Blow-up scenarios for 3D NSE exhibiting sub-criticality with respect to the scaling of one-dimensional local sparseness

It is shown that, if the vorticity magnitude associated with a (presumed singular) three-dimensional incompressible Navier-Stokes flow blows-up in a manner exhibiting certain {\em time dependent local structure}, then {\em time independent} estimates on the $L^1$ norm of $|\omega|\log\sqrt{1+ |\omega|^2}$ follow. The implication is that the volume of the region of high vorticity decays at a rate of greater order than a rate connected to the critical scaling of one-dimensional local sparseness and, consequently, the solution becomes sub-critical.Comment: final version, to appear in J. Math. Fluid Mech., 13p

An Operational Calculus Generalization of Ramanujan's Master Theorem

We give a formal extension of Ramanujan's master theorem using operational methods. The resulting identity transforms the computation of a product of integrals on the half-line to the computation of a Laplace transform. Since the identity is purely formal, we show consistency of this operational approach with various standard calculus results, followed by several examples to illustrate the power of the extension. We then briefly discuss the connection between Ramanujan's master theorem and identities of Hardy and Carr before extending the latter identities in the same way we extended Ramanujan's. Finally, we generalize our results, producing additional interesting identities as a corollary.Comment: 20 pages, 1 figur

Global Navier-Stokes flows in intermediate spaces

We construct global weak solutions of the three dimensional incompressible Navier-Stokes equations in intermediate spaces between the space of uniformly locally square integrable functions and Herz-type spaces which involve weighted integrals centered at the origin. Our results bridge the existence theorems of Lemari\'e-Rieusset and of Bradshaw, Kukavica and Tsai. An application to eventual regularity is included which generalizes the prior work of Bradshaw, Kukavica and Tsai as well as Bradshaw, Kukavica and Ozanski.Comment: We moved Lemma 3.1 to Lemma 2.1, and added Lemma 2.11 and several reference

ConTEXTual Net: A Multimodal Vision-Language Model for Segmentation of Pneumothorax

Radiology narrative reports often describe characteristics of a patient's disease, including its location, size, and shape. Motivated by the recent success of multimodal learning, we hypothesized that this descriptive text could guide medical image analysis algorithms. We proposed a novel vision-language model, ConTEXTual Net, for the task of pneumothorax segmentation on chest radiographs. ConTEXTual Net utilizes language features extracted from corresponding free-form radiology reports using a pre-trained language model. Cross-attention modules are designed to combine the intermediate output of each vision encoder layer and the text embeddings generated by the language model. ConTEXTual Net was trained on the CANDID-PTX dataset consisting of 3,196 positive cases of pneumothorax with segmentation annotations from 6 different physicians as well as clinical radiology reports. Using cross-validation, ConTEXTual Net achieved a Dice score of 0.716$\pm$0.016, which was similar to the degree of inter-reader variability (0.712$\pm$0.044) computed on a subset of the data. It outperformed both vision-only models (ResNet50 U-Net: 0.677$\pm$0.015 and GLoRIA: 0.686$\pm$0.014) and a competing vision-language model (LAVT: 0.706$\pm$0.009). Ablation studies confirmed that it was the text information that led to the performance gains. Additionally, we show that certain augmentation methods degraded ConTEXTual Net's segmentation performance by breaking the image-text concordance. We also evaluated the effects of using different language models and activation functions in the cross-attention module, highlighting the efficacy of our chosen architectural design