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Luis Vega

Publications and source records attributed to Luis Vega.

At least 19 recordsLinked to original sources
arXiv2026

Evolution of viscous vortex filaments and soliton-type propagation

We study the evolution of a viscous incompressible fluid whose initial vorticity is supported on a smooth open curve. We show that, for $\nu t\ll 1$ and sufficiently small Reynolds number $\Gamma/\nu$, the vorticity is described at leading order by a Lamb--Oseen type vortex concentrated around a curve evolving according to the binormal flow predicted by the localized induction approximation. The solution is written as an explicit leading-order profile plus a lower-order perturbation, which is controlled in a Morrey $\mathcal{M}^{\infty}$ norm. Then, we apply this construction to the Hasimoto soliton. In this case, the estimates are uniform with respect to the torsion parameter, allowing us to consider a large-torsion regime in which the soliton undergoes a macroscopic displacement. We show that the corresponding Navier--Stokes solution contains a localized portion of the kinetic-energy distribution, associated with the binormal velocity, which remains concentrated inside a moving physical region and undergoes an order-one displacement during an admissible time interval.

math.AP↗
arXiv2026

Bifurcation and global continuation of travelling-rotating Schr\"odinger maps on the sphere

We study travelling-rotating solutions of the Schr\"odinger map equation into the sphere, viewed as tangent profiles of rigid vortex filaments. Two first integrals reduce the profile equation to a scalar cubic equation for the vertical component, giving an elliptic-function description and explicit closure conditions. We prove bifurcation from the equatorial branch at $\lambda_k=R\sqrt{k^2-1}$, $k\ge2$, and establish a global continuation alternative inside the regular non-polar class. The possible boundary mechanisms are pole contact, vertical collapse, and double-root degeneration. Numerical continuation of the equatorial branches suggests convergence to the north-pole boundary. Up to gauge, the reconstructed vortex filaments are of Kida type.

math.AP↗
arXiv2026

Nemotron 3 Nano Omni: Efficient and Open Multimodal Intelligence

We introduce Nemotron 3 Nano Omni, the latest model in the Nemotron multimodal series and the first to natively support audio inputs alongside text, images, and video. Nemotron 3 Nano Omni delivers consistent accuracy improvements over its predecessor, Nemotron Nano V2 VL, across all modalities, enabled by advances in architecture, training data and recipes. In particular, Nemotron 3 delivers leading results in real-world document understanding, long audio-video comprehension, and agentic computer use. Built on the highly efficient Nemotron 3 Nano 30B-A3B backbone, Nemotron 3 Nano Omni further incorporates innovative multimodal token-reduction techniques to deliver substantially lower inference latency and higher throughput than other models of similar size. We are releasing model checkpoints in BF16, FP8, and FP4 formats, along with portions of the training data and codebase to facilitate further research and development.

cs.LG↗
arXiv2025

NVIDIA Nemotron Nano V2 VL

We introduce Nemotron Nano V2 VL, the latest model of the Nemotron vision-language series designed for strong real-world document understanding, long video comprehension, and reasoning tasks. Nemotron Nano V2 VL delivers significant improvements over our previous model, Llama-3.1-Nemotron-Nano-VL-8B, across all vision and text domains through major enhancements in model architecture, datasets, and training recipes. Nemotron Nano V2 VL builds on Nemotron Nano V2, a hybrid Mamba-Transformer LLM, and innovative token reduction techniques to achieve higher inference throughput in long document and video scenarios. We are releasing model checkpoints in BF16, FP8, and FP4 formats and sharing large parts of our datasets, recipes and training code.

cs.LG↗
arXiv2025

Fraudulent Publishing in the Mathematical Sciences

This report is the first of two publications of a joint Working Group of the International Mathematical Union (IMU) and the International Council of Industrial and Applied Mathematics (ICIAM). In it, we shall analyze the current state of publishing in the mathematical sciences and explain the resulting problems. Our second publication will offer concrete recommendations, guidelines, and best practices for researchers, policymakers, and evaluators of mathematical research. It will explain how to detect and counteract attempts to game bibliometric measures, empowering the community to reclaim control over research evaluation and drive necessary change.

math.HO↗
arXiv2025

A theorem concerning Fourier transforms: A survey

In this note, we highlight the impact of the paper G. H. Hardy, A theorem concerning Fourier transforms, J. Lond. Math. Soc. (1) 8 (1933), 227--231 in the community of harmonic analysis in the last 90 years, reviewing, on the one hand, the direct generalizations of the main results and, on the other, the different connections to related areas and new perspectives.

math.CA↗
arXiv2025

On low regularity well-posedness of the binormal flow

We focus on a class of solutions of the binormal flow, model of the evolution of vortex filaments, that generate several corner singularities in finite time. This phenomenon has been studied earlier in the regular case, which in this context is in terms of the summability of the angles of the corners generated. Our goal here is to investigate the lower regularity case, using further the Hasimoto approach that allows to use the 1D cubic nonlinear Schr\"odinger to study the binormal flow. We first obtain a deterministic result by proving an existence result for general binormal flow solutions at low regularity. Then we obtain improved results on the above class of solutions by a suitable randomization of the curvature and torsion of the vortex filament. To do so, we prove a scattering result for a quasi-invariance measure associated with a suitable 1D cubic nonlinear Schr\"odinger equation that we consider of independent interest. An interesting feature of this result is that we are able to identify a limit measure, which is usually not possible when working on quasi-invariant Gaussian measures for Hamiltonian PDEs on bounded domains.

math.AP↗
arXiv2025

Nemotron-H: A Family of Accurate and Efficient Hybrid Mamba-Transformer Models

As inference-time scaling becomes critical for enhanced reasoning capabilities, it is increasingly becoming important to build models that are efficient to infer. We introduce Nemotron-H, a family of 8B and 56B/47B hybrid Mamba-Transformer models designed to reduce inference cost for a given accuracy level. To achieve this goal, we replace the majority of self-attention layers in the common Transformer model architecture with Mamba layers that perform constant computation and require constant memory per generated token. We show that Nemotron-H models offer either better or on-par accuracy compared to other similarly-sized state-of-the-art open-sourced Transformer models (e.g., Qwen-2.5-7B/72B and Llama-3.1-8B/70B), while being up to 3$\times$ faster at inference. To further increase inference speed and reduce the memory required at inference time, we created Nemotron-H-47B-Base from the 56B model using a new compression via pruning and distillation technique called MiniPuzzle. Nemotron-H-47B-Base achieves similar accuracy to the 56B model, but is 20% faster to infer. In addition, we introduce an FP8-based training recipe and show that it can achieve on par results with BF16-based training. This recipe is used to train the 56B model. We are releasing Nemotron-H base model checkpoints with support in Hugging Face and NeMo.

cs.CL↗
arXiv2025

Asymptotic fractional uncertainty principle for the Helmholtz equation with periodic scattering data

We investigate the fractional dispersion of solutions to the Helmholtz equation with periodic scattering data. We show that, under appropriate rescaling, the interaction between the different frequencies exhibits the same fluctuating behavior as for the Schr\"{o}dinger equation. To achieve this, we first establish an asymptotic fractional uncertainty principle for solutions to the Helmholtz equation.

math.AP↗
arXiv2024

Turbulent solutions of the binormal flow and the 1D cubic Schr\"odinger equation

In the last three decades there has been an intense activity on the exploration of turbulent phenomena of dispersive equations, as for instance the growth of Sobolev norms since the work of Bourgain in the 90s. In general the 1D cubic Schr\"odinger equation has been left aside because of its complete integrability. In a series of papers of the last six years that we survey here for the special issue of the ICMP 2024 ([12],[13],[14],[15],[16],[7],[8]), we considered, together with the 1D cubic Schr\"odinger equation, the binormal flow, which is a geometric flow explicitly related to it. We displayed rigorously a large range of complex behavior as creation of singularities and unique continuation, Fourier growth, Talbot effects, intermittency and multifractality, justifying in particular some previous numerical observations. To do so we constructed a class of well-posedness for the 1D cubic Schr\"odinger equation included in the critical Fourier-Lebesgue space $\mathcal FL^\infty$ and in supercritical Sobolev spaces with respect to scaling. Last but not least we recall that the binormal flow is a classical model for the dynamics of a vortex filament in a 3D fluid or superfluid, and that vortex motions are a key element of turbulence.

math.AP↗
arXiv2024

Multifractality and polygonal vortex filaments

In this proceedings article we survey the results in [5] and their motivation, as presented at the 50th Journ\'ees EDP 2024. With the aim of quantifying turbulent behaviors of vortex filaments, we study the multifractality of a family of generalized Riemann's non-differentiable functions. These functions represent, in a certain limit, the trajectory of regular polygonal vortex filaments that evolve according to the binormal flow, the classical model for vortex filaments dynamics. We explain how we determined their spectrum of singularities through a careful design of Diophantine sets, which we study by using the Duffin-Schaeffer theorem and the Mass Transference Principle.

math.AP↗
arXiv2023

Blow-up for the 1D cubic NLS

We consider the 1D cubic NLS on $\mathbb R$ and prove a blow-up result for functions that are of borderline regularity, i.e. $H^s$ for any $s<-\frac 12$ for the Sobolev scale and $\mathcal F L^\infty$ for the Fourier-Lebesgue scale. This is done by identifying at this regularity a certain functional framework from which solutions exit in finite time. This functional framework allows, after using a pseudo-conformal transformation, to reduce the problem to a large-time study of a periodic Schrödinger equation with non-autonomous cubic nonlinearity. The blow-up result corresponds to an asymptotic completeness result for the new equation. We prove it using Bourgain's method and exploiting the oscillatory nature of the coefficients involved in the time-evolution of the Fourier modes. Finally, as an application we exhibit singular solutions of the binormal flow. More precisely, we give conditions on the curvature and the torsion of an initial smooth curve such that the constructed solutions generate several singularities in finite time.

math.AP↗
arXiv2023

Evolution of viscous vortex filaments and desingularization of the Biot-Savart integral

We consider a viscous fluid with kinematic viscosity $ν$ and initial data consisting of a smooth closed vortex filament with circulation $Γ$. We show that, for short enough time, the solution consists of a deformed Lamb-Oseen vortex whose center (a filament) follows the binormal flow dynamics plus leading order corrections that depend locally on the filament curvature and the nonlocal interactions with distant parts of the filament. In order to achieve this scale separation we require $Γ/ν$ to be sufficiently small.

math.AP↗
arXiv2023

Multifractality and intermittency in the limit evolution of polygonal vortex filaments

With the aim of quantifying turbulent behaviors of vortex filaments, we study the multifractality and intermittency of the family of generalized Riemann's non-differentiable functions \begin{equation} R_{x_0}(t) = \sum_{n \neq 0} \frac{e^{2\pi i ( n^2 t + n x_0 ) } }{n^2}, \qquad x_0 \in [0,1]. \end{equation} These functions represent, in a certain limit, the trajectory of regular polygonal vortex filaments that evolve according to the binormal flow. When $x_0$ is rational, we show that $R_{x_0}$ is multifractal and intermittent by completely determining the spectrum of singularities of $R_{x_0}$ and computing the $L^p$ norms of its Fourier high-pass filters, which are analogues of structure functions. We prove that $R_{x_0}$ has a multifractal behavior also when $x_0$ is irrational. The proofs rely on a careful design of Diophantine sets that depend on $x_0$, which we study by crucially using the Duffin-Schaeffer theorem and the Mass Transference Principle.

math.AP↗
arXiv2023

Reconnection of infinitely thin antiparallel vortices and coherent structures

One of the characteristic features of turbulent flows is the emergence of many vortices which interact, deform, and intersect, generating a chaotic movement. The evolution of a pair of vortices, e.g. condensation trails of a plane, can be considered as a basic element of a turbulent flow. This simple example nevertheless demonstrates very rich behavior which still lacks a complete explanation. In particular, after the reconnection of the vortices some coherent structures with the shape of a horseshoe emerge. They have a high level of complexity generated by the interaction of waves running from the reconnection region. These structures also turn to be very reminiscent to the ones obtained from the localized induction approximation applied to a polygonal vortex. It can be considered as an evidence that a pair of vortices creates a corner singularity during the reconnection. In this work we focus on a study of the reconnection phenomena and the emerged structures. In order to do it we present a new model based on the approximation of an infinitely thin vortex, which allows us to focus on the chaotic movement of the vortex center line. The main advantage of the developed model consists in the ability to go beyond the reconnection time and to see the coherent structures. It is also possible to define the reconnection time by analyzing the fluid impulse.

math.AP↗
arXiv2023

New Conservation Laws and Energy Cascade for 1d Cubic NLS and the Schrödinger map

We review some recent results concerning the Initial Value Problem of 1d-cubic non-linear Schrödinger equation (NLS) and other related systems as the Schrödinger Map. For the latter we prove the existence of a cascade of energy. Finally, some new examples of the Talbot effect at the critical level of regularity are given.

math.AP↗
arXiv2023

Steady solutions for the Schrödinger map equation

In this paper we use bifurcation methods to construct a new family of solutions of the binormal flow, also known as the vortex filament equation, which do not change their form. Our examples are complementary to those obtained by S. Kida in 1981, and therefore they are also related, thanks to the so-called Hasimoto transformation, to travelling wave solutions of the 1d cubic non-linear Schrödinger equation.

math.AP↗