arXiv research

Study proves uniqueness of asymptotic limits for specific manifolds.

Uniqueness of asymptotic limits for Ricci-flat manifolds with linear volume growth is proven.

In this paper, we will show that the limit of some quasilocal mass integrals of the coordinate spheres in an asymptotically hyperbolic (AH) manifold is the mass integral of the AH manifold. This is the analogue of the well known result that the limit of the Brown-York mass of coordinate spheres is the ADM mass in an as…

Study on Teichmüller rays' asymptotic behavior and distances.

The study examines the asymptotic behavior of extremal length in Teichmüller space.

New insights into tSNE for large datasets.

We show that the limit at infinity of the vector-valued Brown-York-type quasi-local mass along any coordinate exhaustion of an asymptotically hyperbolic $3$-manifold satisfying the relevant energy condition on the scalar curvature has the conjectured causal character. Our proof uses spinors and relies on a Witten-type …

In this article, we consider the limit of quasi-local conserved quantities [31,9] at the infinity of an asymptotically hyperbolic initial data set in general relativity. These give notions of total energy-momentum, angular momentum, and center of mass. Our assumption on the asymptotics is less stringent than any previo…

We characterize sequences of Kleinian surface groups with convergent subsequences in terms of the asymptotic behavior of the ending invariants of the associated hyperbolic 3-manifolds. Asymptotic behavior of end invariants in a convergent sequence predicts the parabolic locus of the algebraic limit as well as how the a…

Study shows how learning and analytical models affect reneging and jockeying in a dual M/M/1 system.

We study some asymptotic behavior of the first nonzero eigenvalue of the Lalacian along the normalized Ricci flow and give a direct short proof for an asymptotic upper limit estimate.

The paper analyzes the excess risk of PCA and provides a precise characterization.

Paper relates asymptotic dimension to cofinal dimension using coarse proximities.

We discuss several aspects of the relation between asymptotically AdS and asymptotically dS spacetimes including: the continuation between these types of spaces, the global stability of asymptotically dS spaces and the structure of limits within this class, holographic renormalization, and the maximal mass conjecture o…

The time average of geometric Brownian motion plays a crucial role in the pricing of Asian options in mathematical finance. In this paper we consider the asymptotics of the discrete-time average of a geometric Brownian motion sampled on uniformly spaced times in the limit of a very large number of averaging time steps.…

If the initial hypersurface of an immortal mean curvature flow is asymptotic to a regular cone whose entropy is small, the flow will become asymptotically self-expanding. Moreover, the expander that gives rise to the limiting flow is asymptotically stable as an equilibrium solution of the normalized mean curvature flow…

Study geodesic Lie groups' convergence to limits with quantitative estimates.

Regularized kernel methods such as, e.g., support vector machines and least-squares support vector regression constitute an important class of standard learning algorithms in machine learning. Theoretical investigations concerning asymptotic properties have manly focused on rates of convergence during the last years bu…

In this work we obtain the limit of the Hawking energy of a large class of foliations along general null hypersurfaces $Ω$ satisfying a weak notion of asymptotic flatness. The foliations are not required to be either geodesic or approaching large spheres at infinity. The limit is obtained in terms of a reference backgr…

Study shows Roller compactification's median graph has limited asymptotic dimension.

We discuss asymptotic behavior of the eigenvalue distribution of the differential form Laplacian on a Riemannian foliated manifold when the metric on the ambient manifold is blown up in directions normal to the leaves (in the adiabatic limit). Motivated by analogies with semiclassical spectral asymptotics, we use ideas…

Study shows limits of Fuchsian surfaces in hyperbolic 3-manifolds.

We propose a novel time discretization for the log-normal SABR model which is a popular stochastic volatility model that is widely used in financial practice. Our time discretization is a variant of the Euler-Maruyama scheme. We study its asymptotic properties in the limit of a large number of time steps under a certai…

This paper introduces time-uniform CLT-based confidence intervals for statistical inference.

Study abelian varieties' Weil-Petersson metric asymptotics.

Constructs instantons on a specific G_2-manifold limit.

This paper investigates asymptotic behaviors of gradient descent algorithms (particularly accelerated gradient descent and stochastic gradient descent) in the context of stochastic optimization arising in statistics and machine learning where objective functions are estimated from available data. We show that these alg…

Bayesian hierarchical clustering (BHC) is an agglomerative clustering method, where a probabilistic model is defined and its marginal likelihoods are evaluated to decide which clusters to merge. While BHC provides a few advantages over traditional distance-based agglomerative clustering algorithms, successive evaluatio…

New asymptotic formula for option prices with interest rates and dividend yield effects.

A new proof of Friedrich's theorem on the existence and stability of asymptotically de Sitter spaces in 3+1 dimensions is given, which extends to all even dimensions. In addition, we characterize the possible limits of spaces which are globally asymptotically de Sitter, to the past and future.

Study equi-affine invariants for convex domains with asymptotes.

We show that any grafting ray in Teichmüller space is (strongly) asymptotic to some Teichmüller geodesic ray. As an intermediate step we introduce surfaces that arise as limits of these degenerating Riemann surfaces. Given a grafting ray, the proof involves a Teichmüller ray with a conformally equivalent limit, and bui…

Study on L2-boosting behavior as learning rate approaches zero.

New method achieves superlinear convergence rate with limited memory.

In this short note, we study the asymptotic property of Huisken's functional for mean curvature flow on the minimal submanifolds of Euclidean space. We prove that the limit of Huisken's functional equals to the extrinsic asymptotic volume ratio on the minimal submanifold of Euclidean space.

In this paper, we will show that the limit of the Brown-York mass of a family of convex revolution surfaces in an asymptotically Schwarzschild manifold is the ADM mass.

Gradient descent dynamics in wide neural networks are analyzed using a dynamical CLT.

Asymptotic Laplace transform for geometric Brownian motion applied to bond pricing.

We consider Ricci flow of complete Riemannian manifolds which have bounded non-negative curvature operator, non-zero asymptotic volume ratio and no boundary. We prove scale invariant estimates for these solutions. Using these estimates, we show that there is a limit solution, obtained by scaling down this solution at a…

The paper studies Hawkes processes under mean-field limits and criticality conditions.

Dynamic hedging of an European option under a general local volatility model with small linear transaction costs is studied. A continuous control version of Leland's strategy that asymptotically replicates the payoff is constructed. An associated central limit theorem of hedging error is proved. The asymptotic error va…

We use a Lagrangian perspective to show the limiting absorption principle on Riemannian scattering, i.e. asymptotically conic, spaces, and their generalizations. More precisely we show that, for non-zero spectral parameter, the `on spectrum', as well as the `off-spectrum', spectral family is Fredholm in function spaces…

For general Riemannian foliations, spectral asymptotics of the Laplacian is studied when the metric on the ambient manifold is blown up in directions normal to the leaves (adiabatic limit). The number of ``small'' eigenvalues is given in terms of the differentiable spectral sequence of the foliation. The asymptotics of…

We study the spherical cap packing problem with a probabilistic approach. Such probabilistic considerations result in an asymptotic sharp universal uniform bound on the maximal inner product between any set of unit vectors and a stochastically independent uniformly distributed unit vector. When the set of unit vectors …

Study the limiting shape of solutions to the L_p-Minkowski problem as p approaches negative infinity.

We prove non-asymptotic lower bounds on the expectation of the maximum of $d$ independent Gaussian variables and the expectation of the maximum of $d$ independent symmetric random walks. Both lower bounds recover the optimal leading constant in the limit. A simple application of the lower bound for random walks is an (…

We consider a jump-type Cox--Ingersoll--Ross (CIR) process driven by a standard Wiener process and a subordinator, and we study asymptotic properties of the maximum likelihood estimator (MLE) for its growth rate. We distinguish three cases: subcritical, critical and supercritical. In the subcritical case we prove weak …

PPBoot simplifies prediction-powered inference.