arXiv research

The paper finds optimal threshold strategies for insurance companies with a positive terminal value at creeping ruin.

Researchers find spectral gaps in quantum flag manifolds using twisted operators.

This paper considers magnitude, asymptotics and duration of drawdowns for some Lévy processes. First, we revisit some existing results on the magnitude of drawdowns for spectrally negative Lévy processes using an approximation approach. For any spectrally negative Lévy process whose scale functions are well-behaved at …

Spectral embedding is a procedure which can be used to obtain vector representations of the nodes of a graph. This paper proposes a generalisation of the latent position network model known as the random dot product graph, to allow interpretation of those vector representations as latent position estimates. The general…

New bounds on Bartnik mass for surfaces with non-negative first eigenvalue.

Signed networks allow to model positive and negative relationships. We analyze existing extensions of spectral clustering to signed networks. It turns out that existing approaches do not recover the ground truth clustering in several situations where either the positive or the negative network structures contain no noi…

The paper compares spectral geometry in hyperbolic and spherical manifolds.

In this paper, we give a sharp spectral characterization of conformally compact Einstein manifolds with conformal infinity of positive Yamabe type in dimension $n+1>3$. More precisely, we prove that the largest real scattering pole of a conformally compact Einstein manifold $(X,g)$ is less than $\ndemi -1$ if and only …

We investigate analytic and geometric implications of non-constant Ricci curvature bounds. We prove a Lichnerowicz eigenvalue estimate and finiteness of the fundamental group assuming that $L+2 Ric$ is a positive operator where $L$ is the graph Laplacian. Assuming that the negative part of the Ricci curvature is small …

Improved spectral projection estimates on manifolds of non-positive curvature.

Regularized spectral methods improve clustering in signed graphs, especially for sparse data.

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…

This paper analyzes optimal stopping regions for American options with Poisson exercise opportunities.

Let $(M,g)$ be a complete non-compact Riemannian surface. We consider operators of the form $Δ+ aK + W$, where $Δ$ is the non-negative Laplacian, $K$ the Gaussian curvature, $W$ a locally integrable function, and $a$ a positive real number. Assuming that the positive part of $W$ is integrable, we address the question "…

This paper studies game-type credit default swaps that allow the protection buyer and seller to raise or reduce their respective positions once prior to default. This leads to the study of an optimal stopping game subject to early default termination. Under a structural credit risk model based on spectrally negative Le…

Length spectral rigidity is the question of under what circumstances the geometry of a surface can be determined, up to isotopy, by knowing only the lengths of its closed geodesics. It is known that this can be done for negatively curved Riemannian surfaces, as well as for negatively-curved cone surfaces. Steps are tak…

New method calculates eta invariant without analytic continuation.

Signed graphs encode positive (attractive) and negative (repulsive) relations between nodes. We extend spectral clustering to signed graphs via the one-parameter family of Signed Power Mean Laplacians, defined as the matrix power mean of normalized standard and signless Laplacians of positive and negative edges. We pro…

Researchers approximate spectral targets on manifolds with constant negative curvature.

Geodesic rays and chordal distances link algebraic and geometric properties of positive metrics.

The study finds sparse sets that uniquely determine metrics on negatively curved manifolds.

Let $(M,g)$ be a complete non-compact Riemannian manifold. We consider operators of the form $Δ_g + V$, where $Δ_g$ is the non-negative Laplacian associated with the metric $g$, and $V$ a locally integrable function. Let $ρ: (\hat{M},\hat{g}) \to (M,g)$ be a Riemannian covering, with Laplacian $Δ_{\hat{g}}$ and potenti…

New proof shows certain 3D spaces are essentially like infinite space.

In this paper, we introduce the concept of \emph{Poissonian occupation times} below level $0$ of spectrally negative Lévy processes. In this case, occupation time is accumulated only when the process is observed to be negative at arrival epochs of an independent Poisson process. Our results extend some well known conti…

In this note we give, for a spectrally negative Levy process, a compact formula for the Parisian ruin probability, which is defined by the probability that the process exhibits an excursion below zero, with a length that exceeds a certain fixed period r. The formula involves only the scale function of the spectrally ne…

The study proves Strichartz and spectral projection theorems on specific types of curved surfaces.

Sharp spectral estimates for negatively curved foliations.

The optimal dividend problem by De Finetti (1957) has been recently generalized to the spectrally negative Lévy model where the implementation of optimal strategies draws upon the computation of scale functions and their derivatives. This paper proposes a phase-type fitting approximation of the optimal strategy. We con…

Study magnetic potentials on Anosov manifolds using spectral data.

Paper generalizes Bakry-Émery calculus for curvature and applies to Markov chains.

In this paper, we revisit the optimal periodic dividend problem, in which dividend payments can only be made at the jump times of an independent Poisson process. In the dual (spectrally positive Lévy) model, recent results have shown the optimality of a periodic barrier strategy, which pays dividends at Poissonian divi…

Proves accuracy guarantees for self-supervised learning with correlated positive pairs.

Spectro-Riemannian Graph Neural Networks integrate spectral and curvature signals for better graph representation learning.

Spectral Clustering is a popular technique to split data points into groups, especially for complex datasets. The algorithms in the Spectral Clustering family typically consist of multiple separate stages (such as similarity matrix construction, low-dimensional embedding, and K-Means clustering as post processing), whi…

The paper extends stability theorem for Navier-Stokes equations to negatively curved manifolds.

We introduce a principled and theoretically sound spectral method for $k$-way clustering in signed graphs, where the affinity measure between nodes takes either positive or negative values. Our approach is motivated by social balance theory, where the task of clustering aims to decompose the network into disjoint group…

Global propagator for massless Dirac operator defined and analyzed.

It was asked by J.Birman, Williams, and L.Rudolph whether nontrivial Lorentz knots have always positive signature. Lorentz knots are examples of positive braids (in our convention they have all crossings negative so they are negative links). It was shown by L.Rudolph that positive braids have positive signature (if the…

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…

Optimal estimates for spectral projection norms on compact manifolds.

Sharp Hardy and spectral gap inequalities found on special irreversible Finsler manifolds.

Infinite volume requires no atoms at the bottom of the spectrum for certain groups.

In this paper we consider the optimal dividend problem for an insurance company whose risk process evolves as a spectrally negative Lévy process in the absence of dividend payments. The classical dividend problem for an insurance company consists in finding a dividend payment policy that maximizes the total expected di…

New method learns from either positive or negative feedback alone.

Efficient private matrix analysis algorithms for recent variants.

Affirm Lord Rayleigh's conjecture on curved spaces for clamped plates.

The study examines symmetries in spaces with positive or non-negative curvature.

New DKPP family controls positive and negative dependence in random subsets.