arXiv research

Call a smooth knot (or smooth link) in the unit sphere in $\mathbb{C}^2$ analytic (respectively, smoothly analytic) if it bounds a complex curve (respectively, a smooth complex curve) in the complex ball. Let $K$ be a smoothly analytic knot. For a small tubular neighbourhood of $K$ we give a sharp lower bound for the 4…

New method to parametrize infinite Riemann surfaces with bounded triangulations.

Analytic networks with bounded coefficients can't outperform polynomial approximations.

Paper proves all Lagrangians unobstructed if one is, using non-archimedean analytic structure.

Let C be a real-analytic Jordan curve in $R^3$. Then C cannot bound infinitely many disk-type minimal surfaces which provide relative minima of area.

A classical result due to Blaschke states that for every analytic self-map $f$ of the open unit disk of the complex plane there exists a Blaschke product $B$ such that the zero sets of $f$ and $B$ agree. In this paper we show that there is an analogue statement for critical sets, i.e. for every analytic self-map $f$ of…

Wide neural networks can learn complex functions like gravitational force law.

Given a sequence of complete(compact or noncompact) Kähler manifolds $M^n_i$ with bisectional curvature lower bound and noncollapsed volume, we prove that the pointed Gromov-Hausdorff limit is homeomorphic to a normal complex analytic space. The complex analytic structure is the natural "limit" of complex structure of …

Study extends holomorphic forms on noncompact Kahler manifolds.

The paper defines functions that induce bounded composition operators on RKHSs with analytic positive definite functions.

We give upper bounds for the Bergman kernels associated to tensor powers of a smooth positive line bundle in terms of the rate of growth of the Taylor coefficients of the Kähler potential. As applications, we obtain improved off-diagonal rate of decay for the classes of analytic, quasi-analytic, and more generally Gevr…

The paper calculates bounds on the local Lipschitz constants of neural network layers.

The study bounds quantum eigenfunctions on complex manifolds.

For analytic functions in the unit disk, general bounds on the Schwarzian derivative in terms of Nehari functions are shown to imply uniform local univalence and in some cases finite and bounded valence. Similar results are obtained for the Weierstrass--Enneper lifts of planar harmonic mappings to their associated mini…

We construct a continuous time model for price-mediated contagion precipitated by a common exogenous stress to the banking book of all firms in the financial system. In this setting, firms are constrained so as to satisfy a risk-weight based capital ratio requirement. We use this model to find analytical bounds on the …

We prove the analyticity in time for solutions of two parabolic equations in the whole space, without any decaying or vanishing conditions. One of them involves solutions to the heat equation of exponential growth of order $2$ on $\M$. Here $\M$ is $\R^d$ or a complete noncompact manifold with Ricci curvature bounded f…

Study geometric bounds on generalized Ricci flow.

The combination of the re-parameterization trick with the use of variational auto-encoders has caused a sensation in Bayesian deep learning, allowing the training of realistic generative models of images and has considerably increased our ability to use scalable latent variable models. The re-parameterization trick is …

New Hessian estimators for Riemannian manifolds with reduced bias.

The paper bounds payoffs and option prices in discrete models.

New bounds for quantile aggregation unify and clarify existing methods.

Let p and l be two distinct prime numbers and let G be a group. We study the asymptotic behaviour of the mod-l Betti numbers in p-adic analytic towers of finite index subgroups. If X is a finite l-group of automorphisms of G, our main theorem allows to lift lower bounds for the mod-l cohomology growth in the fixed poin…

Given a choice of metric on the Riemann surface, the regularized determinant of Laplacian (analytic torsion) is defined via the complex power of elliptic operators: $\det(Δ)=\exp(-ζ'(0))$ In this paper we gave an asymptotic effective estimate of analytic torsion under Arakelov metric. In particular, after taking th…

New proof of harmonic map uniqueness with analytic targets.

Polynomials' roots count tied to surface umbilics.

Torsion invariants for manifolds which are not simply connected were introduced by K. Reidemeister and generalized to higher dimensions by W. Franz. The Reidemeister torsion, was the first invariant of manifolds which was not a homotopy invariant. The analytic counterpart of the combinatorial Reidemeister torsion was i…

Extends machine learning models for analytic boundary conditions in differential equations.

We prove that smooth critical points of the Möbius energy parametrized by arc-length are analytic. Together with the main result in \cite{BRS16} this implies that critical points of the Möbius energy with merely bounded energy are not only $C^\infty$ but also analytic. Our proof is based on Cauchy's method of majorants…

It is well known that generic solutions of the heat equation are not analytic in time in general. Here it is proven that ancient solutions with exponential growth are analytic in time in ${\M} \times (-\infty, 0]$. Here $\M=\R^n$ or is a manifold with Ricci curvature bounded from below. Consequently a necessary and suf…

The paper proves ACC for local volumes under boundedness conditions.

Analyzes packing of circles in bounded and unbounded planes using mathematical formulas.

The paper calculates upper bounds on ReLU network Lipschitz constants.

The paper proves a conjecture about the Bergman metric of real analytic domains.

In this paper we prove the strong Sard conjecture for sub-Riemannian structures on 3-dimensional analytic manifolds. More precisely, given a totally nonholonomic analytic distribution of rank 2 on a 3-dimensional analytic manifold, we investigate the size of the set of points that can be reached by singular horizontal …

Study on Einstein solitons with bounds and asymptotic behavior.

The Kähler-Ricci flow near conical singularities is described with a $C/t$ curvature bound.

Let $Ω\subset R^2$ be a bounded piecewise smooth domain and $φ_λ$ be a Neumann (or Dirichlet) eigenfunction with eigenvalue $λ^2$ and nodal set ${ N}_{φ_λ} = {x \in Ω; φ_λ(x) = 0}.$ Let $H \subset Ω$ be an interior $C^ω$ curve. Consider the intersection number $n(λ,H):= \# (H \cap N_{φ_λ} ).$ We first prove that fo…

Geodesics in 3D space with 1-2 analytic obstacles, proving geodesic independence.

Deep neural networks approximate analytic functions in high dimensions with exponential rates.

Geodesically complete spaces with curvature bounded above have maps with finite energy that are Lipschitz.

New formulas estimate life insurance benefits with less computation.

Paper improves MMD estimation for analytical mean embeddings.

Lower bound proves ridgeless regression performs poorly near interpolation threshold.

Analyticity of heat equation extended to Bakry-Émery Ricci curvature manifolds.

Semi-implicit variational inference (SIVI) is introduced to expand the commonly used analytic variational distribution family, by mixing the variational parameter with a flexible distribution. This mixing distribution can assume any density function, explicit or not, as long as independent random samples can be generat…

New portfolios outperform traditional methods by using factor weights.

New bounds for neural networks ensure robustness and accuracy.

A coupling method and an analytic one allow us to prove new lower bounds for the spectral gap of reversible diffusions on compact manifolds. Those bounds are based on the a notion of curvature of the diffusion, like the coarse Ricci curvature or the Bakry--Emery curvature-dimension inequalities. We show that when this …