Some model reduction techniques for multiple time-scale dynamical systems make use of the identification of low dimensional slow invariant attracting manifolds (SIAM) in order to reduce the dimensionality of the phase space by restriction to the slow flow. The focus of this work is on a proposition and discussion of a …
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48 results for “differential-geometric formulation”
This note shows the compatibility of the differential geometric and the topological formulations of equivariant characteristic classes for a compact connected Lie group action.
This paper generalizes Batchelor's theorem in -superschemes.
problem Establishing global splittings in -superschemes.
method Structural generalization of Batchelor's theorem.
result Global splittings of -superspaces can be characterized by Euler vector fields.
Abstract: Geometrically reformulates estimation theory for finite-dimensional C*-algebras.
problem Estimation theory for finite-dimensional C*-algebras.
method Geometrical formulation of estimation theory.
result Derivation of Cramer-Rao and Helstrom bounds.
The theory of slow invariant manifolds (SIMs) is the foundation of various model-order reduction techniques for dissipative dynamical systems with multiple time-scales, e.g. in chemical kinetic models. The construction of SIMs and many approximation methods exploit the restrictive requirement of an explicit time-scale …
Existence of a conjugate point in the incompressible Euler flow on a sphere and an ellipsoid is considered. Misiolek (1996) formulated a differential-geometric criterion (we call M-criterion) for the existence of a conjugate point in a fluid flow. In this paper, it is shown that no zonal flow (stationary Euler flow) sa…
We formulate the laws governing the dynamics of a crystalline solid in which a continuous distribution of dislocations is present. Our formulation is based on new differential geometric concepts, which in particular relate to Lie groups. We then consider the static case, which describes crystalline bodies in equilibriu…
Canonical formalism for quasi-classical particle "Zitterbewegung" in Ostrohrads'kyj mechanicsmath-ph
The homogeneous canonical formalism of Rund is applied to the second-order Lagrangian model of the self-interacting particle of Bopp. The quasi-classical free spinning particle of Mathisson appears then as a constrained subsystem of the previous system. Differential-geometric mechanisms offered in this work are formula…
The closure conditions of the inexact exterior differential form and dual form (an equality to zero of differentials of these forms) can be treated as a definition of some differential-geometrical structure. Such a connection discloses the properties and specific features of the differential-geometrical structures. The…
We study differential geometric properties of cuspidal edges with boundary. There are several differential geometric invariants which are related with the behavior of the boundary in addition to usual differential geometric invariants of cuspidal edges. We study the relation of these invariants with several other invar…
Geometrically reformulates GENERIC stochastic dynamics.
problem Unified treatment of reversible and dissipative dynamics.
method Introduces degenerate Poisson structure, co-metric, and volume form.
result Preserves Boltzmann measure, conserves energy, reduces to deterministic limit.
A local instability mechanism of the Navier-Stokes flow with swirl on the no-slip flat boundarymath.AP
Using numerical simulations of the axisymmetric Navier-Stokes equations with swirl on a no-slip flat boundary, Hsu-Notsu-Yoneda [J. Fluid Mech. 2016] observed the creation of a high-vorticity region on the boundary near the axis of symmetry. In this paper, using a differential geometric approach, we prove that such flo…
CMS formulation solves Poincare conjecture for all dimensions.
problem Poincaré Conjecture in higher dimensions.
method Calculus of moving surfaces (CMS) for evolving hypersurfaces.
result Compact simply connected hypersurfaces relax to constant mean curvature (CMC) manifolds.
This paper treats the theory of Mukai duality on K3 surfaces from the differential geometric perspective, taylored to the need of the author's companion paper about Mukai duality of adiabatic coassociative K3 fibrations.
We compute the differential geometric invariants of cuspidal edges on flat surfaces in hyperbolic -space and in de Sitter space. Several dualities of invariants are pointed out.
Only Bayes should learn a manifold (on the estimation of differential geometric structure from data)stat.ML
We investigate learning of the differential geometric structure of a data manifold embedded in a high-dimensional Euclidean space. We first analyze kernel-based algorithms and show that under the usual regularizations, non-probabilistic methods cannot recover the differential geometric structure, but instead find mostl…
Covariant geometric characterization of slow invariant manifolds: New concepts and viewpointsmath.DS
We point out a new view on slow invariant manifolds (SIM) in dynamical systems which departs from a purely geometric covariant characterization implying coordinate independency. The fundamental idea is to treat the SIM as a well-defined geometric object in phase space and elucidate characterizing geometric properties t…
Geometric AD framework simplifies derivative computation in JAX.
problem Efficient and accurate automatic differentiation.
method Jet functors and Weil algebras for geometric analysis.
result Unified view of derivative propagation with algebraic exactness.
Based on the theory of Poisson vertex algebras we calculate skew-symmetry conditions and Jacobi identities for a class of third-order nonlocal operators of differential-geometric type. Hamiltonian operators within this class are defined by a Monge metric and a skew-symmetric two-form satisfying a number of differential…
Geometric Phasesmath-ph
The differential geometric aspects of Geometric Phases are reviewed.
Evolutionary forms, as well as exterior forms, are skew-symmetric differential forms. But in contrast to the exterior forms, the basis of evolutionary forms is deforming manifolds (with unclosed metric forms). Such forms possess a peculiarity, namely, the closed inexact exterior forms are obtained from that. The closur…
Introduces Q-structures for mechanics using advanced geometry.
problem Challenges in classical differential geometric constructions in mechanics.
method Explains the use of Q-structures and differential graded manifolds.
result Q-structure preserving integrators can be useful in mechanics.
Study geometric properties of S1 singularities and their deformations.
problem Understanding differential geometric properties of S1 singularities and deformations.
method Representing deformation using diffeomorphisms and isometries, studying geometric properties.
result Differential geometric properties of S1 singularities and Whitney umbrellas in deformations.
The exposition has been significantly altered, hopefully improved.
We implement a differential-geometric approach to normal forms for contracting measurable cocycles to $\mbox{Diff}^q({\bf R}^n, {\bf 0})$, . We obtain resonance polynomial normal forms for the contracting cocycle and its centralizer, via changes of coordinates. These are interpreted as nonstationary inv…
Differential geomtrical methods for deriving the Dirac equation in Curved Spacetime are presented. Einstein's field equation is applied in a novel manner; in the most current standard reference, Birrell and Davies, 1994 [1], the suggestions for deriving the Dirac equation in Curved Spacetime make no mention of employin…
The paper simplifies FLRW photon propagators using geometric embeddings.
problem Understanding Friedmann-Lemaître-Robertson-Walker (FLRW) spaces.
method Differential-geometric methods applied to FLRW spaces as submanifolds in \(\mathbb{R}^{n+2}\).
result New and simplified expressions for the photon propagator in four dimensions.
In this paper we research the differential geometric and algebro-geometric proper- ties of the noncollasping limit in the conical continuity equation.
In this paper, we give a geometric expression for the multiplicities of the equivariant index of a spin-c Dirac operator.
On manifold-like polyfolds as differential geometrical objects with applications in complex geometrymath.DG
Advances M-polyfolds for complex geometry applications.
problem Complex geometry challenges in differential geometry.
method Introduces and proves geometric structures within M-polyfolds.
result Establishes M-polyfolds as useful differential geometric objects.
Smooth complex surfaces with triple intersections using differential geometry.
problem Smooth complex surfaces with trivial canonical bundle and triple intersections.
method Explicit construction of local smoothings and solutions to nonlinear elliptic PDEs.
result Existence of smoothings for -semistable SNC complex surfaces with trivial canonical bundle.
The paper generalizes a theorem for quantum flag manifolds.
problem Developing a noncommutative differential geometric presentation of quantum coordinate rings.
method Using quantum principal bundles and the Heckenberger-Kolb first-order differential calculus.
result A novel noncommutative differential geometric presentation of quantum coordinate rings of irreducible quantum flag manifolds.
Variational approach to basic manifold structures.
problem Understanding basic differential geometric structures.
method Variational description of geometric structures.
result Variational formulation of manifold structures.
Purely real space versions of the differential equations describing the kinematics of a dislocated crystalline medium are considered. The differential geometric structures associated with them are revealed.
The purpose of the present paper is to study the differential geometric properties of a quaternion CR-submanifold in a locally conformal quaternion Kaehler manifold.
The aim of this paper is to propose an unambiguous intrinsic formalism for higher-order field theories which avoids the arbitrariness in the generalization of the conventional description of field theories, which implies the existence of different Cartan forms and Legendre transformations. We propose a differential-geo…
In this paper we investigate the differential geometric and algebro-geometric properties of the noncollapsing limit in the continuity method that was introduced by the first two named authors in \cite{LaTi14}.
We give a differential geometric description of the Cartan (or tractor) bundle and its canonical connection in CR geometry, thus offering a direct, alternative, definition to the usual abstract approach.
We survey several mathematical developments in the holonomy approach to gauge theory. A cornerstone of this approach is the introduction of group structures on spaces of based loops on a smooth manifold, relying on certain homotopy equivalence relations -- such as the so-called thin homotopy -- and the resulting interp…
We study the differential geometric properties of the manifold of non-singular symmetric real matrices endowed with the trace metric; in case of positive definite matrices we describe the full group of isometries
This article gives an exposition of the deformation theory for pairs , where is a compact complex manifold and is a holomorphic vector bundle over , adapting an analytic viewpoint à la Kodaira-Spencer. By introducing and exploiting an auxiliary differential operator, we derive the Maurer--Cartan equa…
Certain solutions of a sextic sigma-model Lagrangian reminiscent of Skyrme model correspond to perfect fluids with stiff matter equation of state. We analyse from a differential geometric perspective this correspondence extended to general barotropic fluids.
In this paper, Hamiltonian monodromy is studied from the point of view of geometric quantization abd theta functions, and various differential geometric aspects thereof are dealt with, all related to holonomies of suitable flat connections.
GeoPhy uses geometric gradients to efficiently infer phylogenetic trees from molecular data.
problem Challenges in accurately inferring species relationships from molecular data due to combinatorially vast tree topologies.
method Introduces a novel, fully differentiable formulation of phylogenetic inference using geometric spaces and variational Bayesian methods.
result Significantly outperforms other approximate Bayesian methods in inferring phylogenetic trees.
In 1870, R. Clausius found the virial theorem which amounts to introduce the trace of the stress tensor when studying the foundations of thermodynamics, as a way to relate the absolute temperature of an ideal gas to the mean kinetic energy of its molecules. In 1901, H. Poincar{é} introduced a duality principle in analy…
Paper develops equivariant basic cohomology for Lie groupoids.
problem Equivariant cohomology for Lie groupoids with weak actions.
method Using Kan fibrations and fiber structures, constructing models and comparing with existing theories.
result Equivariant basic cohomology theory for orbifolds and Lie groupoids.
We characterize singularities of focal surfaces of wave fronts in terms of differential geometric properties of the initial wave fronts. Moreover, we study relationships between geometric properties of focal surfaces and geometric invariants of the initial wave fronts.
Study of lightcone framed surfaces in Lorentz-Minkowski 3-space, focusing on curvature behavior.
problem Investigate differential geometric properties of lightcone framed surfaces.
method Introduced modified frame to study the properties of lightcone framed surfaces.
result Showed behavior of Gaussian and mean curvatures at lightlike and singular points.