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Closure for Microscale Fluctuations in Chaotic Mixing Fronts
Heyman Joris, Le Borgne Tanguy, Lester Daniel
https://arxiv.org/abs/2506.15255 https://…

Closure for Microscale Fluctuations in Chaotic Mixing Fronts
Scalar mixing fronts are formed under the stirring and spreading action of a random velocity field. The prediction of concentration statistics below the velocity length scale is an open problem. Here, we show that when fluid agitation is caused by a smooth chaotic flow with weak persistence, the evolution of scalar variance can be uniquely related to the stirring and spreading properties of the flow and the diffusivity of the scalar. Using the spectral description of microscale scalar fluctuati…

An Operator Theoretic Derivation of a Bernoulli Stirling Identity: A Novel Use of the Euler Operator and Integration
Abdelhay Benmoussa
https://arxiv.org/abs/2505.22819

An Operator Theoretic Derivation of a Bernoulli Stirling Identity: A Novel Use of the Euler Operator and Integration
We present a novel proof of the classical identity relating Bernoulli numbers \( B_n \) and Stirling numbers of the second kind \( S(n,k) \), given by \[ B_n = \sum_{k=1}^n S(n,k)\frac{(-1)^k k!}{k+1}. \] Unlike traditional derivations based on generating functions or purely combinatorial arguments, our method leverages the Euler operator \( \vartheta = t \frac{d}{dt} \) and its algebraic action on analytic functions. By applying this operator repeatedly to a suitably chosen function and integr…