arXiv research

Proves Morse index theorem for geodesics in conic Finsler manifolds.

The computation of the index of the Hessian of the action functional in semi-Riemannian geometry at geodesics with two variable endpoints is reduced to the case of a fixed final endpoint. Using this observation, we give an elementary proof of the Morse Index Theorem for Riemannian geodesics with two variable endpoints,…

We prove a semi-Riemannian version of the celebrated Morse Index Theorem for geodesics in semi-Riemannian manifolds; we consider the general case of both endpoints variable on two submanifolds. The key role of the theory is played by the notion of the {\em Maslov index} of a semi-Riemannian geodesic, which is a homolog…

Prove Gromov's Euclidean endpoint $C^0$ rigidity conjecture for positive mass theorem.

Paper proves unique energy-minimizing curves in constrained spaces.

In Carnot-Caratheodory or sub-Riemannian geometry, one of the major open problems is whether the conclusions of Sard's theorem holds for the endpoint map, a canonical map from an infinite-dimensional path space to the underlying finite-dimensional manifold. The set of critical values for the endpoint map is also known …

The paper finds curves minimizing elastic energy pinned at endpoints.

FlexServe simplifies deployment of PyTorch models as REST endpoints.

It is well known that plane curves with the same endpoints are homotopic. An analogous claim for plane curves with the same endpoints and bounded curvature still remains open. In this work we find necessary and sufficient conditions for two plane curves with bounded curvature to be deformed, one to another, by a contin…

Investigates VaR behavior for sums of one-sided random variables, showing impossibilities and conditions for super-additivity.

Let M be a possibly non compact smooth manifold. We study genericity in the C^k-topology (3<=k<=+infty) of nondegeneracy properties of semi-Riemannian geodesic flows on M. Namely, we prove a new version of the Bumpy Metric Theorem for a such M and also genericity of metrics that do not possess any degenerate geodesics …

Given a rank-two sub-Riemannian structure $(M,Δ)$ and a point $x_0\in M$, a singular curve is a critical point of the endpoint map $F:γ\mapstoγ(1)$ defined on the space of horizontal curves starting at $x_0$. The typical least degenerate singular curves of these structures are called \emph{regular singular curves}; the…

The paper classifies and analyzes the stability of elastic curves with fixed endpoints.

New models for short rates show longer periods at higher rates.

Burq-Gérard-Tzvetkov and Hu established $L^p$ estimates ($2\le p\le \infty$) for the restriction of eigenfunctions to submanifolds. The estimates are sharp, except for the log loss at the endpoint $L^2$ estimates for submanifolds of codimension 2. It has long been believed that the log loss at the endpoint can be remov…

For a Riemannian manifold $(M,g)$ with strictly convex boundary $\partial M$, the lens data consists in the set of lengths of geodesics $γ$ with endpoints on $\partial M$, together with their endpoints $(x_-,x_+)\in \partial M\times \partial M$ and tangent exit vectors $(v_-,v_+)\in T_{x_-} M\times T_{x_+} M$. We show …

We investigate the evolution of open curves with fixed endpoints under the curve shortening flow, which evolves curves in proportion to their curvature. Using a distance comparison of Huisken, we determine the long-term behavior of open curves with fixed endpoints evolving in certain convex domains on surfaces of const…

Study on Hausdorff dimension of lamination endpoints for fully irreducible automorphisms.

We prove that the curvature flow of an embedded planar network of three curves connected through a triple junction, with fixed endpoints on the boundary of a given strictly convex domain, exists smooth until the lengths of the three curves stay far from zero. If this is the case for all times, then the evolution exists…

We introduce a Kauffman-Jones type polynomial $\mathcal{L}_γ(A)$ for a curve $γ$ on an oriented surface, whose endpoints are on the boundary of the surface. The polynomial $\mathcal{L}_γ(A)$ is a Laurent polynomial in one variable $A$ and is an invariant of the homotopy class of $γ$. As an application, we obtain an est…

TA-CQR predicts regression intervals with exact coverage, splitting miscoverage between endpoints.

QGMS framework detects market endpoints using geometric patterns.

We review and organize some results describing the behavior of a Teichmüller geodesic and draw several applications: 1) We show that Teichmüller geodesics do not back track. 2) We show that a Teichmüller geodesic segment whose endpoints are in the thick part has the fellow travelling property. This fails when the endpo…

DistillKac generates images quickly using damped wave equations.

Develops local population-risk certificates for model updates

Generative AI models improve clinical trial data by generating survival outcomes.

Study Dirac operators on finite warped cylinders with gauge fields.

Curve diffusion flow straightens curves with endpoints on intersecting lines.

New Virasoro-like structures for circle diffeomorphisms with breaks.

The study of smoothing arcs and curves on surfaces, proving tautness and arc length spectrum properties.

We investigate the Tits boundary of locally compact CAT(0) 2-complexes. In particular we show that away from the endpoints, a geodesic segment in the Tits boundary is the ideal boundary of an isometrically embedded Euclidean sector. As applications, we provide sufficient conditions for two points in the Tits boundary t…

The paper computes a knot's Kauffman bracket polynomial using recursive concatenation of a 4-tangle shadow.

We discuss homotopy properties of endpoint maps for affine control systems. We prove that these maps are Hurewicz fibrations with respect to some $W^{1,p}$ topology on the space of trajectories, for a certain $p>1$. We study critical points of geometric costs for these affine control systems, proving that if the base m…

A tangle is an oriented 1-submanifold of the cylinder whose endpoints lie on the two disks in the boundary of the cylinder. Using an algebraic tool developed by Lescop, we extend the Burau representation of braids to a functor from the category of oriented tangles to the category of Z[t,t^{-1}]-modules. For (1,1)-tangl…

We propose a non-parametric link prediction algorithm for a sequence of graph snapshots over time. The model predicts links based on the features of its endpoints, as well as those of the local neighborhood around the endpoints. This allows for different types of neighborhoods in a graph, each with its own dynamics (e.…

The availability of a large amount of electronic health records (EHR) provides huge opportunities to improve health care service by mining these data. One important application is clinical endpoint prediction, which aims to predict whether a disease, a symptom or an abnormal lab test will happen in the future according…

Let M be a possibly noncompact manifold. We prove, generically in the C^k-topology (k=2,...,\infty), that semi-Riemannian metrics of a given index on M do not possess any degenerate geodesics satisfying suitable boundary conditions. This extends a result of Biliotti, Javaloyes and Piccione for geodesics with fixed endp…

A new sampler for FLMs improves token-level decoding controls.

ACP-UCB1 ranks arms based on upper-tail performance, improving stochastic bandit algorithms.

The focal locus $Σ_X$ of an affine variety $X$ is roughly speaking the (projective) closure of the set of points $O$ for which there is a smooth point $x \in X$ and a circle with centre $O$ passing through $x$ which osculates $X$ in $x$. Algebraic geometry interprets the focal locus as the branching locus of the endpoi…

Extends knotoid theory to include multiple poles and intervals.

QDSB accelerates Schrödinger bridge learning with quantized approximations.

This work provides uncertainty intervals for semantic latent variables in disentangled latent spaces.

The multi dimensional string objects are introduced as a new alternative for an application of string models for time series forecasting in trading on financial markets. The objects are represented by open string with 2-endpoints and D2-brane, which are continuous enhancement of 1-endpoint open string model. We show ho…

Most approaches to machine learning from electronic health data can only predict a single endpoint. Here, we present an alternative that uses unsupervised deep learning to simulate detailed patient trajectories. We use data comprising 18-month trajectories of 44 clinical variables from 1908 patients with Mild Cognitive…

Study Sard problem in step 2 and filiform Carnot groups.

New geodesic rays found on hyperbolic surfaces with unique halo.

Improved Strichartz estimates for Schrödinger equation on negatively curved manifolds.