The paper defines flows on Z-graded manifolds and proves unique maximal flows for vector fields.
problem Lack of a treatment for flows on Z-graded manifolds. method Definition and proof of maximal flows for vector fields on Z-graded manifolds. result Every vector field admits a unique maximal flow, with conditions for vector fields invariant under flows and commuting flows.
Vector fields on schemes have flows if rings are finitely generated.
problem Understanding vector fields and flows on schemes.
method Analyzing vector fields on affine C∞-schemes with finitely generated rings. result Vector fields on affine C∞-schemes with finitely generated rings have flows and are groupoid internal maps. The paper examines the limit of harmonic flow on flat vector bundles.
problem Understanding the limiting behavior of harmonic flow on flat complex vector bundles.
method Analyzes the harmonic flow and proves the limit is isomorphic to a graded flat complex vector bundle.
result The limit of the harmonic flow on flat complex vector bundles is isomorphic to a graded flat complex vector bundle.
Proofs for flows of linear vector fields and their applications.
problem Existence of flows for linear vector fields and related properties.
method Detailed proofs and flow construction techniques.
result Smooth triviality of vector bundles over contractible bases and isomorphy of fibers of transitive Lie algebroids.
Study shows limitations of Lie bracket commutation for nonsmooth vector fields.
problem Limitations of Lie bracket commutation for nonsmooth vector fields.
method Analysis of nonsmooth vector fields, focusing on commutation of flows and Lie bracket conditions.
result Lie bracket commutation cannot be extended to general a.e. differentiable vector fields, but holds for certain Sobolev regular fields.
Study vector fields and flows on singular spaces like submanifolds.
problem Understanding vector fields and flows on singular spaces.
method Integrate derivations of the C∞-ring of global smooth functions into flows. result Derivations integrate to smooth flows on subcartesian spaces.
This paper gives quantitative global estimates between a time dependent flow on a Riemannian manifold (M) and the flow of a vector field constructed by truncating the formal Magnus expansion for the logarithm of the flow. As a corollary, we also find quantitative estimates between the composition of the …
In this paper we introduce and study a geometric heat flow to find Killing vector fields on closed Riemannian manifolds with positive sectional curvature. We study its various properties, prove the global existence of the solution of this flow, discuss its convergence and possible applications, and its relation to the …
The geodesic flow on the tangent bundle is the flow of a certain vector field which is called the spray S:TM→TTM. The flow lines of the vector field $\ka_{TM}øTS:TTM\to TTTM$ project to the Jacobi fields on TM. This could be called the Jacobi flow.
Study Kähler-Ricci flow on manifolds with singularities.
problem Behavior of Kähler-Ricci flow on manifolds with finite-time singularities.
method Use of holomorphic vector fields to prove estimates.
result Proves estimates related to previous work on the flow.
Study geometric flows with varying parameters and prove continuous dependence.
problem Continuous dependence of flows on parameters in geometric settings.
method Derived suitable topologies for vector fields and flows, proved new continuous dependence.
result Proved continuous dependence of flows on parameters in a general topological space.
Flow of curves with curvature and forcing vector field exists.
problem Existence of a curve flow with curvature and forcing.
method Proved existence through Brakke motion law.
result Non-trivial flow of curves exists through singularities.
Two proofs of Kalman Theorem using flows of vector fields.
problem Classical result of Control Theory (Kalman Theorem).
method Two proofs using flows of vector fields.
result New criteria for local controllability of non-linear systems.
Researchers identify surfaces with special fluid flow fields.
problem Understanding fluid flows on curved surfaces.
method Defined and analyzed hydrodynamic Killing vector fields (HKVF) on surfaces.
result Any connected, orientable surface with HKVF is conformally equivalent to one of 14 canonical Riemann surfaces.
Analyzes vector fields in polytope decompositions, proving curve finiteness.
problem Analyzing vector fields in polytope decompositions.
method Proves integral curves are chopped into finitely many pieces by polytope decompositions.
result Finiteness of edge flips in discrete Yamabe flow.
Geodesic completeness and flow properties of compact Brinkmann spacetimes proven.
problem Geodesic completeness and flow properties of compact Brinkmann spacetimes.
method Proof of geodesic completeness and flow properties of isotropic parallel vector fields in compact Brinkmann spaces.
result Geodesic completeness and flow properties of compact Brinkmann spacetimes proven.
DFM simplifies CNF training without interpolants.
problem Efficiently training CNFs with computationally expensive ODE solving.
method DFM optimizes dual vector fields for bijective transformations.
result DFM outperforms CNF trained with FM or ML objectives.
Paper extends isoperimetric inequalities for non-starshaped hypersurfaces.
problem Isoperimetric inequalities for non-starshaped hypersurfaces.
method Volume preserving and area decreasing mean curvature flow with conformal Killing vector fields.
result Established isoperimetric inequalities for a broader class of hypersurfaces.
Smooth solutions found for modified mean curvature flow in Riemannian manifolds.
problem Existence of smooth solutions for modified mean curvature flow.
method A priori estimates for modified mean curvature flow in Riemannian manifolds with Killing vector field.
result Existence of smooth, entire, longtime solutions for modified mean curvature flow with smooth initial data.
This paper introduces the notions of vector field and flow on a general differentiable stack. Our main theorem states that the flow of a vector field on a compact proper differentiable stack exists and is unique up to a uniquely determined 2-cell. This extends the usual result on the existence and uniqueness of flows o…
In this paper we find solutions uε to a certain class of vector-valued parabolic Allen-Cahn equation that as ε→0 develops as interface a given triod evolving under curve shortening flow.
A new framework enhances generative modeling by learning local flows over complex manifolds.
problem Limited expressivity of current normalizing flows for low-dimensional manifolds.
method Vector quantized local normalizing flows (VQ-Flows) using a VQ-AE atlas and conditional flows.
result Enhanced modeling of complex data distributions over manifolds.
Lyapunov 1-forms on orbifolds help understand flows on compact spaces.
problem Understanding flows on orbifolds using Lyapunov 1-forms.
method Introducing Lyapunov 1-forms, using asymptotic cycles and chain-recurrent sets.
result Existence of a Lyapunov 1-form in a prescribed cohomology class for compact orbifolds.
The central object of study of this thesis is inverse mean curvature vector flow of two-dimensional surfaces in four-dimensional spacetimes. Being a system of forward-backward parabolic PDEs, inverse mean curvature vector flow equation lacks a general existence theory. Our main contribution is proving that there exist …
RFM uses tangent vector fields to match data on manifolds, analyzing TV convergence for Euler discretization.
problem Matching data on curved manifolds using flow-based models.
method Developed a nonasymptotic TV convergence analysis for RFM samplers using Euler discretization.
result Explicit bounds on TV convergence separating numerical discretization and learning errors.
New proof confirms periodic orbit conjecture for Eulerisable flows.
problem Periodic orbit conjecture for non-vanishing vector fields on closed manifolds.
method Characterization of Eulerisable flows and use of strongly adapted one-forms.
result Periodic orbit conjecture holds for Eulerisable flows.
From the paper "Formality Conjecture" (Ascona 1996): "I am aware of only one such a class, it corresponds to simplest good graph, the complete graph with 4 vertices (and 6 edges). This class gives a remarkable vector field on the space of bi-vector fields on Rd. The evolution with respect to the t…
Floer constructs homology from flow lines in generalized dynamical systems and combinatorial vector fields.
problem Computing homology in discrete and smooth dynamical systems.
method Counting flow lines between orbits and critical points.
result Directly recovers Z2 homology from flow lines. Exact universal interpolation property for landmark configurations in Euclidean space.
problem Representing and deforming landmark configurations through flows of vector fields.
method Explicitly describe vector fields for exact universal interpolation property in all dimensions.
result Achieve controllability by combining constant and polynomial vector fields.
In this paper we prove the following result: if two 2-dimensional 2-homogeneous rational vector fields commute, then either both vector fields can be explicitly integrated to produce rational flows with orbits being lines through the origin, or both flows can be explicitly integrated in terms of algebraic functions. In…
Improved flow matching using Gaussian processes for better sample quality.
problem Training continuous normalizing flows with reduced variance and flexibility.
method Extending conditional flow matching to streams modeled with Gaussian processes.
result Improved quality of generated samples with moderate computational cost.
Develops a new exponential map for time-varying vector fields.
problem Lack of global flows for general time-varying vector fields.
method Categorical development of spaces of vector fields and flows, allowing for systematic localisation.
result Derives the homeomorphism of the exponential map for vector fields with measurable time-dependence.
Study the isoperimetric problem in Riemannian manifolds with non-trivial conformal vector fields.
problem Isoperimetric problem in Riemannian manifolds with non-trivial conformal vector fields.
method Introduced a mean curvature type flow to study the isoperimetric problem.
result Established the isoperimetric inequality for star-shaped hypersurfaces in such manifolds.
We study the twisted Ruelle zeta function ζX(s) for smooth Anosov vector fields X acting on flat vector bundles over smooth compact manifolds. In dimension 3, we prove Fried conjecture, relating Reidemeister torsion and ζX(0). In higher dimensions, we show more generally that ζX(0) is locally constant with…
Proves properties of Morse vector fields on compact manifolds.
problem Properties of gradient vector fields of Morse functions.
method Analyzes connectedness of critical points and shrinkage of flow.
result Shows connectedness of critical points through orbits and exponential shrinkage.
Surveying recent progress on flows of G2-structures on 7-manifolds.
problem Preserving metrics while modifying G2-structures on 7-manifolds. method Heat flows and other approaches in terms of 3-forms, octonions, vector fields, and geometric structures. result Comparison of different perspectives on G2-structure flows. In this paper, we introduce an α-flow for the Yang-Mills functional in vector bundles over four dimensional Riemannian manifolds, and establish global existence of a unique smooth solution to the α-flow with smooth initial value. We prove that the limit of solutions of the α-flow as α\to 1 is a weak solution to the Yan…
The study finds solitons for curve shortening flow on hyperbolic plane.
problem Characterizing solitons for curve shortening flow on hyperbolic plane.
method Characterization using geodesic curvature and inner product with fixed vector in Minkowski space.
result Existence of 2-parameter family of soliton solutions on 2D hyperbolic plane.
We consider 3D flow equations inspired by the renormalization group (RG) equations of string theory with a three dimensional target space. By modifying the flow equations to include a U(1) gauge field, and adding carefully chosen De Turck terms, we are able to extend recent 2D results of Bakas to the case of a 3D Riema…
Rescaling expansiveness proven for k*-expansive vector fields.
problem Proving rescaling expansiveness for k*-expansive vector fields.
method Introducing and exploring singular-expansive flows.
result Rescaling expansiveness established for k*-expansive vector fields.
The paper classifies solitons in a curved product space.
problem Classifying solitons in a curved product space.
method Examined vector fields tangent to fibers and rotations, classified solitons under specific symmetries.
result A classification of solitons in s2imesR under certain symmetries. We introduce the conical Kähler-Ricci flow modified by a holomorphic vector field. We construct a long-time solution of the modified conical Kähler-Ricci flow as the limit of a sequence of smooth Kähler-Ricci flows.
Geometric equation defines canonical metrics on vector bundle families.
problem Finding canonical metrics on families of holomorphic vector bundles.
method Introducing a geometric partial differential equation for families of holomorphic vector bundles.
result Construction of Hermite--Einstein metrics in adiabatic classes on product manifolds and proof of the existence of a unique solution for the Dirichlet problem.
Study inverse curve shortening flow on hyperbolic plane, classifying solitons.
problem Understanding the behavior of curves in hyperbolic geometry under a specific flow.
method Classifying solitons with respect to vector fields and studying their properties.
result Parabolic solitons are graphs on the y-axis, conformal solitons on the x-axis.
I classify spacelike self-similar shrinking solutions of the mean curvature flow in pseudo-euclidean space in arbitrary codimension, if the mean curvature vector is not a null vector and the principal normal vector is parallel in the normal bundle. Moreover, I exclude the existence of such self-shrinkers in several cas…
A new method for reconstructing flows from perturbed distributions.
problem Reconstructing flows from perturbed probability distributions.
method Integrable vector fields and Green's functions.
result A nonparametric flow can be computed to generate samples from a perturbed distribution.
Constructs coordinates to diagonalize Toda flow on matrices with simple spectrum.
problem Diagonalizing the Toda flow on matrices with simple spectrum.
method Lie theoretic methods applied to complex semisimple Lie algebras and their real forms.
result Decouples the Toda vector field into simpler components.
In this paper a data analytical approach featuring support vector machines (SVM) is employed to train a predictive model over an experimentaldataset, which consists of the most relevant studies for two-phase flow pattern prediction. The database for this study consists of flow patterns or flow regimes in gas-liquid two…