arXiv research

We show that asymptotically hyperbolic solutions of the Einstein constraint equations with constant mean curvature can be glued in such a way that their asymptotic regions are connected.

We construct solutions with prescribed asymptotics to the Einstein constraint equations using a cut-off technique. Moreover, we give various examples of vacuum asymptotically flat manifolds whose center of mass and angular momentum are ill-defined.

VRPG algorithm optimizes convex constraints with non-asymptotic guarantees.

Neural-guided symbolic regression uses asymptotic constraints to find unknown functions.

We follow the approach employed by Y. Choquet-Bruhat, J. Isenberg and D. Pollack in the case of closed manifolds and establish existence and non-existence results for the Einstein-scalar field constraint equations on asymptotically hyperbolic manifolds.

Proves existence of solutions to Einstein constraints with specific boundary conditions and verifies Penrose inequality.

We construct low regularity solutions of the vacuum Einstein constraint equations. In particular, on 3-manifolds we obtain solutions with metrics in $H^s\loc$ with $s>{3\over 2}$. The theory of maximal asymptotically Euclidean solutions of the constraint equations descends completely the low regularity setting. Moreove…

The extended constraint equations arise as a special case of the conformal constraint equations that are satisfied by an initial data hypersurface $Z$ in an asymptotically simple spacetime satisfying the vacuum conformal Einstein equations developed by H. Friedrich. The extended constraint equations consist of a quasi-…

Given asymptotically flat initial data on M^3 for the vacuum Einstein field equation, and given a bounded domain in M, we construct solutions of the vacuum constraint equations which agree with the original data inside the given domain, and are identical to that of a suitable Kerr slice (or identical to a member of som…

Optimizes resource allocation in MAB models with resource constraints.

We study the portfolio selection problem of a long-run investor who is maximising the asymptotic growth rate of her expected utility. We show that, somewhat surprisingly, it is essentially not affected by introduction of a floor constraint which requires the wealth process to dominate a given benchmark at all times. We…

A new method SLIDE ensures fairness in AI models.

We construct asymptotically Euclidean solutions of the vacuum Einstein constraint equations with an apparent horizon boundary condition. Specifically, we give sufficient conditions for the constant mean curvature conformal method to generate such solutions. The method of proof is based on the barrier method used by Ise…

In this dissertation, we prove a number of results regarding the conformal method of finding solutions to the Einstein constraint equations. These results include necessary and sufficient conditions for the Lichnerowicz equation to have solutions, global supersolutions which guarantee solutions to the conformal constra…

We prove that there are no restrictions on the spatial topology of asymptotically flat solutions of the vacuum Einstein equations in (n+1)-dimensions. We do this by gluing a solution of the vacuum constraint equations on an arbitrary compact manifold to an asymptotically Euclidean solution of the constraints on R^n. Fo…

Study optimizes growth rate for investors with long-only constraints.

Study on harmonic maps from surfaces to homogeneous spaces, focusing on bubble formation and geometric constraints.

We develop asymptotically optimal policies for the multi armed bandit (MAB), problem, under a cost constraint. This model is applicable in situations where each sample (or activation) from a population (bandit) incurs a known bandit dependent cost. Successive samples from each population are iid random variables with u…

We construct solutions of the constraint equation with non constant mean curvature on an asymptotically hyperbolic manifold by the conformal method. Our approach consists in decreasing a certain exponent appearing in the equations, constructing solutions of these sub-critical equations and then in letting the exponent …

New methods for efficient exploration under unknown linear constraints in bandits.

MINTS uses a minimalist Bayesian framework to tackle multi-armed bandits with structural constraints.

Given a collection of N solutions of the (3+1) vacuum Einstein constraint equations which are asymptotically Euclidean, we show how to construct a new solution of the constraints which is itself asymptotically Euclidean, and which contains specified sub-regions of each of the N given solutions. This generalizes earlier…

The paper proves positive energy-momentum theorems for charged AdS initial data sets.

Researchers create a method to join hyperboloidal data sets without violating the shear-free condition.

This work removes logarithmic singularities from hyperboloidal initial data without creating new ones.

Under the assumption of asymptotic relative Chow-stability for polarized algebraic manifolds $(M, L)$, a series of weighted balanced metrics $ω_m$, $m \gg 1$, called polybalanced metrics, are obtained from complete linear systems $|L^m|$ on $M$. Then the asymptotic behavior of the weights as $m \to \infty$ will be stud…

Study proposes initial data sets for solving gravitational equations, proving energy estimates.

We prove the existence of a large class of initial data for the vacuum Einstein equations which possess a finite number of asymptotically Euclidean and asymptotically conformally cylindrical or periodic ends. Aside from being asymptotically constant, only mild conditions on the mean curvature of these initial data sets…

We consider the problem of option hedging in a market with proportional transaction costs. Since super-replication is very costly in such markets, we replace perfect hedging with an expected loss constraint. Asymptotic analysis for small transactions is used to obtain a tractable model. A general expansion theory is de…

We survey some results on scalar curvature and properties of solutions to the Einstein constraint equations. Topics include an extended discussion of asymptotically flat solutions to the constraint equations, including recent results on the geometry of the center of mass of such solutions. We also review methods to con…

In this paper we consider the Allen-Cahn equation with constraint. In 1994, Chen and Elliott studied the asymptotic behavior of the solution of the Allen-Cahn equation with constraint. They proved that the zero level set of the solution converges to the classical solution of the mean curvature flow under the suitable c…

The goal of this paper is to establish the existence of a foliation of the asymptotic region of an asymptotically flat manifold with nonzero mass by surfaces which are critical points of the Willmore functional subject to an area constraint. Equivalently these surfaces are critical points of the Geroch-Hawking mass. Th…

We study local complexity measures for stochastic convex optimization problems, providing a local minimax theory analogous to that of Hájek and Le Cam for classical statistical problems. We give complementary optimality results, developing fully online methods that adaptively achieve optimal convergence guarantees. Our…

We provide a Hilbert manifold structure {à} la Bartnik for the space of asymptotically hyperbolic initial data for the vacuum constraint equations. The adaptation led us to prove new weighted Poincar{é} and Korn type inequalities for AH manifolds with inner boundary and weakly regular metric.

Study wSAA for contextual decisions, improving uncertainty quantification under computational constraints.

We extend the study of the vacuum Einstein constraint equations on manifolds with ends of cylindrical type initiated by Chruściel and Mazzeo by finding a class of solutions to the fully coupled system on such manifolds. We show that given a Yamabe positive metric g, which is conformally asymptotically cylindrical on ea…

Study on high-codimensional minimal surfaces in hyperbolic space.

Subbagging estimation for big data reduces memory usage while maintaining statistical consistency.

Given a geodesic inside a simply-connected, complete, non-positively curved Riemannian (NPCR) manifold M, we get an associated geodesic inside the asymptotic cone Cone(M). Under mild hypotheses, we show that if the latter is contained inside a bi-Lipschitz flat, then the original geodesic supports a non-trivial, orthog…

Optimal policy for multi-hypothesis testing with controlled sensing to minimize delay and error.

Researchers simplify Einstein-scalar field equations on specific manifolds.

Stabilized neural differential equations enforce constraints on dynamical systems.

We characterize the asymptotic performance of nonparametric goodness of fit testing. The exponential decay rate of the type-II error probability is used as the asymptotic performance metric, and a test is optimal if it achieves the maximum rate subject to a constant level constraint on the type-I error probability. We …

Optimal tests for nonparametric one- and two-sample testing are derived using MMD and KSD.

Study minimizes Willmore energy with constraints on surface properties.

We establish a general gluing theorem for constant mean curvature solutions of the vacuum Einstein constraint equations. This allows one to take connected sums of solutions or to glue a handle (wormhole) onto any given solution. Away from this handle region, the initial data sets we produce can be made as close as desi…

Study characterizes conformal boundaries of de Sitter spacetimes.

The paper examines how ESG constraints affect portfolio optimization in large datasets.