arXiv research

We construct homology theories with coefficients in L-spectra on the category of ball complexes and we define products in this setting. We also obtain signatures of geometric situations in these homology groups and prove product formulae which we hope will clarify products used in the theory of the total surgery obstru…

The paper constructs free boundary minimal surfaces in product spaces using eigenvalue methods.

Paper finds new ball quotients from curve products.

The paper proves Schauder estimates on cone products and characterizes harmonic functions.

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…

Study shows fundamental gap of horoconvex domains in hyperbolic space has no positive lower bound.

We first study holomorphic isometries from the Poincaré disk into the product of the unit disk and the complex unit $n$-ball for $n\ge 2$. On the other hand, we observe that there exists a holomorphic isometry from the product of the unit disk and the complex unit $n$-ball into any irreducible bounded symmetric domain …

The paper studies isoperimetric inequalities on warped product manifolds.

We prove explicit upper and lower bounds for the $L^1$-moment spectra for the Brownian motion exit time from extrinsic metric balls of submanifolds $P^m$ in ambient Riemannian spaces $N^{n}$. We assume that $P$ and $N$ both have controlled radial curvatures (mean curvature and sectional curvature, respectively) as view…

The purpose of this note is to provide some volume estimates for Einstein warped products similar to a classical result due to Calabi and Yau for complete Riemannian manifolds with nonnegative Ricci curvature. To do so, we make use of the approach of quasi-Einstein manifolds which is directly related to Einstein warped…

Estimates kernel eigenvalues for compositional dot-product kernels.

Study on Vapnik-Chervonenkis dimension of product intervals in R^d.

New method shows stochastic momentum can converge quickly on optimization problems.

We study properties of generic elements of groups of isometries of hyperbolic spaces. Under general combinatorial conditions, we prove that loxodromic elements are generic (i.e. they have full density with respect to counting in balls for the word metric) and translation length grows linearly. We provide applications t…

We give a complete enumeration of all combinatorial 3-manifolds with 10 vertices: There are precisely 247882 triangulated 3-spheres with 10 vertices as well as 518 vertex-minimal triangulations of the sphere product $S^2\times S^1$ and 615 triangulations of the twisted sphere product $S^2_\times_S^1$. All the 3-spheres…

Sharp upper bounds found for Steklov eigenvalues of warped products.

For any closed Riemannian manifold $X$ we prove that large isoperimetric regions in $X\times{\mathbb R}^n$ are of the form $X\times$(Euclidean ball). We prove that if $X$ has non-negative Ricci curvature then the only soap bubbles enclosing a large volume are the products $X\times$(Euclidean sphere). We give an example…

High-dimensional handlebodies are shown to be products of simpler shapes.

Proves Hodge-Riemann relations for mixed valuations and strengthens geometric inequalities.

This is primarily an exposition, combining work of several authors (Curtis, Hsiang, Freedman, Stong, Matveyev, and Bizaca), of the proof that a smooth 5-dimensional h-cobordism between simply connected 4-manifolds is a product off of a contractible piece which itself is diffeomorphic to the 5-ball.

In this paper, we give a new sharp generalization bound of lp-MKL which is a generalized framework of multiple kernel learning (MKL) and imposes lp-mixed-norm regularization instead of l1-mixed-norm regularization. We utilize localization techniques to obtain the sharp learning rate. The bound is characterized by the d…

Defines a distance function on a manifold using symplectic embeddings and recovers the metric.

The paper resolves a problem about metric inequivalence and characterizes proper holomorphic maps.

We give two characterizations of varieties whose universal cover is a bounded symmetric domain without ball factors in terms of the existence of a holomorphic endomorphism \s of the tensor product T\otimes T' of the tangent bundle T with the cotangent bundle T'. To such a curvature type tensor \s one associates the fir…

Asymptotic factorizations for the small-ball probability (SmBP) of a Hilbert valued random element $X$ are rigorously established and discussed. In particular, given the first $d$ principal components (PCs) and as the radius $\varepsilon$ of the ball tends to zero, the SmBP is asymptotically proportional to (a) the joi…

We obtain a family of matrix integrals which decompose to a product of Gamma-functions (they have some relations with S.G.Gindikin 'Beta', but generally speaking essentially differ from it). We obtain Plancherel formula for Berezin representations for all series of classical groups (for large values of parameters of re…

The paper classifies critical metrics on manifolds with positive isotropic curvature.

We give a geometric realization of the polyhedra governed by the structure of associative algebras with co-inner products, or more precisely, governed by directed planar trees. Our explicit realization of these polyhedra, which include the associahedra in a special case, shows in particular that these polyhedra are hom…

Study investigates minimal surfaces in spherical caps, extending previous findings.

New tube manifolds model hyperbolic crystallography with dense ball packings.

In this paper, we give a sharp lower bound for the first eigenvalue of the basic Laplacian acting on basic $1$-forms defined on a compact manifold whose boundary is endowed with a Riemannian flow. The limiting case gives rise to a particular geometry of the flow and the boundary. Namely, the flow is a local product and…

Estimates latent norms and Gram matrices for graphs on Euclidean balls.

Study on Serrin's problem in convex cones with rigidity results and geometric inequalities.

In this paper, we consider the Tensor Robust Principal Component Analysis (TRPCA) problem, which aims to exactly recover the low-rank and sparse components from their sum. Our model is based on the recently proposed tensor-tensor product (or t-product). Induced by the t-product, we first rigorously deduce the tensor sp…

3-balls in 4-sphere become isotopic in 5-ball.

A hyperbolic conjugacy class in the modular group PSL(2,Z) corresponds to a closed geodesic in the modular orbifold. Some of these geodesics virtually bound immersed surfaces, and some do not; the distinction is related to the polyhedral structure in the unit ball of the stable commutator length norm. We prove the foll…

In Minkowski geometry the metric features are based on a compact convex body containing the origin in its interior. This body works as a unit ball with its boundary formed by the unit vectors. Using one-homogeneous extension we have a so-called Minkowski functional to measure the lenght of vectors. The half of its squa…

Minimal surfaces with complex branching structures constructed using various methods.

In this short note, we show the rigidity of a trace estimate for Steklov eigenvalues with respect to functions in our previous work (Trace and inverse trace of Steklov eigenvalues. J. Differential Equations 261 (2016), no. 3, 2026--2040.). Namely, we show that equality of the estimate holds if and only if the manifold …

We examine the vertical component of surface area in the warped product of a Euclidean interval and a fiber manifold with product density. We determine general conditions under which vertical fibers minimize vertical surface area among regions bounding the same volume and use these results to conclude that in many such…

Study on ball widths and minimal submanifolds in space forms.

We study the asymptotic Dirichlet problem for Killing graphs with prescribed mean curvature $H$ in warped product manifolds $M\times_\varrho \mathbb{R}$. In the first part of the paper, we prove the existence of Killing graphs with prescribed boundary on geodesic balls under suitable assumptions on $H$ and the mean cur…

Let $M^n=[0,R)\times \mathbb{S}^{n-1}$ be an $n$-dimensional ($n\geq 2$) smooth Riemannian manifold equipped with the warped product metric $g=dr^2+h^2(r)g_{\mathbb{S}^{n-1}}$ and diffeomorphic to a Euclidean ball. Assume that $M$ has strictly convex boundary. First, for the classical Steklov eigenvalue problem, we obt…

This paper describes a method to construct standard 4-balls from homotopy 4-balls in $\mathbb{C}^2$.

Let $X=S\times E \times B$ be the metric product of a symmetric space $S$ of noncompact type, a Euclidean space $E$ and a product $B$ of Euclidean buildings. Let $Γ$ be a discrete group acting isometrically and cocompactly on $X$. We determine a family of quasi-isometry invariants for such $Γ$, namely the $k$-dimension…

An explicit construction of closed, orientable, smooth, aspherical 4-manifolds with any odd Euler characteristic greater than 12 is presented. The manifolds constructed here are all Haken manifolds in the sense of B. Foozwell and H. Rubinstein and can be systematically reduced to balls by suitably cutting them open alo…

We introduce a novel and efficient sampling algorithm for the Multiplicative Attribute Graph Model (MAGM - Kim and Leskovec (2010)}). Our algorithm is \emph{strictly} more efficient than the algorithm proposed by Yun and Vishwanathan (2012), in the sense that our method extends the \emph{best} time complexity guarantee…

Two minimal hypersurfaces in a ball intersect in any half-ball.