相关论文: Quantifying Homology Classes II: Localization and …
In this note we study the homology of configuration spaces of closed manifolds. We sharpen the eventual periodicity results of Nagpal and Cantero-Palmer, prove integral homological stability for configuration spaces of odd-dimensional…
This paper proposes a geometric interpretation of the angles and scales which the orientation- and scale-covariant feature detectors, e.g. SIFT, provide. Two new general constraints are derived on the scales and rotations which can be used…
Regularization is a well-established technique in machine learning (ML) to achieve an optimal bias-variance trade-off which in turn reduces model complexity and enhances explainability. To this end, some hyper-parameters must be tuned,…
Dimensionless ratios characterizing many-body systems are a powerful tool to reveal the main universal quantities involved. The recently-introduced localisation parameter allow to study the occurrence of crystal, clusterisation, and quantum…
We identify two recursively defined polynomial conditions for FI-modules in the literature. We characterize these conditions using homological invariants of FI-modules (namely the local degree and regularity, together with the stable…
Numerical homogenization aims to efficiently and accurately approximate the solution space of an elliptic partial differential operator with arbitrarily rough coefficients in a $d$-dimensional domain. The application of the inverse operator…
We prove homological stability for both general linear groups of modules over a ring with finite stable rank and unitary groups of quadratic modules over a ring with finite unitary stable rank. In particular, we do not assume the modules…
In this paper we prove quantitative regularity results for stationary and minimizing extrinsic biharmonic maps. As an application, we determine sharp, dimension independent $L^p$ bounds for $\nabla^k f$ that do not require a small energy…
Many of the large structures of the cell, such as the cytoskeleton, are assembled and maintained far from equilibrium. We study the stabilities of various structures for a simple model of such a far-from-equilibrium organized assembly in…
We extend the notion of the distance to a measure from Euclidean space to probability measures on general metric spaces as a way to do topological data analysis in a way that is robust to noise and outliers. We then give an efficient way to…
This paper develops a process-based account of scientific explanation that reconceives grounding in terms of stabilisation. Grounding theories capture hierarchical dependence but lack criteria for when explanations remain adequate under…
We establish an optimal regularity result for parametrized two-dimensional stationary varifolds. Namely, we show that the parametrization map is a smooth minimal branched immersion and that the multiplicity function is constant. We provide…
We investigate local minimizers of Ginzburg--Landau-type functionals in dimension $n\geq 3$ that satisfy logarithmic energy bounds, assuming the potential has a vacuum manifold with a finite fundamental group. We show that the normalized…
Small corrections to the uncertainty relations, with effects in the ultraviolet and/or infrared, have been discussed in the context of string theory and quantum gravity. Such corrections lead to small but finite minimal uncertainties in…
We study the stability in finite times of the trajectories of interacting particles. Our aim is to show that in average and uniformly in the number of particles, two trajectories whose initial positions in phase space are close, remain…
We study the problem of signal source localization using received signal strength measurements. We begin by presenting verifiable geometric conditions for sensor deployment that ensure the model's asymptotic localizability. Then we…
The homology of configuration spaces of point-particles in manifolds has been studied intensively since the 1970s; in particular it is known to be stable if the underlying manifold is connected and open. Closely related to configuration…
We prove a motivic stabilization result for the cohomology of the local systems on configuration spaces of varieties over $\mathbb{C}$ attached to character polynomials. Our approach interprets the stabilization as a probabilistic…
Regularization is a core component of modern inverse problems, as it helps establish the well-posedness of the solution of interest. Popular regularization approaches include variational regularization and iterative regularization. The…
The main goal of the present paper is to convince that it is feasible to construct a `periodic orbit theory' of localization by extending the idea of classical action correlations. This possibility had been questioned by many researchers in…