi><m:mn>3</m:mn></m:msub><m:mo>,</m:mo><m:mi mathvariant="normal">…</m:mi><m:mo>,</m:mo><m:msub><m:mi>X</m:mi><m:mi>n</m:mi></m:msub></m:mrow></m:math> X_{1},X_{2},X_{3},\ldots,X_{n} vanishes, but none of its moments <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:msub><m:mo largeop="true" symmetric="true">∫</m:mo><m:msup><m:mi mathvariant="double-struck">R</m:mi><m:mi>n</m:mi></m:msup></m:msub><m:mrow><m:mi>ψ</m:mi><m:mo></m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mi>x</m:mi><m:mo stretchy="false">)</m:mo></m:mrow><m:mo></m:mo><m:mpadded width="+1.7pt"><m:msup><m:mi>x</m:mi><m:mi>m</m:mi></m:msup></m:mpadded><m:mo></m:mo><m:mrow><m:mo mathvariant="italic" rspace="0pt">d</m:mo><m:mi>x</m:mi></m:mrow></m:mrow></m:mrow></m:math> \int_{\mathbb{R}^{n}}\psi(x)x^{m}\,dx is zero; here <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:msup><m:mi>x</m:mi><m:mi>m</m:mi></m:msup><m:mo>=</m:mo><m:mrow><m:mpadded width="+1.7pt"><m:msubsup><m:mi>x</m:mi><m:mn>1</m:mn><m:msub><m:mi>m</m:mi><m:mn>1</m:mn></m:msub></m:msubsup></m:mpadded><m:mo></m:mo><m:msubsup><m:mi>x</m:mi><m:mn>2</m:mn><m:msub><m:mi>m</m:mi><m:mn>2</m:mn></m:msub></m:msubsup><m:mo></m:mo><m:mi mathvariant="normal">…</m:mi><m:mo></m:mo><m:msubsup><m:mi>x</m:mi><m:mi>n</m:mi><m:msub><m:mi>m</m:mi><m:mi>n</m:mi></m:msub></m:msubsup></m:mrow></m:mrow></m:math> x^{m}=x_{1}^{{m_{1}}}\,x_{2}^{{m_{2}}}\ldots x_{n}^{{m_{n}}} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mrow><m:mi>d</m:mi><m:mo></m:mo><m:mi>x</m:mi></m:mrow><m:mo>=</m:mo><m:mrow><m:mi>d</m:mi><m:mo></m:mo><m:mpadded width="+1.7pt"><m:msub><m:mi>x</m:mi><m:mn>1</m:mn></m:msub></m:mpadded><m:mo></m:mo><m:mi>d</m:mi><m:mo></m:mo><m:msub><m:mi>x</m:mi><m:mn>2</m:mn></m:msub><m:mo></m:mo><m:mi mathvariant="normal">…</m:mi><m:mo></m:mo><m:mi>d</m:mi><m:mo></m:mo><m:msub><m:mi>x</m:mi><m:mi>n</m:mi></m:msub></m:mrow></m:mrow></m:math> dx=dx_{1}\,dx_{2}\ldots dx_{n} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>m</m:mi><m:mo>=</m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:msub><m:mi>m</m:mi><m:mn>1</m:mn></m:msub><m:mo>,</m:mo><m:msub><m:mi>m</m:mi><m:mn>2</m:mn></m:msub><m:mo>,</m:mo><m:mi mathvariant="normal">…</m:mi><m:mo>,</m:mo><m:msub><m:mi>m</m:mi><m:mi>n</m:mi></m:msub><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> m=(m_{1},m_{2},\ldots,m_{n}) and each of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:msub><m:mi>m</m:mi><m:mn>1</m:mn></m:msub><m:mo>,</m:mo><m:msub><m:mi>m</m:mi><m:mn>2</m:mn></m:msub><m:mo>,</m:mo><m:mi mathvariant="normal">…</m:mi><m:mo>,</m:mo><m:msub><m:mi>m</m:mi><m:mi>n</m:mi></m:msub></m:mrow></m:math> m_{1},m_{2},\ldots,m_{n} is at least 1. The set of such kernel will be denoted by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:msub><m:mi mathvariant="script">D</m:mi><m:mi>m</m:mi></m:msub><m:mo></m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:msup><m:mi mathvariant="double-struck">R</m:mi><m:mi>n</m:mi></m:msup><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> \mathcal{D}_{m}(\mathbb{R}^{n}) . But the uniqueness theorem for our wavelet inversion formula is valid for the space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:msubsup><m:mi mathvariant="script">D</m:mi><m:mi>F</m:mi><m:mo>′</m:mo></m:msubsup><m:mo></m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:msup><m:mi mathvariant="double-struck">R</m:mi><m:mi>n</m:mi></m:msup><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> \mathcal{D}_{F}^{\prime}(\mathbb{R}^{n}) obtained by filtering (deleting) (i) all non-zero constant distributions from the space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:msup><m:mi mathvariant="script">D</m:mi><m:mo>′</m:mo></m:msup><m:mo></m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:msup><m:mi mathvariant="double-struck">R</m:mi><m:mi>n</m:mi></m:msup><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> \mathcal{D}^{\prime}(\mathbb{R}^{n}) , (ii) all non-zero constants that appear with a distribution as a union as for example for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mfrac><m:mrow><m:msubsup><m:mi>x</m:mi><m:mn>1</m:mn><m:mn>2</m:mn></m:msubsup><m:mo>+</m:mo><m:msubsup><m:mi>x</m:mi><m:mn>2</m:mn><m:mn>2</m:mn></m:msubsup><m:mo>+</m:mo><m:mrow><m:mi mathvariant="normal">⋯</m:mi><m:mo></m:mo><m:msubsup><m:mi>x</m:mi><m:mi>n</m:mi><m:mn>2</m:mn></m:msubsup></m:mrow></m:mrow><m:mrow><m:mn>1</m:mn><m:mo>+</m:mo><m:msubsup><m:mi>x</m:mi><m:mn>1</m:mn><m:mn>2</m:mn></m:msubsup><m:mo>+</m:mo><m:msubsup><m:mi>x</m:mi><m:mn>2</m:mn><m:mn>2</m:mn></m:msubsup><m:mo>+</m:mo><m:mrow><m:mi mathvariant="normal">⋯</m:mi><m:mo></m:mo><m:msubsup><m:mi>x</m:mi><m:mi>n</m:mi><m:mn>2</m:mn></m:msubsup></m:mrow></m:mrow></m:mfrac><m:mo>=</m:mo><m:mrow><m:mn>1</m:mn><m:mo>-</m:mo><m:mfrac><m:mn>1</m:mn><m:mrow><m:mn>1</m:mn><m:mo>+</m:mo><m:msubsup><m:mi>x</m:mi><m:mn>1</m:mn><m:mn>2</m:mn></m:msubsup><m:mo>+</m:mo><m:msubsup><m:mi>x</m:mi><m:mn>2</m:
Récupéré en direct depuis OpenAlex et désinversé. Les résumés ne sont pas conservés dans cette base de données : les index inversés représentent 8,6 Go des 9,3 Go de texte de la base, et le serveur dispose de 13 Go libres.
Prédiction distillée sur la base complète
Imitation des enseignantsNi prévalence calibrée, ni vérité terrain. Validation humaine à venir. Apprise à partir de 10 348 étiquettes directes de Codex et de 10 348 étiquettes directes de Gemma. Le mode candidate est l'union des têtes enseignantes seuillées; le consensus est leur intersection. Ces sorties portent le statut machine_predicted_unvalidated et ne sont ni des étiquettes humaines ni des étiquettes directes de modèles de pointe.
Scores Codex et Gemma par catégorie
| Catégorie | Codex | Gemma |
|---|---|---|
| Métarecherche | 0,002 | 0,000 |
| Méta-épidémiologie (sens strict) | 0,000 | 0,000 |
| Méta-épidémiologie (sens large) | 0,001 | 0,001 |
| Bibliométrie | 0,001 | 0,004 |
| Études des sciences et des technologies | 0,000 | 0,000 |
| Communication savante | 0,000 | 0,000 |
| Science ouverte | 0,000 | 0,000 |
| Intégrité de la recherche | 0,000 | 0,000 |
| Charge utile insuffisante (le modèle a refusé de juger) | 0,005 | 0,000 |
Scores machine (provisoires)
Les deux têtes enseignantes du modèle étudiant, lues sur ce travail. Un score ordonne la base pour la relecture; il n'affirme jamais une catégorie, et le statut de validation accompagne chaque rangée tel quel.
Scores de référence d'un modèle non mature (critères de maturité non atteints, 7 itérations). Un score ordonne; il n'affirme jamais une catégorie.
score_only:v0-immature-baseline · tel quel depuis la passe de notation : score_only signifie que le nombre peut ordonner les travaux, et qu'aucune étiquette de catégorie n'en découle