Extends Infinitesimal Jackknife for model covariance, enhancing ensemble model analysis.
problem Estimating uncertainty in combinations of models.
method Extends Infinitesimal Jackknife to estimate covariance between models.
result Theoretical consistency of Infinitesimal Jackknife covariance estimate demonstrated.
Synthetic Differential Geometry modifies local space structures with infinitesimal curvature.
problem Infinitesimal curvature in categorical spaces.
method Developed differential geometry on infinitesimal formal manifolds, constructed models and studied their properties.
result Riemann curvature tensor is infinitesimal in infinitesimal level.
New examples of rigid Lie foliations with dense leaves found.
problem Infinitesimal rigidity of Lie foliations with dense leaves.
method Construction of specific Lie foliations.
result First examples of infinitesimally rigid Riemannian foliations with dense leaves.
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.
Study infinitesimal Einstein deformations on compact manifolds.
problem Existence of infinitesimal Einstein deformations under curvature conditions.
method Analyzing deformations on compact flat and product manifolds, proving refinements of existing results.
result Obstructions on the existence of infinitesimal Einstein deformations under certain curvature conditions.
Study on bending knots and energy changes in 3D space.
problem Understanding energy changes in knots under small deformations.
method Analyzes infinitesimal bending of knots and energy changes using Willmore and Mobius energies.
result Changes in energy under small deformations of knots have been quantified.
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…
Cauchy used infinitesimals in differential geometry and integral geometry.
problem Applying infinitesimals in differential and integral geometry.
method Using infinitesimals as numbers in differential and integral geometry.
result Valid application of infinitesimals in geometric probability, differential geometry, elasticity, and Dirac delta functions.
Metric spaces with certain curvature properties are universally infinitesimally Hilbertian.
problem Analyzing the infinitesimal geometry of metric spaces with curvature bounds.
method Proving a metric space with a Gromov-Hausdorff tangent splitting property is universally infinitesimally Hilbertian.
result Metric spaces with curvature bounds are universally infinitesimally Hilbertian.
Develops second order infinitesimal structures on Teichmüller space.
problem Understand the infinitesimal structures of Teichmüller space.
method Formulated second order infinitesimal structures over Teichmüller space.
result Affirmative answers to two folklore problems on Teichmüller space.
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…
Study infinitesimal automorphisms of VB-groupoids and algebroids.
problem Understand transformations preserving the structure of VB-groupoids and algebroids.
method Examine vector fields generating flows that preserve both linear and groupoid/algebroid structures.
result Infinitesimal automorphisms of a special class are multiplicative sections of a derivation groupoid/algebroid.
A general model for geometric structures on differentiable manifolds is obtained by deforming infinitesimal symmetries. Specifically, this model consists of a Lie algebroid, equipped with an affine connection compatible with the Lie algebroid structure. The curvature of this connection vanishes precisely when the struc…
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.
problem Exploring properties of infinitesimal groups and affine connections.
method Introducing second-order infinitesimal groups and using them to define Lie brackets and connections.
result Generalized correspondence between symmetric and non-symmetric affine connections.
Deformation theory for holomorphic Cartan geometries studied.
problem Understanding the deformations of holomorphic Cartan geometries.
method Computed infinitesimal automorphisms and deformations, proved semi-universal deformation existence.
result Existence of semi-universal deformation of holomorphic Cartan geometries.
Ancient formula connects volume forms and infinitesimal square volumes in manifolds.
problem Relating volume forms and infinitesimal square volumes in Riemannian manifolds.
method Uses Heron's formula to link these concepts.
result Established a connection between volume forms and infinitesimal square volumes.
Infinitesimal boosting converges to a deterministic process in large sample limit.
problem Characterizing the asymptotic behavior of infinitesimal gradient boosting in large sample sizes.
method Proving convergence to a deterministic process using large sample theory and differential equations.
result The test error decreases over time in the population limit.
Discussing Lie and Cartan theory on infinitesimal differential systems.
problem Understanding invariant differential systems on infinitesimal level.
method Infinitesimal approach to discuss Parts I and II of Lie and Cartan theory.
result Simpler calculations on infinitesimal level compared to finite level.
Lie algebroids are like infinitesimal Lie groupoids.
problem None explicitly stated in the abstract.
method Overview article.
result Lie algebroids are infinitesimal counterparts of Lie groupoids.
New classification of hypersurfaces with conformal variations.
problem Classifying hypersurfaces with conformal infinitesimal variations.
method Analyzing hypersurfaces in conformal geometry, extending previous work by Cartan and Sbrana.
result The class of hypersurfaces with conformal infinitesimal variations is larger than previously known.
Using standard analysis only, we present an extension ∙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…
The paper constructs new structures for manifolds using connections and combinations.
problem Understanding smooth manifolds with precise infinitesimal affine structures.
method Constructing new infinitesimal structures for higher-order neighbourhoods of the diagonal.
result Any symmetric affine connection on a manifold extends to a second-order infinitesimally affine structure.
New findings on bendable hypersurfaces in higher dimensions.
problem Characterizing infinitesimal bendable complete hypersurfaces.
method Analyzing non-flat infinitesimal bendable hypersurfaces in Euclidean spaces.
result Infinitesimal bendable complete hypersurfaces are only along ruled strips.
The paper studies infinitesimal variations of submanifolds in Euclidean space.
problem Understanding infinitesimal variations of submanifolds of arbitrary dimension and codimension.
method Proving a system of three equations as integrability conditions for the differential equation determining infinitesimal variations.
result Established a Fundamental theorem for infinitesimal variations of submanifolds.
Study of Teichmüller space geometry using infinitesimal and global methods.
problem Understanding the geometry of Teichmüller space and its tangent/cotangent spheres.
method Systematic study of Thurston metric's infinitesimal and global properties.
result Rigidity statements for the Thurston metric analogous to Royden theorem.
Reconstructing hypersurfaces from infinitesimal data using Lie algebroids.
problem Reconstructing hypersurfaces from infinitesimal data.
method Definition and use of Lie algebroids.
result Motivation and application of Lie algebroids.
In this paper, an obstruction against the integrability of certain infinitesimal solitonic deformations is given. Using this obstruction, we show that the complex projective spaces of even complex dimension are rigid as Ricci solitons although they have infinitesimal solitonic deformations.
In this paper we study infinitesimal and finite flexibility for generic semidiscrete surfaces. We prove that generic 2-ribbon semidiscrete surfaces have one degree of infinitesimal and finite flexibility. In particular we write down a system of differential equations describing isometric deformations in the case of exi…
New framework discovers non-affine continuous symmetries in neural networks.
problem Lack of efficient methods for detecting non-affine continuous symmetries in neural networks.
method Computational framework for discovering infinitesimal generators of multi-parameter group actions.
result Framework can discover non-affine continuous symmetries in neural networks.
Study of infinitesimal rigidity in hyperbolic manifolds.
problem Proving infinitesimal rigidity of geometrically infinite hyperbolic manifolds.
method Developed a strategy to study infinitesimal rigidity of cyclic coverings of manifolds colored by right-angled polytopes.
result Proved infinitesimal rigidity of some hyperbolic 4- and 5-manifolds.
The paper proposes a method to compute higher infinitesimals in numerical and symbolic analysis.
problem Computing higher-order derivatives with higher infinitesimals.
method Automatic differentiation in terms of C-infinity rings and Weil algebras.
result A unifying theoretical framework for multivariate higher-order derivatives.
Study infinitesimal symmetries in planar differential equations.
problem Qualitative study of differential equations and their integral curves.
method Use meromorphic connections methods to analyze infinitesimal symmetries.
result Infinitesimal symmetries of planar d-webs are understood. New method classifies hypersurfaces that can bend infinitesimally.
problem Classifying hypersurfaces that can be slightly deformed while preserving the Möbius metric.
method Introducing infinitesimal Moebius variations and characterizing isometric immersions.
result Classification of hypersurfaces that can be infinitesimally Moebius bendable.
Geometrically constructs BGG resolutions in Lie algebra type A.
problem Constructing BGG resolutions in Lie algebras of type A.
method Geometric construction in singular infinitesimal character.
result BGG resolutions constructed in Lie algebra type A.
Study of infinitesimal variations and moduli spaces in Strominger system and Killing spinors.
problem Understanding the moduli space of solutions to the Strominger system and related Killing spinor equations.
method Elliptic operator theory and generalized geometry to construct and describe the infinitesimal structure of the moduli space.
result Proposed a unifying framework for metrics with holonomy SU(3) and solutions of the Strominger system.
Study of how small changes in triangulated surfaces affect their geometry.
problem Understanding how small changes in triangulated surfaces affect their geometry.
method Investigates infinitesimal conformal deformations of triangulated surfaces in Euclidean space.
result There is a one-to-one correspondence between infinitesimal conformal deformations and infinitesimal isometric deformations of the stereographic image on the sphere.
Study variational properties of cone structures with infinitesimal symmetry.
problem Variational properties of cone structures with infinitesimal symmetry.
method Establishing a correspondence between cone structures and geometric structures via symmetry reduction and quasi-contactification.
result Invariant conditions for specific properties of cone structures.
Study multiplicative tensors on Lie groupoids using infinitesimal data.
problem Understanding tensors on Lie groupoids with groupoid structure.
method Complete description through infinitesimal data.
result Unified description of multiplicative forms, multivector fields, and holomorphic structures.
The paper tests hypotheses on two Lévy process-driven streams of observations.
problem Testing hypotheses on two Lévy process-driven streams of observations.
method Infinitesimal generators and super/sub-solutions are used to compute bounds and analyze the model.
result Bounds for infinitesimal generators are computed in terms of super/sub-solutions.
Characterizes infinitesimally homogeneous manifolds with specific structure groups.
problem Understanding manifolds with specific structure groups and properties.
method Analyzes triples (M, nabla, P) and characterizes infinitesimally homogeneous manifolds for various structure groups.
result Characterizes classes of infinitesimally homogeneous manifolds for specific structure groups.
Continuous Lorentzian metrics yield infinitesimal Minkowskian spacetimes.
problem Understanding spacetime properties from continuous Lorentzian metrics.
method Proving infinitesimal Minkowskianity for causally simple metric measure spacetimes.
result Continuous Lorentzian metrics result in spacetimes that are infinitesimally Minkowskian.
Survey of nonstandard analysis approaches and their applications.
problem Working with infinitesimals and infinite numbers.
method Exploring several approaches to nonstandard analysis, including Robinson's and Nelson's.
result Highlighting applications such as Loeb's approach to Lebesgue measure and true infinitesimal differential geometry.
We study non-flat planar 3-webs with infinitesimal symmetries. Using multi-dimensional Schwarzian derivative we give a criterion for linearization of such webs and present a projective classification thereof. Using this classification we show that the Gronwall conjecture is true for 3-webs admitting infinitesimal symme…
Study infinitesimally tight Lagrangian orbits in symplectic manifolds.
problem Understanding Lagrangian orbits in symplectic manifolds.
method Analyzing isotropic orbits of Lie group actions with equivariant moment maps.
result Examples of Lagrangian orbits in complex flag manifolds and cotangent bundles.
The study examines non-continuous Riemannian metrics on manifolds and their infinitesimal properties.
problem Investigating non-continuous Riemannian metrics and their infinitesimal structure.
method Constructing examples of metric measure spaces with discontinuous metrics.
result Examples show failure of infinitesimal Hilbertian or quasi-Riemannian properties.
The paper extends Noether's theorem to contact systems, finding dissipated quantities instead of conserved ones.
problem Noether's theorem for contact systems does not produce conserved quantities.
method Classification of infinitesimal symmetries in contact Lagrangian systems, leading to dissipated quantities.
result Infinitesimal symmetries in contact dynamics lead to dissipated quantities rather than conserved ones.
The paper computes infinitesimals for group actions on a multispace of curves.
problem Examining group actions on a multispace of curves.
method Formal study using recursion relations, closely mimicking jet space prolongation.
result Produces a recursion relation for infinitesimals in the multispace.