arXiv research

Study geometric flow on curves with positive torsion, finding stationary solutions and their stability.

Noncompact Ricci-flat solutions have infinite unstable dimensions.

Smooth solutions found for Hamiltonian stationary equations in low dimensions.

This is the second part of the investigation started in [Stationary solutions and asymptotic flatness I]. We prove here that Strongly Stationary ends having cubic volume growth are Weakly Asymptotically Flat. Combined with the results of the previous paper this shows that Strongly Stationary ends are Asymptotically Fla…

We derive a local curvature estimate for four-dimensional stationary solutions to the inheriting Einstein-Maxwell-Klein-Gordon equations. In particular, it implies that any such stationary geodesically complete solution with vanishing Poynting vector and proper coupling constants (like dark energy) is flat. We also gen…

It is proved that the only geodesically complete stationary vacuum solution of the Einstein equations is the empty Minkowski space, or a quotient of it by a discrete group of isometries, generalizing a classical result of Lichnerowicz. In addition, we obtain an apriori bound on the curvature of stationary vacuum soluti…

We characterize stationary solutions to McKean-Vlasov equations on the circle.

Unified approach for non-stationary and clustered bandits.

We classify all Hamiltonian stationary Lagrangian surfaces in complex Euclidean plane which are self-similar solutions of the mean curvature flow.

Develops methods for constructing exact, non-stationary solutions to Euler equations.

New black hole solutions cannot be rotated without breaking their structure.

We find various exact solutions for a new stochastic volatility (SV) model: the transition probability density, European-style option values, and (when it exists) the martingale defect. This may represent the first example of an SV model combining exact solutions, GBM-type volatility noise, and a stationary volatility …

We consider smooth radial solutions to the Hamiltonian stationary equation which are defined away from the origin. We show that in dimension two all radial solutions on unbounded domains must be special Lagrangian. In contrast, for all higher dimensions there exist non-special Lagrangian radial solutions over unbounded…

Proves $C^{1,1}$ regularity for multiple membrane solutions.

In [LW], we construct examples of two-dimensional Hamiltonian stationary self-shrinkers and self-expanders for Lagrangian mean curvature flows, which are asymptotic to the union of two Schoen-Wolfson cones. These self-shrinkers and self-expanders can be glued together to yield solutions of the Brakke flow - a weak form…

We prove partial regularity of stationary solutions and minimizers $u$ from a set $Ω\subset \mathbb R^n$ to a Riemannian manifold $N$, for the functional $\int_ΩF(x,u,|\nabla u|^2) dx$. The integrand $F$ is convex and satisfies some ellipticity and boundedness assumptions. We also develop a new monotonicity formula and…

Generalising the results in arXiv:1612.00281, we construct infinite-dimensional families of non-singular stationary space times, solutions of Yang-Mills-Higgs-Einstein-Maxwell-Chern-Simons-dilaton-scalar field equations with a negative cosmological constant. The families include an infinite-dimensional family of soluti…

Constructs solutions to Einstein-Maxwell-current system using Sasakian manifolds.

We analyze here Hamiltonian stationary surfaces in the complex projective plane as (local) solutions to an integrable system, formulated as a zero curvature on a loop group. As an application, we show in details why such tori are finite type solutions, and eventually describe the simplest of them: the homogeneous ones.

Using generalized Riemann maps, normal forms for almost complex domains (D, J) with singular foliations by stationary disks are defined. Such normal forms are used to construct counterexamples and to determine intrinsic conditions, under which the stationary disks are extremal disks for the Kobayashi metric or determin…

We use an elliptic system of equations with complex coefficients for a set of complex-valued tensor fields as a tool to construct infinite-dimensional families of non-singular stationary black holes, real-valued Lorentzian solutions of the Einstein-Maxwell-dilaton-scalar fields-Yang-Mills-Higgs-Chern-Simons-$f(R)$ equa…

We study global aspects of complete, non-singular asymptotically locally AdS spacetimes solving the vacuum Einstein equations whose conformal infinity is an arbitrary globally stationary spacetime. It is proved that any such solution which is asymptotically stationary to the past and future is itself globally stationar…

Bandit algorithms have been predominantly analyzed in the convex setting with function-value based stationary regret as the performance measure. In this paper, motivated by online reinforcement learning problems, we propose and analyze bandit algorithms for both general and structured nonconvex problems with nonstation…

The alternating gradient descent (AGD) is a simple but popular algorithm which has been applied to problems in optimization, machine learning, data ming, and signal processing, etc. The algorithm updates two blocks of variables in an alternating manner, in which a gradient step is taken on one block, while keeping the …

Paper solves Merton's portfolio problem in a non-Markovian, non-semimartingale model.

Center manifold analysis can be used in order to investigate the stability of the stationary solutions of various PDEs. This can be done by considering the PDE as an ODE between certain Banach spaces and linearising about the stationary solution. Here we investigate the volume preserving mean curvature flow using such …

Proves a higher-order positive energy theorem for stationary solutions in fourth-order gravity.

Polyconvex energies with conformal invariance have smooth stationary points outside a discrete set.

The paper proves optimal smoothness for certain Lagrangian graphs with specific Hölder continuity.

Investigates mean-variance portfolio selection in non-Markovian markets.

A well known result of Da Rios and Levi-Civita says that a closed planar curve is elastic if and only if it is stationary under the localized induction (or smoke ring) equation, where stationary means that the evolution under the localized induction equation is by rigid motions. We prove an analogous result for surface…

We study the regularity problem of the nonlinear sigma model with gravitino fields in higher dimensions. After setting up the geometric model, we derive the Euler--Lagrange equations and consider the regularity of weak solutions defined in suitable Sobolev spaces. We show that any weak solution is actually smooth under…

In this paper we generalize examples of Hamiltonian stationary Lagrangian submanifolds constructed by Lee and Wang in $\mathbb{C}^m$ to toric almost Calabi-Yau manifolds. We construct examples of weighted Hamiltonian stationary Lagrangian submanifolds in toric almost Calabi-Yau manifolds and solutions of generalized La…

Proposes a method to forecast non-stationary time series.

Unified framework for photon and massive particle hypersurfaces in stationary spacetimes.

We present a novel approach for fully non-stationary Gaussian process regression (GPR), where all three key parameters -- noise variance, signal variance and lengthscale -- can be simultaneously input-dependent. We develop gradient-based inference methods to learn the unknown function and the non-stationary model param…

We prove that a single-layer neural network trained with the Q-learning algorithm converges in distribution to a random ordinary differential equation as the size of the model and the number of training steps become large. Analysis of the limit differential equation shows that it has a unique stationary solution which …

In this paper we study general Schatten-$p$ quasi-norm (SPQN) regularized matrix minimization problems. In particular, we first introduce a class of first-order stationary points for them, and show that the first-order stationary points introduced in [11] for an SPQN regularized $vector$ minimization problem are equiva…

We show that the only convex ancient solutions to curve shortening flow are the stationary lines, shrinking circles, Grim Reapers and Angenent ovals, completing the classification initiated by Daskalopoulos, Hamilton and Sesum and X.-J. Wang

In this paper we consider a class of weighted-volume preserving curvature flows acting on hypersurfaces that are trapped within two parallel hyperplanes and satisfy an orthogonal boundary condition. In the author's thesis the stability of cylinders under the flows was considered; it was found that they are stable provi…

Noise balance theory explains SGD's behavior in neural networks.

Integrable flows on null curves in anti-de Sitter 3-space studied.

We study the problem of stationary bi-axially symmetric solutions of the $5$-dimensional minimal supergravity equations. Essentially all possible solutions with nondegenerate horizons are produced, having the allowed horizon cross-sectional topologies of the sphere $S^3$, ring $S^1\times S^2$, and lens $L(p,q)$, as wel…

We prove a Morrey-type theorem for Hamiltonian stationary submanifolds of $\mathbb{C}^{n}$. Namely, if $L$ $\subset$ $\mathbb{C}^{n}$ is a $C^{1}$ Lagrangian submanifold with weakly harmonic Lagrangian phase $θ,$ then $L$ must be smooth. In the process we also discuss a local version of the equation, which is a nonline…

Extends Euler's problem to Lorentz-Minkowski plane.

We construct examples of shrinkers and expanders for Lagrangian mean curvature flows. These examples are Hamiltonian stationary and asymptotic to the union of two Hamiltonian stationary cones found by Schoen and Wolfson. The Schoen-Wolfson cones $C_{p,q}$ are obstructions to the existence problems of special Lagrangian…

New framework analyzes effectiveness of neural network-based combinatorial problem solvers.

Adaptive Bayesian learning agent for non-stationary bandits.