arXiv research

The paper establishes conditions for Bayesian consistency in supremum metric.

We study the supremum of the total mean curvature on the boundary of compact, mean-convex 3-manifolds with nonnegative scalar curvature, and a prescribed boundary metric. We establish an additivity property for this supremum and exhibit rigidity for maximizers assuming the supremum is attained. When the boundary consis…

We obtain supremum of the k-th normalized Steklov eigenvalues of all rotational symmetric conformal metrics on the cylinder with k>1. The case k=1 for all conformal metrics has been completely solved by Fraser and Schoen. We give geometric description in terms of minimal surfaces for metrics attaining the supremum. We …

On a Fano manifold M we study the supremum of the possible t such that there is a Kähler metric in c_1(M) with Ricci curvature bounded below by t. This is shown to be the same as the maximum existence time of Aubin's continuity path for finding Kähler-Einstein metrics. We show that on P^2 blown up in one point this sup…

Let $L_g$ be the subcritical GJMS operator on an even-dimensional compact manifold $(X, g)$ and consider the zeta-regularized trace $\mathrm{Tr}_ζ(L_g^{-1})$ of its inverse. We show that if $\ker L_g = 0$, then the supremum of this quantity, taken over all metrics $g$ of fixed volume in the conformal class, is always g…

Let $M$ be a compact manifold of dimension $n$. In this paper, we introduce the {\em Mass Function} $a \geq 0 \mapsto \xp{M}{a}$ (resp. $a \geq 0 \mapsto \xm{M}{a}$) which is defined as the supremum (resp. infimum) of the masses of all metrics on $M$ whose Yamabe constant is larger than $a$ and which are flat on a ball…

Paper relaxes triangle inequality for KL divergence between Gaussian distributions.

We study the set of volumes of constant scalar curvature one metrics on an atoroidal three-manifold.The infinum of this set is believed to be attained at a hyperbolic metric. We prove that the supremum of this set is always infinity. The technique is: minimal surfaces, Thurston norm in homology and new conformal invari…

Develops a new essential supremum concept for financial models.

We study a particular class of open manifolds. In the category of Riemannian manifolds these are complete manifolds with cylindrical ends. We give a natural setting for the conformal geometry on such manifolds including an appropriate notion of the cylindrical Yamabe constant/invariant. This leads to a corresponding ve…

In this paper we consider a punctured Riemann surface endowed with a Hermitian metric which equals the Poincaré metric near the punctures and a holomorphic line bundle which polarizes the metric. We show that the Bergman kernel can be localized around the singularities and its local model is the Bergman kernel of the p…

The paper bounds the expectation of empirical processes indexed by Hölder classes.

Let $M$ be a compact connected manifold of dimension $n$ endowed with a conformal class $C$ of Riemannian metrics of volume one. For any integer $k\geq0$, we consider the conformal invariant $λ_k ^c (C)$ defined as the supremum of the $k$-th eigenvalue $λ_k (g)$ of the Laplace-Beltrami operator $Δ_g$, where $g$ runs ov…

We show how to compute lower bounds for the supremum Bayes error if the class-conditional distributions must satisfy moment constraints, where the supremum is with respect to the unknown class-conditional distributions. Our approach makes use of Curto and Fialkow's solutions for the truncated moment problem. The lower …

Upper bound on expected supremum of Bernoulli process.

We consider the equivariant Yamabe problem, i.e. the Yamabe problem on the space of G-invariant metrics for a compact Lie group G. The G-Yamabe invariant is analogously defined as the supremum of the constant scalar curvatures of unit volume G-invariant metrics minimizing the total scalar curvature functional in their …

We consider the optimal prediction problem of stopping a spectrally negative Lévy process as close as possible to a given distance $b \geq 0$ from its ultimate supremum, under a squared error penalty function. Under some mild conditions, the solution is fully and explicitly characterised in terms of scale functions. We…

Study optimal control of diffusion processes with infimum or supremum costs.

For every smooth del Pezzo surface $S$, smooth curve $C\in|-K_{S}|$ and $β\in(0,1]$, we compute the $α$-invariant of Tian $α(S,(1-β)C)$ and prove the existence of Kähler--Einstein metrics on $S$ with edge singularities along $C$ of angle $2πβ$ for $β$ in certain interval. In particular we give lower bounds for the inva…

The systole of a compact non simply connected Riemannian manifold is the smallest length of a non-contractible closed curve ; the systolic ratio is the quotient $(\mathrm{systole})^n/\mathrm{volume}$. Its supremum on the set of all the riemannian metrics, is known to be finite for a large class of manifolds, including …

The Yamabe invariant Y(M) of a smooth compact manifold is roughly the supremum of the scalar curvatures of unit-volume constant-scalar-curvature Riemannian metrics g on M. (To be precise, one only considers those constant-scalar-curvature metrics which are Yamabe minimizers, but this technicality does not, e.g. affect …

A compact manifold is called Bieberbach if it carries a flat Riemannian metric. Bieberbach manifolds are aspherical, therefore the supremum of their systolic ratio, over the set of Riemannian metrics, is finite by a fundamental result of M. Gromov. We study the optimal systolic ratio of compact of $3$-dimensional orien…

We prove that, given $|H|<1$, a generic simple closed curve embedded in the asymptotic boundary of $\mathbb{H}^3$ (with respect to the supremum metric) bounds more than one complete surface embedded in $\mathbb{H}^3$ which has constant mean curvature $H$. We remark that this is not true for the space of simple closed $…

Let (M,g) be a compact Riemannian manifold of dimension >2. We show that there is a metric h conformal to g and of volume 1 such that the first positive eigenvalue the conformal Laplacian with repect to h is arbitrarily large. A similar statement is proven for the first positive eigenvalue of the Dirac operator on a sp…

An elementary proof shows submodular functions can be represented as measure suprema.

The first nontrivial eigenvalue of the Laplacian can be considered as a functional on the space of all Riemannian metrics of unit volume on a fixed surface. In this paper we prove that for the surface of genus 2 the supremum of this functional is equal to $16π$. This provides a positive answer to the conjecture by Jako…

A compact Riemannian manifold may be immersed into Euclidean space by using high frequency Laplace eigenfunctions. We study the geometry of the manifold viewed as a metric space endowed with the distance function from the ambient Euclidean space. As an application we give a new proof of a result of Burq-Lebeau and othe…

In this paper we study the supremum of Perelman's λ-functional {λ}_M(g) on Riemannian 4-manifold M by using the Seiberg-Witten equations. We prove among others that, for a compact Kähler-Einstein complex surface (M, J, g_{0}) with negative scalar curvature, (i) If g_{1} is a Riemannian metric on M with λ_{M}(g_{1})= λ_…

New algorithm tackles subgroup fairness in AI with multiple sensitive attributes.

The {\em drawdown} process $Y$ of a completely asymmetric Lévy process $X$ is equal to $X$ reflected at its running supremum $\bar{X}$: $Y = \bar{X} - X$. In this paper we explicitly express in terms of the scale function and the Lévy measure of $X$ the law of the sextuple of the first-passage time of $Y$ over the leve…

Study short-term behavior of up-and-in barrier options using Malliavin calculus.

Researchers prove existence of metrics maximizing Laplace eigenvalue on all closed surfaces.

Let $M$ be a compact manifold with a metric $g$ and with a fixed spin structure $χ$. Let $λ\_1^+(g)$ be the first non-negative eigenvalue of the Dirac operator on $(M,g,χ)$. We set $τ(M,χ):= \sup \inf λ\_1^+(g)$ where the infimum runs over all metrics $g$ of volume 1 in a conformal class $[g\_0]$ on $M$ and where the…

Derives integral representations for a Lévy process and its extremum, hitting time, with fast evaluation.

Paper proves uniform continuity bounds for complex Monge-Ampère solutions.

The Yamabe invariant Y(M) of a smooth compact manifold is roughly the supremum of the scalar curvatures of unit-volume constant-scalar curvature Riemannian metrics g on M. (To be absolutely precise, one only considers constant-scalar-curvature metrics which are Yamabe minimizers, but this does not affect the sign of th…

The Yamabe Invariant of a smooth compact manifold is by definition the supremum of the scalar curvatures of unit-volume Yamabe metrics on the manifold. For an explicit infinite class of 4-manifolds, we show that this invariant is positive but strictly less than that of the 4-sphere. This is done by using spin^c Dirac o…

This paper considers the valuation of exotic path-dependent options in Lévy models, in particular options on the supremum and the infimum of the asset price process. Using the Wiener--Hopf factorization, we derive expressions for the analytically extended characteristic function of the supremum and the infimum of a Lév…

Estimates Bartnik mass for metrics with nonnegative Gauss curvature.

Logistic regression for brain imaging without p-values.

In this paper, we extend the method in [TZhu5] to study the energy level $L(\cdot)$ of Perelman's entropy $λ(\cdot)$ for Kähler-Ricci flow on a Fano manifold. Consequently, we first compute the supremum of $λ(\cdot)$ in Kähler class $2πc_1(M)$ under an assumption that the modified Mabuchi's K-energy $μ(\cdot)$ defined …

The paper constructs optimal confidence bands for kernel gradient flow estimators.

Study on stable translation lengths of surface homeomorphisms and their approximations.

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n$ with smooth boundary $\partial M$. Suppose that $(M,g)$ admits a scalar-flat conformal metric. We prove that the supremum of the isoperimetric quotient over the scalar-flat conformal class is strictly larger than the best constant of the isoperimetric…

In this note we find a formula for the supremum distribution of spectrally positive or negative Lévy processes with a broken linear drift. This gives formulas for ruin probabilities in the case when two insurance companies (or two branches of the same company) divide between them both claims and premia in some specifie…

We prove that if a geodesic metric measure space satisfies a comparison condition for isoperimetric profile and if the observable variance is maximal, then the space is foliated by minimal geodesics, where the observable variance is defined to be the supremum of the variance of 1-Lipschitz functions on the space. Our r…

We prove that on Fano manifolds, the Kähler-Ricci flow produces a "most destabilising" degeneration, with respect to a new stability notion related to the H-functional. This answers questions of Chen-Sun-Wang and He. We give two applications of this result. Firstly, we give a purely algebro-geometric formula for the su…

Vertex distortion measures how far lattice knots deviate from straight lines.