arXiv research

Explicit BCH series radii found for special Banach-Malcev shift algebras.

Study nearest-neighbor radii under dependent sampling, finding they remain informative.

Minimal spectral radii found for specific matrix types.

Grosjean proved that the $(1/p)$-th power of the first eigenvalue of the $p$-Laplacian on a closed Riemannian manifold converges to the twice of the inverse of the diameter of the space, as $p \to \infty$. Before this, a corresponding result for the Dirichlet first eigenvalues was also obtained by Juutinen, Lindqvist a…

In this paper, we obtain two-sided bounds for the volumes of the Aloff-Wallach spaces $W(p,q),$ compute maximal and minimal sectional curvature for the spaces $W(n,n+1),$ and use this information to estimate the injectivity radii: We derive an upper bound for the injectivity radii of $W(p,q)$ and a lower bound for the …

The paper studies geometric structures of curvature radii on Riemannian manifolds.

Paper improves neural network robustness certification with tighter radii estimates.

The paper analyzes geometric densities and compression radii for knot types.

We study one extremal problem on the product of power of generalized inner radii of non-overlapping domains in $\mathbb{C}^{n}$.

We construct hyperbolic integer homology 3-spheres where the injectivity radius is arbitrarily large for nearly all points of the manifold. As a consequence, there exists a sequence of closed hyperbolic 3-manifolds which Benjamini-Schramm converge to H^3 whose normalized Ray-Singer analytic torsions do not converge to …

The paper studies a flow of convex hypersurfaces using anisotropic curvature functions.

Sharp pseudospectral bounds prevent transient amplification in coupled gradient descent.

New proof shows not all Salem numbers are growth rates of Coxeter groups.

The study finds parametrizations for surfaces of revolution with a linear curvature ratio.

Researchers compute the full spectrum of Laplace operator on distance spheres in symmetric spaces.

Paper proves rigidity of Doyle spirals in hexagonal lattice circle packings.

We study a volume/area preserving curvature flow of hypersurfaces that are convex by horospheres in the hyperbolic space, with velocity given by a generic positive, increasing function of the mean curvature, not necessarly homogeneous. For this class of speeds we prove the exponential convergence to a geodesic sphere. …

The paper studies non-integer curvature flows and proves convergence to spheres under specific conditions.

A Gauss equation is proved for subspaces of Alexandrov spaces of curvature bounded above by K. That is, a subspace of extrinsic curvature less than or equal to A, defined by a cubic inequality on the difference of arc and chord, has intrinsic curvature less than or equal to K+A^2. Sharp bounds on injectivity radii of s…

The paper studies a flow of convex hypersurfaces expanding by their support and curvature functions.

The paper bounds radii and curvatures in Riemannian manifolds.

We consider classical curvature flows: 1-parameter families of convex embeddings of the 2-sphere into Euclidean 3-space which evolve by an arbitrary (non-homogeneous) function of the radii of curvature. The associated flow of the radii of curvature is a second order system of partial differential equations which we sho…

Consider a sequence of pointed n-dimensional complete Riemannian manifolds {(M_i,g_i(t), O_i)} such that t in [0,T] are solutions to the Ricci flow and g_i(t) have uniformly bounded curvatures and derivatives of curvatures. Richard Hamilton showed that if the initial injectivity radii are uniformly bounded below then t…

Improves safety region certification for smoothed classifiers without changing smoothing scheme.

The Korányi ellipsoidal ring $\mathcal{E}$ of radii $B$ and $A$, $0<B<A$, is defined as the image of the Korányi spherical ring of the same radii and centred at the origin via a linear contact map $L$ in the Heisenberg group. If $K\ge 1$ is the maximal distortion of $L$ then we prove that the modulus of $\mathcal{E}$ i…

Nonuniform tubular neighborhoods of curves in Euclidean n-space are studied by using weighted distance functions and generalizing the normal exponential map. Different notions of injectivity radii are introduced to investigate singular but injective exponential maps. A generalization of the thickness formula is obtaine…

Study examines large deviations in random walks on hyperbolic spaces.

In this paper, we discuss how a Gromov-Hausdorff-like distance function over the space of all isometric classes of compact $C^k$-Riemannian manifolds should be defined in the aspect of the Riemannan submanifold theory, where $k\geq 1$. The most important fact in this discussion is as follows. The Hausdorff distance fun…

Given a non-compact, simply connected homogeneous three-manifold $X$ and a sequence $\{Ω_n\}_n$ of isoperimetric domains in $X$ with volumes tending to infinity, we prove that as $n\to \infty$: 1. The radii of the $Ω_n$ tend to infinity. 2. The ratios $\{Area} (\partial Ω_n)/\{Vol}(Ω_n)$ converge to the Cheeger consta…

We study the mean curvature flow of hypersurfaces in $\R^{n+1}$, with initial surfaces sufficiently close to the standard $n$-dimensional sphere. The closeness is in the Sobolev norm with the index greater than $\frac{n}{2}+1$ and therefore it does not impose restrictions of the mean curvature of the initial surface. W…

We show that the 2-torus in ${\mathbb R}^3$ is a critical point of a sequence of functionals ${\cal F}_{n}$ ($n=1,2,3, \cdots$) defined over compact 2-surfaces in ${\mathbb R}^3$. When the Lagrange function ${\cal E}$ is a polynomial of degree $n$ of the mean curvature $H$ of the surface, the radii ($a,r$) of the 2-tor…

We prove that the normal curvatures of hyperspheres, the Rund curvature, and the Finsler curvature of circles in Hilbert geometry tend to 1 as the radii tend to infinity

We show that every effective action of a compact Lie group $K$ on a unit sphere $S^n$ admits an explicit orbit whose principal curvatures are bounded from above by $4\sqrt{14}$.

In this paper, it is shown that any surface automorphism of positive mapping-class entropy possesses a virtual homological eigenvalue which lies outside the unit circle of the complex plane.

We study families of submanifolds in symmetric spaces of compact type arising as exponential images of s-orbits of variable radii. Special attention is given to the cases where the s-orbits are symmetric.

Validates neural networks inputs to protect against adversarial examples.

We show that there is no bi-Lipschitz homeomorphism of $\mathbb{R}^2$ that maps a spiral with a sub-exponential decay of winding radii to an unwinded arc. This result is sharp as shows an example of a logarithmic spiral.

We compute the series expansions for the normal curvatures of hyperspheres, the Finsler and Rund curvatures of circles in Funk geometry as the radii tend to infinity. These three curvatures are different at infinity in Funk geometry.

Understanding the exceptional Lie groups as the symmetry groups of simpler objects is a long-standing program in mathematics. Here, we explore one famous realization of the smallest exceptional Lie group, G2. Its Lie algebra acts locally as the symmetries of a ball rolling on a larger ball, but only when the ratio of r…

We will extend partially our previous results about the limit of the Brown-York mass of a family of convex revolution surfaces in the Schwarzschild manifold such that these surfaces may have unbounded ratios of their radii.

We obtain sharp lower bounds on the radii of inscribed balls for strictly convex isoperimetric domains lying in a 2-dimensional Alexandrov metric space of curvature bounded below. We also characterize the case when such bounds are attained.

Unified flow solves $L^p$ Christoffel-Minkowski problem for $p>1$.

We consider a shrinking flow of smooth, closed, uniformly convex hypersurfaces in (n+1)-dimensional Euclidean space with speed fu^{alpha}{sigma}_n^{beta}, where u is the support function of the hypersurface, alpha, beta are two constants, and beta>0, sigma_n is the n-th symmetric polynomial of the principle curvature r…

Pack hyperbolic surfaces with circles or horocycles, noting symmetries.

This note proves that any locally extremal non-self-conjugate geodesic loop in a Riemannian manifold is a closed geodesic. As a consequence, any complete and non-contractible Riemannian manifold with diverging injectivity radii along diverging sequences and without points conjugate to themselves, possesses a minimizing…

We show that the Brill-Lindquist initial data provides a counterexample to a Riemannian Penrose inequality with charge conjectured by G. Gibbons. The observation illustrates a sub-additive characteristic of the area radii for the individual connected components of an outermost horizon as a lower bound of the ADM mass.

The article considers the problem of existence and uniqueness of centrally symmetrical convex body for which the projection curvature radius function coincides with a given flag function. A necessary and sufficient condition is found that ensures a positive answer. An algorithm for construction the body in question is …

Paper extends SMM to weakly convex and multi-convex surrogates for non-convex optimization.