The Fractional Dunkl Laplacian: Extension Problem and Pointwise Formulas
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Bibliographic record
Abstract
<div> The Dunkl Laplacian ∆ k , associated with a finite Coxeter group W in R d , constitutes a deformation of the usual Laplace operator via parameterized differential-difference operators that involve the action of the group W . In this paper, we focus on the fractional Dunkl Laplacian (-∆ k ) α , 0 < α < 1. We prove a Caffarelli-Silvestre characterization for the fractional Dunkl Laplacian through an extension problem. Moreover, we establish that it can be expressed in terms of the ∆ k -volume mean operator as follows This novel geometric pointwise formula provides a new approach to derive several definitions of the fractional Dunkl Laplacian, including integral representations with respect to singular kernels involving the representing probability measure of the Dunkl intertwining operator. As an application, we use a generalized jump kernel to introduce and investigate a Gagliardo-type seminorm on an H α -fractional Sobolev-type space. </div>
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.009 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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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