arXiv research

New steady Euler flows found on 3-sphere and Sasakian manifolds.

Study of 4D Ricci solitons with symmetry, finding precise geometric asymptotics.

FNO-DEQ solves steady-state PDEs as fixed points, outperforming traditional FNOs.

Study ancient solutions on noncompact steady Ricci solitons, proving types of ancient solutions.

We solve Euler equations on graph manifolds, classifying steady flows with Morse-Bott Bernoulli functions.

Non-vanishing steady Euler flows and Beltrami fields found in high dimensions.

A nontrivial smooth steady incompressible Euler flow in three dimensions with compact support is constructed. Another uncommon property of this solution is the dependence between the Bernoulli function and the pressure.

Study steady gradient Ricci solitons with cylindrical tangent flows at infinity.

We point out a duality between steady incompressible Euler flows and solutions of the strongly coupled Faddeev-Skyrme sigma model with potential (mass) term. We supplement this result with various applications and several explicit examples.

We study the convergence of complete non-compact conformally flat solutions to the Yamabe flow to Yamabe steady solitons. We also prove the existence of Type II singularities which develop at either a finite time $T$ or as $t \to +\infty$.

Study on scalar curvature decay in four-dimensional steady solitons.

We characterize, using commuting zero-flux homologies, those volume-preserving vector fields on a $3$-manifold that are steady solutions of the Euler equations for some Riemannian metric. This result extends Sullivan's homological characterization of geodesible flows in the volume-preserving case. As an application, we…

This work concerns with the existence and detailed asymptotic analysis of Type II singularities for solutions to complete non-compact conformally flat Yamabe flow with cylindrical behavior at infinity. We provide the specific blow-up rate of the maximum curvature and show that the solution converges, after blowing-up a…

This paper solves steady-state planning for multichain MDPs.

New curvature condition proves rigidity of Bryant Ricci solitons.

Online learners track optimal solutions with constant step-size.

The paper classifies invariant gradient $k$-Yamabe solitons in pseudo-Euclidean spaces.

We present a steady Euler flow on the round 3-sphere whose velocity vector field has the property of having two independent first integrals, being tangent to the fibres of an almost submersion onto the 2-sphere. This submersion turns out to be a critical point for the quartic Faddeev-Skyrme model with a standard potent…

We study two classes of over-the-counter markets specified by systems of ODE's, in the spirit of Duffie-Garleanu-Pedersen, Econometrica, 2005. We first compute the steady states for many of these ODE's. Then we obtain the prices at which investors trade with each other at these steady states. Finally, we study the stab…

We study closed ancient solutions to gradient flows of elliptic functionals in Riemannian manifolds, including mean curvature flow and harmonic map heat flow. Our work has various consequences. In all dimensions and codimensions, we classify ancient mean curvature flows in S^n with low area: they are steady or shrinkin…

Study of solitons in Laplacian flow on 7-manifolds.

In this paper, we solve the so-called CR Poincaré-Lelong equation by solving the CR Poisson equation on a complete noncompact CR $(2n+1)$-manifold with nonegative pseudohermitian bisectional curvature tensors and vanishing torsion which is an odd dimensional counterpart of Kähler geometry. With applications of this sol…

Classifies polynomial growth solutions to drift-harmonic equations on asymptotically paraboloidal manifolds.

We study the problem of controlling linear time-invariant systems with known noisy dynamics and adversarially chosen quadratic losses. We present the first efficient online learning algorithms in this setting that guarantee $O(\sqrt{T})$ regret under mild assumptions, where $T$ is the time horizon. Our algorithms rely …

Survey classifies singularity models in 3D Ricci flow.

The paper proves convergence of normalized Ricci flow on compact manifolds.

Many structured data-fitting applications require the solution of an optimization problem involving a sum over a potentially large number of measurements. Incremental gradient algorithms offer inexpensive iterations by sampling a subset of the terms in the sum. These methods can make great progress initially, but often…

Study 4D solitons with specific curvature properties, proving curvature bounds and classifying solutions.

We show that a steady-state stock-flow consistent macro-economic model can be represented as a Constraint Satisfaction Problem (CSP).The set of solutions is a polytope, which volume depends on the constraintsapplied and reveals the potential fragility of the economic circuit,with no need to study the dynamics. Several …

Study on steady solitons' curvature behavior near infinity.

In this article we construct a smooth Euler flow supported in a neighborhood of a helix. It may be considered a generalization of a similar solution found by the author for a circle.

A large class of variational equations for geometric objects is studied. The results imply conformal monotonicity and Liouville theorems for steady, polytropic, ideal flow, and the regularity of weak solutions to generalized Yang-Mills and Born-Infeld systems.

New findings on $κ$-solutions with round cylinder as asymptotic shrinker.

In this paper we study potential function of gradient steady Ricci solitons. We prove that infimum of potential function decays linearly; in particular, potential function of rectifiable gradient steady Ricci solitons decays linearly. As a consequence, we show that a gradient steady Ricci soliton with bounded potential…

Exact asymptotic solutions found for nonlinear Hawkes processes.

Let (M,g) be a three-dimensional steady gradient Ricci soliton which is non-flat and κ-noncollapsed. We prove that (M,g) is isometric to the Bryant soliton up to scaling. This solves a problem mentioned in Perelman's first paper.

In this paper, we give a description for steady Ricci solitons with a linear decay of sectional curvature. In particular, we classify all 3-dimensional steady Ricci solitons and 4-dimensional $κ$-noncollpased steady Ricci solitons with nonnegative sectional curvature under the linear curvature decay.

Unique steady and expanding solitons with spherical links identified.

We develop an integral geometry of stationary Euler equations defining some function $w$ on the Grassmannian of affine lines in the space. This function depends on a putative compactly supported solution $v$ of the system, and we deduce a linear differential equation for $w$. We prove also that the purported annulation…

This paper considers a mortgage contract where the borrower pays a fixed mortgage rate and has the choice of making prepayment. Assume the market interest follows the CIR model, a free boundary problem is formulated. Here we focus on the infinite horizon problem. Using variational method, we obtain an analytical soluti…

3D steady gradient Ricci solitons are all O(2)-symmetric.

The study classifies steady Ricci solitons based on geometric conditions.

We study the steady state solutions of a generalized logistic type equation on a complete Riemannian manifold. We provide sufficient conditions for existence, respectively non-existence of positive solutions, which depend on the relative size of the coefficients and their mutual interaction with the geometry of the man…

Paper proves volume growth estimate for steady gradient Ricci solitons.

Existence of singular gradient Ricci solitons proved in higher dimensions.

This research shows that steady solitons in higher dimensions always reduce at infinity.

New classification of gradient steady Ricci solitons with vanishing D-tensor.

We show that any locally conformally flat ancient solution to the Ricci flow must be rotationally symmetric. As a by-product, we prove that any locally conformally flat Ricci soliton is a gradient soliton in the shrinking and steady cases as well as in the expanding case, provided the soliton has nonnegative curvature.