Related papers: The projective translation equation and unramified…
We construct a one-parameter family of singly periodic translating solutions to mean curvature flow that converge as the period tends to $0$ to the union of a grim reaper surface and a plane that bisects it lengthwise. The surfaces are…
Normalizing flows are a powerful tool for building expressive distributions in high dimensions. So far, most of the literature has concentrated on learning flows on Euclidean spaces. Some problems however, such as those involving angles,…
The transport of many kinds of singular structures in a medium, such as vortex points/lines/sheets in fluids, dislocation loops in crystalline plastic solids, or topological singularities in magnetism, can be expressed in terms of the…
Let $p$ and $l$ be rational primes such that $l$ is odd and the order of $p$ modulo $l$ is even. For such primes $p$ and $l$, and for $e=l, 2l$, we consider the non-singular projective curves $aY^e = bX^e + cZ^e$ ($abc \neq 0$) defined over…
Statistical surrogate modeling of fluid flows is hard because dynamics are multiscale and highly sensitive to initial conditions. Conditional diffusion surrogates can be accurate, but usually need hundreds of stochastic sampling steps. We…
In this article we investigate the two-dimensional incompressible rotating and stratified, just rotating, just stratified Euler equations with each other and with the normal Euler equations with the self-similar Ansatz. There are analytic…
The dynamics and statistical properties of two-dimensional (2D) turbulence are often investigated through numerical simulations of incompressible, viscous fluids in doubly periodic domains. A key challenge in 2D turbulence research is…
Normalizing flows are invertible neural networks with tractable change-of-volume terms, which allow optimization of their parameters to be efficiently performed via maximum likelihood. However, data of interest are typically assumed to live…
We discuss a general definition of directional derivative of any tensor flow field and its practical applications in physics. It is shown that both Lagrangian and Eulerian descriptions as complementary types of flow field specifications…
Normalizing flows are a powerful class of generative models demonstrating strong performance in several speech and vision problems. In contrast to other generative models, normalizing flows are latent variable models with tractable…
The results of an analysis of turbulent pipe flow based on a Karhunen-Lo`eve decomposition are presented. The turbulent flow is generated by a direct numerical simulation of the Navier-Stokes equations using a spectral element algorithm at…
Incompressible flows of an ideal two-dimensional fluid on a closed orientable surface of positive genus are considered. Linear stability of harmonic, i.e. irrotational and incompressible, solutions to the Euler equations is shown using the…
In this paper, we study the convexity, interior gradient estimate, Liouville type theorem and asymptotic behavior at infinity of translating solutions to mean curvature flow as well as the nonlinear flow by powers of the mean curvature.
We prove that if a translating soliton can be expressed as the sum of two curves and one of these curves is planar, then the other curve is also planar and consequently the surface must be a plane or a grim reaper. We also investigate…
Many problems in nonlinear and statistical physics are formulated through represented flows, including physical-space vector fields, phase-space drift fields, and truncated renormalization-group beta functions. We introduce a complementary…
We provide in this article a new proof of the uniqueness of the flow solution to ordinary differential equations with $BV$ vector-fields that have divergence in $L^\infty$ (or in $L^1$) and that are nearly incompressible (see the text for…
In this paper, we study a family of curves on $S^2$ that defines a two-dimensional smooth projective plane. We use curve shortening flow to prove that any two-dimensional smooth projective plane can be smoothly deformed through a family of…
We formulate a hierarchical rectified flow to model data distributions. It hierarchically couples multiple ordinary differential equations (ODEs) and defines a time-differentiable stochastic process that generates a data distribution from a…
Fake projective planes are smooth complex surfaces of general type with Betti numbers equal to those of the usual projective plane. They come in complex conjugate pairs and have been classified as quotients of the two-dimensional ball by…
In this paper, we investigate steady Euler flows in a two-dimensional bounded domain. By an adaption of the vorticity method, we prove that for any nonconstant harmonic function $q$, which corresponds to a nontrivial irrotational flow,…