arXiv research

Study local limits of subgroups in $SL_3(\mathbb{R})$.

The article calculates the near horizon limit of Wang--Yau quasi-local mass.

In this article, we study the small sphere limit of the Wang-Yau quasi-local energy defined in [18,19]. Given a point $p$ in a spacetime $N$, we consider a canonical family of surfaces approaching $p$ along its future null cone and evaluate the limit of the Wang-Yau quasi-local energy. The evaluation relies on solving …

Ricci limit spaces are semi-locally simply connected.

In [13], a new quasi-local energy is introduced for spacetimes with a non-zero cosmological constant. In this article, we study the small sphere limit of this newly defined quasi-local energy for spacetimes with a negative cosmological constant. For such spacetimes, the anti de-Sitter space is used as the reference for…

Study shows local topologies of certain geometric spaces.

We provide an easily verifiable condition for local $k$-connectedness of an inverse limit of polyhedra.

The study extends convergence theorems for Ricci-limit spaces with bounded curvature.

Adaptive sampling detects local concept drift with limited labels.

In the paper arXiv:1411.4887 [math.AP] it is shown that the set of Riemannian metrics which do not admit global limiting Carleman weights is open and dense, by studying the conformally invariant Weyl and Cotton tensors. In the paper arXiv:1011.2507 [math.DG] it is shown that the set of Riemannian metrics which do not a…

The paper analyzes phase retrieval under limited samples, ensuring a benign local landscape for convergence.

In general relativity, the local gravitational energy is best characterised by the quasilocal mass. The small sphere limit of quasilocal mass provides us the most local notion of gravitational energy. In four dimensions, the limits were shown be the stress tensor in non-vacuum and the Bel-Robinson tensor in vacuum. We …

We use Ricci flow to obtain a local bi-Holder correspondence between Ricci limit spaces in three dimensions and smooth manifolds. This is more than a complete resolution of the three-dimensional case of the conjecture of Anderson-Cheeger-Colding-Tian, describing how Ricci limit spaces in three dimensions must be homeom…

Study examines Wang-Yau quasi-local energy in strong fields near apparent horizons.

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 …

Study limits of quasi-local angular momentum at infinity of gravitating systems.

In this note, we compute the limit of the Wang-Yau quasi-local mass on unit spheres at spatial infinity of an asymptotically flat initial data set. Similar to the small sphere limit of the Wang-Yau quasi-local mass, we prove that the leading order term of the quasi-local mass recovers the stress-energy tensor. For a va…

Local limit theorem for random walks on hyperbolic groups with parabolic subgroups.

We show that the Gromov-Hausdorff limit of a sequence of leaves in a compact foliation is a covering space of the limiting leaf which is no larger than this leaf's holonomy cover. We also show that convergence to such a limit is smooth instead of merely Gromov-Hausdorff. Corollaries include Reeb's local stability theor…

Many graph clustering quality functions suffer from a resolution limit, the inability to find small clusters in large graphs. So called resolution-limit-free quality functions do not have this limit. This property was previously introduced for hard clustering, that is, graph partitioning. We investigate the resolution-…

Study the topology of Ricci limit spaces using Gromov-Hausdorff limits.

Improved local multivariable regression for better inference with limited data.

Classifies limits of groups of involutions in SL(2,F) over local fields.

We study the rigid limit of a class of hypermultiplet moduli spaces appearing in Calabi-Yau compactifications of type IIB string theory, which is induced by a local limit of the Calabi-Yau. We show that the resulting hyperkahler manifold is obtained by performing a hyperkahler quotient of the Swann bundle over the modu…

Let $X$ be a non-collapsing Ricci limit space and let $x\in X$. We show that for any $ε>0$, there is $r>0$ such that every loop in $B_t(x)$ is contractible in $B_{(1+ε)t}(x)$, where $t\in(0,r]$. In particular, $X$ is semi-locally simply connected.

We study some similarities between almost product Riemannian structures and almost Hermitian structures. Inspired by the similarities, we prove lower eigenvalue estimates for the Dirac operator on compact Riemannian spin manifolds with locally product structures. We also provide some examples (limiting manifolds) for t…

Using the reformulation in divergence form of the Euler-Lagrange equation for the Willmore functional as it was developed in "Analysis of the Willmore Functional" by T. Riviere (Invent. Math. 174), we study the limit of a local Palais-Smale sequence of weak Willmore immersions with locally square-integrable second fund…

Proposes a new contrastive loss for semi-supervised medical image segmentation.

We prove that any limit-interface corresponding to a locally uniformly bounded, locally energy-bounded sequence of stable critical points of the van der Waals--Cahn--Hilliard energy functionals with perturbation parameter tending to 0 is supported by an embedded smooth stable minimal hypersurface in low dimensions and …

Study homogenizes equations on parallelizable manifolds using tensor localization and periodicity.

Finite-precision learning of $anh$ networks is limited by the Monte Carlo rate.

This paper derives a diffusion approximation for a sequence of discrete-time one-sided limit order book models with non-linear state dependent order arrival and cancellation dynamics. The discrete time sequences are specified in terms of an $\R_+$-valued best bid price process and an $L^2_{loc}$-valued volume process. …

This paper is devoted to the framework of direct limit of anchored Banach bundles over a convenient manifold which is a direct limit of Banach manifold. In particular we give a criterion of integrability for distributions on such convenient manifolds which are locally direct limits of particular sequences of Banach anc…

New foliations found for critical surfaces of Hawking energy, resolving discrepancies.

We construct a global homeomorphism from any 3D Ricci limit space to a smooth manifold, that is locally bi-Holder. This extends the recent work of Miles Simon and the second author, and we build upon their techniques. A key step in our proof is the construction of local "pyramid Ricci flows", existing on uniform region…

Study rigidity of Ricci flow limits on nilpotent bundles with zero curvature.

In this article, we study the relationship between the weak limit of a sequence of integral currents in a metric space and the possible Hausdorff limit of the sequence of supports. Due to cancellation, the weak limit is in general supported in a strict subset of the Hausdorff limit. We exhibit sufficient conditions in …

Uncertainty principles such as Heisenberg's provide limits on the time-frequency concentration of a signal, and constitute an important theoretical tool for designing and evaluating linear signal transforms. Generalizations of such principles to the graph setting can inform dictionary design for graph signals, lead to …

New research limits what GNNs can compute and generalizes their performance.

Study limits of Kähler-Einstein metrics with cone singularities on complex projective manifolds.

Graph-based weather prediction adapted for local models.

Accurate identification and localization of abnormalities from radiology images play an integral part in clinical diagnosis and treatment planning. Building a highly accurate prediction model for these tasks usually requires a large number of images manually annotated with labels and finding sites of abnormalities. In …

The paper studies empirical processes from nearest neighbors in regression.

Local convolutions bias neural networks towards high-frequency adversarial examples.

Paper proposes a federated learning method for quantile inference with local differential privacy.

This thesis discusses the Newtonian limit of General Relativity for static isolated systems with compactly supported matter. We call these systems "geometrostatic" to underline their geometric nature. We introduce new quasi-local notions of mass and center of mass that can be read off locally in the vicinity of the mat…

This paper analyzes Local SGD for federated learning, achieving both statistical and communication efficiency.

We study the collapsing behaviour of Ricci-flat Kahler metrics on a projective Calabi-Yau manifold which admits an abelian fibration, when the volume of the fibers approaches zero. We show that away from the critical locus of the fibration the metrics collapse with locally bounded curvature, and along the fibers the re…