Uniformly observable systems can be transformed into a triangular form with non-Lipschitz functions.
problem Uniformly observable and differentially observable systems with higher order than state dimension.
method Established triangular canonical form with non-Lipschitz functions.
result Characterization of points where non-Lipschitzness occurs and its relation to uniform infinitesimal observability.
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.
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.
Extended Regge complex for linearized Riemann-Cartan geometry and cohomology.
problem Cohomology of the Regge complex in three dimensions.
method Constructing a discrete version of linearized Riemann-Cartan geometry on any triangulation.
result The cohomology of the Regge complex is isomorphic to the infinitesimal-rigid-body-motion-valued de~Rham cohomology.
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 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.
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…
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.
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.
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.
We study infinitesimal conformal deformations of a triangulated surface in Euclidean space and investigate the change in its extrinsic geometry. A deformation of vertices is conformal if it preserves length cross-ratios. On one hand, conformal deformations generalize deformations preserving edge lengths. On the other h…
In category theory, logic and geometry cooperate with each other producing what is known under the name Synthetic Differential Geometry (SDG). The main difference between SDG and standard differential geometry is that the intuitionistic logic of SDG enforces the existence of infinitesimal objects which essentially modi…
We study infinitesimal Einstein deformations on compact flat manifolds and on product manifolds. Moreover, we prove refinements of results by Koiso and Bourguignon which yield obstructions on the existence of infinitesimal Einstein deformations under certain curvature conditions.
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.
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…
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.
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.
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. 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.
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.
Conditions are given under which an infinitesimal automorphism of a torsion-free connection preserving a symplectic form is necessarily a symplectic vector field. An example is given of a compact symplectic manifold admitting a flat symplectic connection and an infinitesimal automorphism that is not symplectic.
In many areas, practitioners seek to use observational data to learn a treatment assignment policy that satisfies application-specific constraints, such as budget, fairness, simplicity, or other functional form constraints. For example, policies may be restricted to take the form of decision trees based on a limited se…
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.
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.
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.
The paper classifies and studies conformal variations of submanifolds.
problem Classifying and understanding conformal variations of submanifolds.
method Develops a Fundamental theorem and a rigidity theorem for Euclidean submanifolds.
result Fundamental theorem and rigidity theorem for Euclidean submanifolds.
We study infinitesimal semi-simple extrinsic symmetric spaces and give a classification in the symplectic case.
Study infinitesimal deformations of Killing spinors on nearly parallel G2-manifolds.
problem Infinitesimal deformations of Killing spinors on nearly parallel G2-manifolds.
method Examined using the correspondence between nearly parallel G2-structures and Killing spinors.
result Identified that the space of Rarita-Schwinger fields coincides with a subspace of the eigenspace of the Laplacian.
Study on deformations of holomorphic Cartan geometries, focusing on flat cases.
problem Deformation of holomorphic Cartan geometries on complex manifolds.
method Computing infinitesimal deformations and analyzing the forgetful map.
result The forgetful map from infinitesimal deformations of a flat holomorphic Cartan geometry to the underlying flat principal bundle is an isomorphism.
Study infinitesimal deformations of Lie algebroid pairs.
problem Infinitesimal deformations of Lie algebroid pairs.
method Investigate isomorphism classes of infinitesimal deformations of (L,A) modulo automorphisms from exponentials of derivations of L and those from the exponentials of inner derivations of L. result Find the associated governing L∞-algebras in the sense of extended deformation theory. The paper explores conditions for constructing and extending infinitesimal isometries on special sub-Riemannian manifolds.
problem Finding conditions for infinitesimal isometries on special sub-Riemannian manifolds.
method Introducing $\is^*$-regular and $\is$-regular points to construct and extend infinitesimal isometries.
result Conditions on special sub-Riemannian manifolds allow for the construction and extension of infinitesimal isometries.
We categorify the notion of an infinitesimal braiding in a linear strict symmetric monoidal category, leading to the notion of a (strict) infinitesimal 2-braiding in a linear symmetric strict monoidal 2-category. We describe the associated categorification of the 4-term relation, leading to six categorified relations. …