arXiv research

Functional central limit theorem for kernel gradient flow and infinitesimal gradient boosting

The pentagram map's limit point is related to infinitesimal perturbations of polygons.

Any given surface of revolution embedded in Euclidean three-space can always be perturbed by arbitrarily small ambient isotopies as to admit highly nontrivial vector fields inducing infinitesimal deformations. For this matter Morse Theory is used, clarifying and giving a general response of a problem started with an id…

Estimates derivatives of max-stable fields for risk analysis.

A new method for reconstructing flows from perturbed distributions.

Proves infinitesimal rigidity of Hermitian gravitational instantons.

The paper proposes a method to compute higher infinitesimals in numerical and symbolic analysis.

Study examines deformations of Kerr-(A)dS near horizon geometry.

Using standard analysis only, we present an extension ${^\bullet\R}$ of the real field containing nilpotent infinitesimals. On the one hand we want to present a very simple setting to formalize infinitesimal methods in Differential Geometry, Analysis and Physics. On the other hand we want to show that these infinitesim…

We consider the Zermelo navigation problem on the ellipsoid of revolution (spheroid) in the presence of a perturbation $W$ determined by a mild velocity vector field, $|W|<1$, with application of Finsler metric of Randers type in the context of the corresponding optimal control represented by a time-efficient ship's he…

New framework for higher-order singular-value derivatives of rectangular matrices.

Extends field theory foundations to infinitesimal spaces, simplifying complex concepts.

This is a survey of several approaches to the framework for working with infinitesimals and infinite numbers, originally developed by Abraham Robinson in the 1960s, and their constructive engagement with the Cantor-Dedekind postulate and the Intended Interpretation hypothesis. We highlight some applications including (…

Cauchy used infinitesimals in differential geometry and integral geometry.

On the one hand, we prove that the Clifford torus in $\mathbb{C}^2$ is unstable for Lagrangian mean curvature flow under arbitrarily small Hamiltonian perturbations, even though it is Hamiltonian $F$-stable and locally area minimising under Hamiltonian variations. On the other hand, we show that the Clifford torus is r…

We show that every finite volume hyperbolic manifold of dimension greater or equal to 3 is stable under rescaled Ricci flow, i.e. that every small perturbation of the hyperbolic metric flows back to the hyperbolic metric again. Note that we do not need to make any decay assumptions on this perturbation. It will turn ou…

Extends Infinitesimal Jackknife for model covariance, enhancing ensemble model analysis.

Study analyzes Lévy process structure on manifolds with conjugate points.

We propose a simple approach to a problem introduced by Galatolo and Pollicott, consisting in perturbing a dynamical system in order for its absolutely continuous invariant measure to change in a prescribed way. Instead of using transfer operators, we observe that restricting to an infinitesimal conjugacy already yield…

In shape analysis, the concept of shape spaces has always been vague, requiring a case-by-case approach for every new type of shape. In this paper, we give a general definition for an abstract space of shapes in a manifold. This notion encompasses every shape space studied so far in the literature, and offers a rigorou…

Novel framework for learning infinitesimal generator of stochastic processes.

F.: Good morning Hermann, I would like to talk with you about infinitesimals. G.: Tell me Pierre. F.: I'm fed up of all these slanders about my attitude to be non rigorous, so I've started to study nonstandard analysis (NSA) and synthetic differential geometry (SDG). G.: Yes, I've read something ... F.: Ok, no problem …

New method discovers symmetries in differential equations from data.

Generalized differential geometry uses infinitesimals to solve singularities in differential equations.

Recently, the infinitesimal moduli space of heterotic $G_2$ compactifications was described in supergravity and related to the cohomology of a target space differential. In this paper we identify the marginal deformations of the corresponding heterotic nonlinear sigma model with cohomology classes of a worldsheet BRST …

DBPA assesses LLM perturbations using frequentist hypothesis testing.

The paper presents a method for detecting jump sizes in crude oil prices.

New theory of sensitivity for unbiased estimators using Wasserstein geometry.

The paper introduces extremal perturbations for better attribution analysis in deep networks.

The construction (by Kapranov) of the space of infinitesimal paths on a manifold is extended to include higher dimensional infinitesimal objects, encoding contractions of infinitesimal loops. This full infinitesimal groupoid is shown to have the algebra of polyvector fields as its non-linear cohomology.

Lyapunov's second theorem is an essential tool for stability analysis of differential equations. The paper provides an analog theorem for incremental stability analysis by lifting the Lyapunov function to the tangent bundle. The Lyapunov function endows the state-space with a Finsler structure. Incremental stability is…

New methods for estimating gradients of expectations using pairwise interactions.

Study on bending knots and energy changes in 3D space.

In this paper, we establish an infinitesimal equivariant index formula in the noncommutative geometry framework using Greiner's approach to heat kernel asymptotics. An infinitesimal equivariant index formula for odd dimensional manifolds is also given. We define infinitesimal equivariant eta cochains, prove their regul…

The article reviews how gradient flow systems on hypergraphs connect to information geometry and nonequilibrium physics.

Metric spaces with certain curvature properties are universally infinitesimally Hilbertian.

Weiss and, independently, Mazzeo and Montcouquiol recently proved that a 3--dimensional hyperbolic cone-manifold (possibly with vertices) with all cone angles less than $2π$ is infinitesimally rigid. On the other hand, Casson provided 1998 an example of an infinitesimally flexible cone-manifold with some of the cone an…

Improved online Lasso reduces regret in sparse linear contextual bandits.

In this paper, we prove that infinitesimal equivariant Chern-Connes characters are well-defined. We decompose an equivariant index as a pairing of infinitesimal equivariant Chern-Connes characters with the Chern character of an idempotent matrix. We compute the limit of infinitesimal equivariant Chern- Connes character…

Paper generalizes connections between Lie groups and affine connections.

New proof shows all conformal fields are Killing on specific spaces.

Deformation theory for holomorphic Cartan geometries studied.

New analysis shows temperature guarantees generalization in stochastic training.

Ancient formula connects volume forms and infinitesimal square volumes in manifolds.

Unified analysis of perturbation-based strategies in stochastic and adversarial bandit problems.

We study CR hypersurfaces in C^4 with constant rank Levi form and find their defining equations.

New examples of rigid Lie foliations with dense leaves found.

Lie algebroids are like infinitesimal Lie groupoids.