arXiv research

Study on spherical bodies of constant width on the unit sphere, proving bounds on their relative effective radius.

Paper generalizes inscribed radius estimate to hyperbolic Einstein manifolds.

Paper develops a robust HVA measure for dynamic hedging under liquidity stress.

We define a capacity which measures the size of Weinstein tubular neighbourhoods of Lagrangian submanifolds. In symplectic vector spaces this leads to bounds on the codisc radius for any closed Lagrangian submanifold in terms of Viterbo's isoperimetric inequality. Moreover, we prove a generalization of Gromov's packing…

Time dilation $\frac{1}{\sqrt{1-v^2}}$ and relative velocity $v$ are observationally indistinguishable in the special theory of relativity, a duality that carries over into the general theory under Fermi coordinates along a curve (in coordinate-independent language, in the tangent Minkowski space along the curve). For …

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

Develops a robust hedging valuation adjustment measure for dynamic hedging under liquidity-demand stress.

The paper is concerned with regularity properties of boundaries of causal pasts of points in a 3+1-dimensional Einstein-vacuum spacetime. In a Lorentzian manifold such boundaries play crucial role in propagation of linear and nonlinear waves. We prove a uniform lower bound on the radius of injectivity of these null bou…

Study finds the covering radius of RM(4,8) is 26.

For a polygonal knot K, it is shown that a tube of radius R(K), the polygonal thickness radius, is an embedded torus. Given a thick configuration K, perturbations of size r<R(K) define satellite structures, or local knotting. We explore knotting within these tubes both theoretically and numerically. We provide bounds o…

Motivated by the application to spacetimes of general relativity we investigate the geometry and regularity of Lorentzian manifolds under certain curvature and volume bounds. We establish several injectivity radius estimates at a point or on the past null cone of a point. Our estimates are entirely local and geometric,…

In the presence of model risk, it is well-established to replace classical expected values by worst-case expectations over all models within a fixed radius from a given reference model. This is the "robustness" approach. We show that previous methods for measuring this radius, e.g. relative entropy or polynomial diverg…

Relatively extremal knots are the relative minima of the ropelength functional in C^1 topology. On the set curves of fixed length, they are the relative maxima of thickness (normal injectivity radius) functional, including the ideal knots. We prove that a C^{1,1} relatively extremal knot in R^n has thickness equal to h…

Let $π$ be a finitely presented group. If h is a non trivial homology class in Hn($π$; Z), a theorem of Gromov (see [Gro83], 6) asserts the existence of regular geometric cycles which represent h, whose relative systolic volume is as close as desired to the systolic volume of h, in which we can control the volume of ba…

The paper extends radius estimates for stable hypersurfaces in 2, 3, and 4 dimensions.

We give sharp, effective bounds on the distance between tori of fixed injectivity radius inside a Margulis tube in a hyperbolic 3-manifold.

Solves relative isoperimetric problem for cubes, identifying specific minimizers.

Vacuum gravity shows black holes can form without collapse.

Study on curvature in finitely generated groups, showing positive curvature in specific cases.

The first algorithm for sampling the space of thick equilateral knots, as a function of thickness, will be described. This algorithm is based on previous algorithms of applying random reflections. To prove the existence of the algorithm, we describe a method for turning any knot into the regular planar polygon using on…

Explaining the unreasonable effectiveness of deep learning has eluded researchers around the globe. Various authors have described multiple metrics to evaluate the capacity of deep architectures. In this paper, we allude to the radius margin bounds described for a support vector machine (SVM) with hinge loss, apply the…

The thickness, NIR(K) of a knot or link K is defined to be the radius of the largest solid tube one can put around the curve without any self intersections, which is also known as the normal injectivity radius of K. For C^{1,1} curves K, NIR(K)=min{(1/2)DCSC(K),(1/(supkappa(K))))}, where kappa(K) is the generalized cur…

The Kapustin-Witten equations on R^4 are equations for a pair of connection on the product principle SU(2) bundle and 1-form with values in the product Lie algebra bundle. The 1-form is the Higgs field. A dichotomy is proved to the effect that either the averaged norm of the Higgs field on large radius spheres grows fa…

Paper introduces robust market making using Wasserstein distance and entropy regularization.

Estimates Betti numbers of loop spaces of compact manifolds.

The paper assesses text classification robustness through maximal safe radius computation.

Let $Γ$ be a relatively hyperbolic group and let $μ$ be an admissible symmetric finitely supported probability measure on $Γ$. We extend Floyd-Ancona type inequalities up to the spectral radius of $μ$. We then show that when the parabolic subgroups are virtually abelian, the Martin boundary of the induced random walk o…

In this paper we prove the following pointwise and curvature-free estimates on convexity radius, injectivity radius and local behavior of geodesics in a complete Riemannian manifold $M$: 1) the convexity radius of $p$, $\operatorname{conv}(p)\ge \min\{\frac{1}{2}\operatorname{inj}(p),\operatorname{foc}(B_{\operatorname…

Positive injectivity radius for manifolds with Lie structure at infinity.

A reservoir computer is a complex dynamical system, often created by coupling nonlinear nodes in a network. The nodes are all driven by a common driving signal. In this work, three dimension estimation methods, false nearest neighbor, covariance and Kaplan-Yorke dimensions, are used to estimate the dimension of the res…

The ratio of convexity radius over injectivity radius may be made arbitrarily small within the class of compact Riemannian manifolds of any fixed dimension at least two. This is proved using Gulliver's method of constructing manifolds with focal points but no conjugate points. The approach is suggested by a characteriz…

Paper optimizes hyperparameters for high-dimensional regression models.

Upper bound for conjugate radius in open manifolds with scalar curvature and spectrum constraints.

Proves upper bound on filling radius for manifolds with positive scalar curvature.

Compact theorem for minimal surfaces with lower injectivity radius.

Injectivity radius on Stiefel manifold is π.

Lower bound on boundary injectivity radius for specific tubes.

We consider statistical estimation of superhedging prices using historical stock returns in a frictionless market with d traded assets. We introduce a plugin estimator based on empirical measures and show it is consistent but lacks suitable robustness. To address this we propose novel estimators which use a larger set …

Study gives bounds on filling radius for Riemannian manifolds.

The paper proves estimates and theorems for Kähler manifolds.

The study examines how average scalar curvature influences geometric properties of Riemannian manifolds.

Clarifies definition of polarized canonical radius in Kahler Ricci flow.

The paper establishes lower bounds for the relative volume of Poincaré-Einstein manifolds.

Enhances robustness of deep neural networks with randomized smoothing.

The paper bounds bandwidth and focal radius for manifolds with positive isotropic curvature.

Upper bound on Stiefel manifold's injectivity radius found.

Upper estimate of Dirac eigenvalue linked to hyperspherical radius.

Adversarial training improves linear regression solutions, offering robustness against small perturbations.