arXiv research

Study finds a limiting distribution for free path lengths on flat surfaces with circular obstacles.

Each free homotopy class of directed closed curves on a surface with boundary can be described by a cyclic reduced word in the generators of the fundamental group and their inverses. The word length is the number of letters of the cyclic word. If the surface has a hyperbolic metric with geodesic boundary, the geometric…

Study on length distribution of random multicurves on large genus surfaces converging to Poisson-Dirichlet distribution.

Study h-principles for non-integrable distributions on manifolds.

We present the first treatment of the arc length of the Gaussian Process (GP) with more than a single output dimension. GPs are commonly used for tasks such as trajectory modelling, where path length is a crucial quantity of interest. Previously, only paths in one dimension have been considered, with no theoretical con…

Analyzes geodesic lengths in sparse networks, deriving a distribution.

Improved phylogenetic tree reconstruction using flexible branch length distributions.

Study finds saddle connections on random surfaces follow Poisson distribution.

This work studies the chord length distribution, in the case where both ends lie on a $N$-dimensional hypersphere ($N \geq 2$). Actually, after connecting this distribution to the recently estimated surface of a hyperspherical cap \cite{SLi11}, closed-form expressions of both the probability density function and the cu…

This research proves guarantees on sequence models' generalization to longer and novel sequences.

We investigate the distribution of lengths obtained by intersecting a random geodesic with a geodesic lamination. We give an explicit formula for the distribution for the case of a maximal lamination and show that the distribution is independent of the surface and lamination. We also show how the moments of the distrib…

Study on lengths of random multicurves on hyperbolic surfaces.

Study finds numerical moduli in special 2-flags of length 5.

The study of isospectral surfaces in Euclidean and hyperbolic geometries.

Study on length spectrum of random hyperbolic 3-manifolds.

Study geodesics on flat tori, focusing on orthogonal lengths and their distribution.

Motivated by the study of billiards in polygons, we prove fine results for the distribution of gaps of directions of saddle connections on translation surfaces. As an application we prove that for almost every holomorphic differential $ω$ on a Riemann surface of genus $g \geq 2$ the smallest gap between saddle connecti…

Transformer pretraining yields strong EB performance without explicit adaptation.

In the recent years, a number of issues concerning distributions generating 1- flags (called also Goursat flags) has been analyzed. Presently similar questions are discussed as regards distributions generating multi-flags. (In fact, only so-called special multi-flags, to avoid functional moduli.) In particular and fore…

We analyze the joint probability distribution on the lengths of the vectors of hidden variables in different layers of a fully connected deep network, when the weights and biases are chosen randomly according to Gaussian distributions, and the input is in $\{ -1, 1\}^N$. We show that, if the activation function $φ$ sat…

Improved phylogenetic inference using VBPI-Mixtures for tree topology and branch length.

Study on abnormal curves in sub-Riemannian manifolds, proving length-minimizing properties.

Model learns brevity by exposing to easy problems, improving efficiency without explicit length penalties.

Improved phylogenetic inference with normalizing flows.

New statistical models for predicting ranked preferences from partial orders.

Randomized positional encodings boost transformer performance on longer sequences.

Short geodesics are important in the study of the geometry and the spectra of Riemann surfaces. Bers' theorem gives a global bound on the length of the first $3g-3$ geodesics. We use the construction of Brooks and Makover of random Riemann surfaces to investigate the distribution of short ($< \log (g)$) geodesics on a …

Study on stable commutator length in RAAGs and Coxeter groups, proving spectral gaps and hardness results.

Let $S=Γ\backslash \mathbb{H}$ be a hyperbolic surface of finite topological type, such that the Fuchsian group $Γ\le \operatorname{PSL}_2(\mathbb{R})$ is non-elementary, and consider any generating set $\mathfrak S$ of $Γ$. When sampling by an $n$-step random walk in $π_1(S) \cong Γ$ with each step given by an element…

This paper attempts to find out numerically the distribution of the queue-length ratio in the context of a model of preferential attachment. Here we consider two restaurants only and a large number of customers (agents) who come to these restaurants. Each day the same number of agents sequentially arrives and decides w…

New bounds show transformers need longer training for length generalization.

A random Heegaard splitting is a 3-manifold obtained by using a random walk of length n on the mapping class group as the gluing map between two handlebodies. We show that the joint distribution of random walks of length n and their inverses is asymptotically independent, and converges to the product of the harmonic an…

ATLAS adapts HMC step size and trajectory length for complex geometries.

Let $S$ be a closed oriented surface of genus at least $2$, and denote by $\mathcal{T}(S)$ its Teichm{ü}ller space. For any isotopy class of closed curves $γ$, we compute the first three derivatives of the length function $\ell\_γ:\mathcal{T}(S)\rightarrow\mathbf{R}\_+$ in the shearing coordinates associated to a maxim…

Study explains Zipf's law using geometric mechanisms from a finite alphabet.

Study geodesics on graphs with random lengths, proving bi-infinite paths exist.

Geometric characterization of sub-Riemannian geodesics on frame bundles.

Study on multiplicities in length spectrum of Salem numbers.

In this paper, we numerically investigate the length spectra and the low-lying eigenvalue spectra of the Laplace-Beltrami operator for a large number of small compact(closed) hyperbolic (CH) 3-manifolds. The first non-zero eigenvalues have been successfully computed using the periodic orbit sum method, which are compar…

Transformer struggles with arithmetic length but improves with explicit structure encoding.

Critical graphs of quadratic differentials equidistribute in moduli space.

The flat trace of geodesic Koopman operators varies with negatively curved surfaces.

A general approach to proving that the length spectrum of a compact Riemannian manifold is an invariant of the Laplace spectrum comes from considering the wave trace, a spectrally determined tempered distribution. The Poisson relation states that the singularities of the wave trace can only occur at lengths of closed g…

Study shows Transformers can generalize to varying task lengths.

Based on empirical financial time-series, we show that the "silence-breaking" probability follows a super-universal power law: the probability of observing a large movement is inversely proportional to the length of the on-going low-variability period. Such a scaling law has been previously predicted theoretically [R. …

A simple text model shows word lengths follow Zipf's law.

This work challenges the assumption that shorter conformal prediction intervals are always better.

We study the dynamics of the linear and non-linear serial dependencies in financial time series in a rolling window framework. In particular, we focus on the detection of episodes of statistically significant two- and three-point correlations in the returns of several leading currency exchange rates that could offer so…