arXiv research

The paper proves new inequalities on the unit ball in higher dimensions.

Stable capillary surfaces in weighted balls are disks.

New surface area measures defined for ball-convex bodies, leading to entropy and inequalities.

Three new efficient algorithms project vectors onto weighted l1 ball.

Paper proves Faber-Krahn inequalities for weighted Laplacian eigenvalues.

The paper proves cohomology vanishing for a specific type of minimal submanifolds in a weighted Euclidean ball.

Study on optimal ReLU networks with weight decay for interpolation.

The paper sets up eigenvalue comparison theorems for specific Laplacians on manifolds.

New illumination bodies defined for ball-convex shapes, proving convexity and establishing surface area measures.

Paper refines Alesker-Bernig-Schuster theorem, proving Hodge-Riemann relations for Euclidean balls.

We study Deraux's non arithmetic orbifold ball quotient surfaces obtained as birational transformations of a quotient $X$ of a particular Abelian surface $A$. Using the fact that $A$ is the Jacobian of the Bolza genus $2$ curve, we identify $X$ as the weighted projective plane $\mathbb{P}(1,3,8)$. We compute the equati…

No radial balanced metrics found on Kepler manifold unit ball with mild boundary conditions.

We obtain upper estimates for the bottom (that is, greatest lower bound) of the essential spectrum of weighted Laplacian operator of a weighted manifold under assumptions of the volume growth of their geodesic balls and spheres. Furthermore, we find examples where the equality occurs in the estimates obtained. As a con…

We study forgetful maps between Deligne-Mostow moduli spaces of weighted points on P^1, and classify the forgetful maps that extend to a map of orbifolds between the stable completions. The cases where this happens include the Livné fibrations and the Mostow/Toledo maps between complex hyperbolic surfaces. They also in…

Wide neural networks become linear, but adding bottlenecks makes them bilinear or multilinear.

Study connects weighted isoperimetric problems to nonlocal elliptic operator extensions.

New isoperimetric inequalities in the plane with radial weights identified.

In constant curvatures spaces, there are a lot of characterizations of geodesic balls as optimal domain for shape optimization problems. Although it is natural to expect similar characterizations in rank one symmetric spaces, very few is known in this setting. In this paper we prove that, in a non-compact rank one symm…

The paper aims at proving global height estimates for Killing graphs defined over a complete manifold with nonempty boundary. To this end, we first point out how the geometric analysis on a Killing graph is naturally related to a weighted manifold structure, where the weight is defined in terms of the length of the Kil…

The study improves Bochner inequality on Finsler manifolds to derive important inequalities.

Let $(M,g)$ be a complete non-compact Riemannian manifold together with a function $e^h$, which weights the Hausdorff measures associated to the Riemannian metric. In this work we assume lower or upper radial bounds on some weighted or unweighted curvatures of $M$ to deduce comparisons for the weighted isoperimetric qu…

Study finds formulas for minimal submanifolds using Möbius transformations.

Let $G=(V,E,w)$ be a finite, connected graph with weighted edges. We are interested in the problem of finding a subset $W \subset V$ of vertices and weights $a_w$ such that $\frac{1}{|V|}\sum_{v \in V}^{}{f(v)} \sim \sum_{w \in W}{a_w f(w)}$ for functions $f:V \rightarrow \mathbb{R}$ that are `smooth' with respect t…

The paper studies Finsler manifolds with a new curvature concept.

We prove a spanning result for vector-valued Poincaré series on a bounded symmetric domain. We associate a sequence of holomorphic automorphic forms to a submanifold of the domain. When the domain is the unit ball in ${\Bbb{C}}^n$, we provide estimates for the norms of these automorphic forms and we find asymptotics of…

Survey of rigidity and gap phenomena in sphere-ball submanifolds.

Let $\mathcal{O}(D)$ be an equivariant line bundle which is big and nef on a complex projective nonsingular toric variety $X$. Given a continuous toric metric $\|\cdot\|$ on $\mathcal{O}(D)$, we define the energy at equilibrium of $(X,φ_{\bar{D}})$ where $φ_{\bar{D}}$ is the weight of the metrized toric divisor $\bar{D…

The paper finds large Steklov eigenvalues on manifolds using homogenization.

We prove some old and new isoperimetric inequalities with the best constant using the ABP method applied to an appropriate linear Neumann problem. More precisely, we obtain a new family of sharp isoperimetric inequalities with weights (also called densities) in open convex cones of $\mathbb{R}^n$. Our result applies to…

We present a method to extract a weighted finite automaton (WFA) from a recurrent neural network (RNN). Our algorithm is based on the WFA learning algorithm by Balle and Mohri, which is in turn an extension of Angluin's classic \lstar algorithm. Our technical novelty is in the use of \emph{regression} methods for the s…

The paper explores conditions for real holomorphic gradient fields on Kähler and conformally Kähler manifolds.

Study geometric and analytical properties of $ρ$-Einstein solitons.

Sharp upper bounds derived for Alexandrov-Fenchel deficit using weighted Minkowski integral formulas.

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…

This paper provides an explicit form for symmetric differentials and their corresponding holomorphic functions.

In the first part of this article we obtain an identity relating the radial spectrum of rotationally invariant geodesic balls and an isoperimetric quotient $\sum 1/λ_{i}^{\rm rad}=\int V(s)/S(s)ds$. We also obtain upper and lower estimates for the series $\sum λ_{i}^{-2}(Ω)$ where $Ω$ is an extrinsic ball of a proper m…

The paper studies the curvature behavior near the boundary of certain domains.

Orbifold uniformization of complex algebraic variety via polystable parabolic Higgs bundle

Develops a local Fokker--Planck geometric framework for more accurate score estimation.

In an earlier paper we explained how to convert the problem of symplectically embedding one 4-dimensional ellipsoid into another into the problem of embedding a certain set of disjoint balls into \CP^2 by using a new way to desingularize orbifold blow ups Z of the weighted projective space \CP^2_{1,m,n}. We now use a r…

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

Every connected, weighted graph with non-negative curvature has exactly two ends.

It was conjectured by Escobar [J. Funct. Anal. 165 (1999), 101-116] that for an $n$-dimensional ($n\geq 3$) smooth compact Riemannian manifold with boundary, which has nonnegative Ricci curvature and boundary principal curvatures bounded below by $c>0$, the first nonzero Steklov eigenvalue is greater than or equal to $…

Improves few-shot learning for hierarchical data using hyperbolic space.

K-means clustering improved for robustness to outliers and distribution shifts.

Study on ball widths and minimal submanifolds in space forms.

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

Integrating adaptive learning rate and momentum techniques into SGD leads to a large class of efficiently accelerated adaptive stochastic algorithms, such as AdaGrad, RMSProp, Adam, AccAdaGrad, \textit{etc}. In spite of their effectiveness in practice, there is still a large gap in their theories of convergences, espec…