CDFD analyzes circularity and directionality in weighted directed networks.
problem Analyzing circularity and directionality in weighted directed networks.
method CDFD framework separates flow into circular and acyclic components.
result CDFD yields a normalized circularity index capturing flow in cycles and directionality.
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.
The paper shows how gradient flow on over-parametrized tensor decomposition behaves like deflation.
problem Understanding the training dynamics of gradient flow on tensor decomposition.
method Empirical observation and mathematical proof of gradient flow dynamics for orthogonally decomposable tensors.
result Gradient flow dynamics for orthogonally decomposable tensors follows a tensor deflation process, recovering all tensor components.
The paper solves curvature problems on infinite hyperbolic surfaces.
problem Prescribed curvature flow on hyperbolic surfaces with infinite topological type.
method Introduced a prescribed curvature flow adapted to infinite cellular decompositions.
result Established well-posedness of the flow and proved convergence under certain conditions.
The space of probability densities is an infinite-dimensional Riemannian manifold, with Riemannian metrics in two flavors: Wasserstein and Fisher--Rao. The former is pivotal in optimal mass transport (OMT), whereas the latter occurs in information geometry---the differential geometric approach to statistics. The Rieman…
Algorithm decides if pseudo-Anosov flows have perfect fits.
problem Determining if pseudo-Anosov flows have specific asymptotic properties.
method Algorithm based on box decompositions and universal cover analysis.
result Algorithmic decision on pseudo-Anosov flows' perfect fit status.
This paper uses combinatorial Ricci flow to tackle Thurston's triangulation conjecture.
problem Thurston's triangulation conjecture for hyperbolic 3-manifolds.
method Combinatorial Ricci flow approach to prove convergence and geometric decompositions.
result Combinatorial Ricci flow converges if and only if the triangulation is geometric.
Study controls curvature in Ricci flows using necks.
problem Controlling curvature in Ricci flows.
method Introducing necks of maximal symmetry and decomposing curvature into uniform bounds.
result Established L1-bounds on Riemann curvature tensor. Paper shows how SFA fits into FBM framework for time series separation.
problem Identifying time series decomposition in flow-based models.
method Combining SFA and FBM to make time series decomposition identifiable.
result Time series decomposition becomes identifiable using SFA and FBM.
Decomposes flows with jumps into simpler components.
problem Understanding dynamics of flows with discontinuities.
method Extension of Itô-Ventzel-Kunita formula for stochastic flows with jumps.
result Explicit equations for each component of the decomposition.
Transverse one dimensional foliations play an important role in the study of codimension one foliations. In \cite{KR2}, the authors introduced the notion of flow box decomposition of a 3-manifold M. This is a decomposition of M that reflects both the structure of a given codimension one foliation and that of a give…
Develops methods to analyze feature-outcome associations in subpopulations.
problem Challenges in understanding feature-outcome associations in high-dimensional data.
method Geometric decomposition framework using gradient flow and co-monotonicity decomposition.
result Identifies context-dependent patterns and improves statistical power and interpretability.
Gradient flow solves optimal mass transport for covariance matrices.
problem Optimal mass transport for covariance matrices.
method Gradient flow on fiber bundle structure.
result Global convergence to polar decomposition.
The study connects geodesic flows on Riemann surfaces to random walks on their dual graphs.
problem Understanding ergodicity of geodesic flows on infinite Riemann surfaces.
method Analyzing random walks on the dual graph of pants decompositions.
result Equivalence between ergodicity of geodesic flows and recurrence of random walks.
Continuous time analysis of bubble formation in harmonic maps.
problem Understanding bubble formation in harmonic map heat flow.
method Continuous time approach to analyze bubbling sequences.
result Solutions approach multi-bubble configurations in continuous time.
Graph neural networks are explained through energy gradient flow and framelet decomposition.
problem Understanding and improving graph neural networks.
method Viewing framelet-based models as gradient flows of energy, proposing a generalized energy via framelet decomposition.
result The proposed model leads to more flexible dynamics, enhancing graph neural networks.
Unified framework for analyzing gradient flows of measures with exponential decay of entropy.
problem Analyzing exponential decay of entropy functionals in gradient flows of measures.
method Characterization of global exponential decay behaviors using Hellinger-Kantorovich geometry, shape-mass decomposition, and Polyak-Łojasiewicz-type inequalities.
result Unified theoretical framework for gradient flows with complete analysis of exponential decay behaviors.
We exhibit a concentration-collapse decomposition of singularities of fourth order curvature flows, including the L2 curvature flow and Calabi flow, in dimensions n≤4. The proof requires the development of several new a priori estimates. First, we develop a smoothing result for initial metrics with small ener…
This paper explores vortices and harmonic flows on compact surfaces, using Hodge decomposition.
problem Understanding the interplay between vortices and harmonic flows on compact surfaces.
method Hodge decomposition of Euler's equations, focusing on point vortices on compact Riemann surfaces.
result The harmonic part of the flow is constant on flat tori but not on non-flat tori.
Geometry arising from two diffusion operators (smooth semi-elliptic, second order differential operators) on different spaces but intertwined by a smooth map is described. Particular cases arise from Riemannian submersions when the operators are Laplace-Beltrami operators, from equivariant operators on the total space …
RFM simplifies generative modeling on complex geometries without simulation.
problem Training generative models on non-Euclidean geometries is challenging.
method Riemannian Flow Matching (RFM) constructs a premetric for efficient vector field computation.
result RFM achieves state-of-the-art performance on various non-Euclidean datasets.
Non-ergodic geodesic flow on Cantor tree surfaces found.
problem Determining when geodesic flow on Cantor tree surfaces is non-ergodic.
method Interpolating between two rates of convergence of cuff lengths to zero to prove non-ergodicity.
result Cantor tree surfaces with certain rates of cuff length convergence are non-parabolic.
Decompositions on manifolds appear in various geometric structures. Necessary and sufficient conditions for quotient spaces of decompositions to be manifolds are widely characterized. We characterize necessary and sufficient conditions to be k-manifolds (k=1,2), which generalize characterizations in the codimens…
Let S be a closed, connected, orientable surface of genus at least 2, and let C(S) denote the deformation space of convex real projective structures S. In this article, we introduce two new flows on C(S), which we call the internal bulging flow and the eruption flow. These are geometrically defined flows associated to …
Study reveals structure of Bitcoin's crypto flow network.
problem Understanding crypto flows among Bitcoin users.
method Blockchain data, user identification, network construction, bow-tie structure, Hodge decomposition, non-negative matrix factorization.
result Users are located in upstream, downstream, and core of the crypto flow network.
We prove a uniform Sobolev inequality along the Sasaki-Ricci flow. In the process, we develop the theory of basic Lebesgue and Sobolev function spaces, and prove some general results about the decomposition of the heat kernel for a class of elliptic operators on a Sasaki manifold.
Study geometric flows of G2-structures, determining curvature and torsion invariants.
problem Investigate geometric flows of G2-structures and their invariants.
method Explicitly compute differential invariants, decompose curvature and torsion, analyze principal symbols.
result Established short-time existence and uniqueness for geometric flows of G2-structures.
Harmonic map flow's singularity properties proven with Lojasiewicz inequalities.
problem Finite-time singularities of harmonic map flow in critical dimensions.
method Proving a weighted Lojasiewicz inequality.
result Continuity of body map and no-neck property for bubble-tree decompositions.
New GPs model edge functions on complex networks, capturing divergence and curl.
problem Modeling flow data on networks with independent learning of Hodge components.
method Developed Hodge-compositional edge GPs using Hodge decomposition.
result Hodge-compositional edge GPs can represent any edge function and capture flow relevance.
Gradient flow solves multi-index regression for high-dimensional Gaussian data.
problem Learning multi-index functions from high-dimensional Gaussian data.
method Two-timescale algorithm with non-parametric link function learning.
result Global convergence of Grassmannian population gradient flow dynamics.
Investigates nearly Kähler and parallel G2 manifolds using Hitchin functionals.
problem Stability analysis of nearly Kähler and parallel G2 manifolds.
method Gradient flow of Hitchin functionals, spectral decomposition of Hessians, Hitchin index.
result Hitchin index provides a lower bound for the Einstein co-index.
COMBO network improves optical flow estimation by combining deep learning with brightness constancy.
problem Optical flow estimation using deep learning requires complex training schemes.
method COMBO network explicitly exploits brightness constancy and combines it with a data-driven approach.
result COMBO network outperforms state-of-the-art methods on various benchmarks.
We produce infinitely many examples of Anosov flows in closed 3-manifolds where the set of periodic orbits is partitioned into two infinite subsets. In one subset every closed orbit is freely homotopic to infinitely other closed orbits of the flow. In the other subset every closed orbit is freely homotopic to only one …
Given an open book decomposition (Σ,τ) of a three manifold Y, Thurston and Winkelnkemper [TW] construct a specific contact form a on Y. Given a spin-c Dirac operator D on Y, the contact form naturally associates a one parameter family of Dirac operators $D_r = D - \frac{ir}{2}\cl(a)$ for r≥0. When $r>>…
We make a conjecture about mean curvature flow of Lagrangian submanifolds of Calabi-Yau manifolds, expanding on \cite{Th}. We give new results about the stability condition, and propose a Jordan-Hölder-type decomposition of (special) Lagrangians. The main results are the uniqueness of special Lagrangians in hamiltonian…
In this paper we study the Föllmer-Schweizer decomposition of a square integrable random variable ξ with respect to a given semimartingale S under restricted information. Thanks to the relationship between this decomposition and that of the projection of ξ with respect to the given information flow, we characteri…
We prove that every flat nonlinear discrete-time system can be decomposed by coordinate transformations into a smaller-dimensional subsystem and an endogenous dynamic feedback. For flat continuous-time systems, no comparable result is available. The advantage of such a decomposition is that the complete system is flat …
Study infinite combinatorial Ricci flow on spherical surfaces.
problem Investigate infinite combinatorial Ricci flow with spherical background.
method Establish existence and convergence of solution for infinite cellular decompositions.
result Existence and convergence of solution for infinite combinatorial Ricci flow in spherical geometry.
HFNO enhances interpretability of turbulent flows through parallel wavenumber bin processing.
problem Opaque inner workings of Fourier Neural Operators (FNOs) hinder physical interpretability.
method Introduces HFNO, a novel FNO-based architecture that processes wavenumber bins in parallel, enhancing interpretability.
result HFNO decomposes turbulent flows across various scales, enabling increased interpretability and multiscale modeling.
Let M be a closed oriented three-manifold, whose prime decomposition contains no aspherical factors. We show that for any initial riemannian metric on M the solution to the Ricci flow with surgery, defined in our previous paper math.DG/0303109, becomes extinct in finite time. The proof uses a version of the minimal dis…
A new method splits surface flow discretizations into streamfunctions and harmonic fields.
problem Discretizing incompressible flows on surfaces with pressure and saddle-point structure.
method Discrete Helmholtz-Hodge decomposition for BDM elements on surfaces.
result Eliminates pressure and saddle-point structure, ensuring exact tangentiality and divergence-freeness.
We propose a faster and more accurate method for learning classification trees.
problem Learning optimal binary classification trees is challenging and slow.
method We introduce a stronger MIP formulation and Benders' decomposition method.
result Our method is 50 times faster and improves out-of-sample performance.
Study controls bifurcations in Eulerian flows with multiple Hopf singularities.
problem Bifurcation analysis and control of nonlinear Eulerian flows with non-resonant n-tuple Hopf singularities.
method Analysis of CW complex bifurcations of flow-invariant Clifford hypertori, using leaf-bifurcation varieties.
result Tertiary toral CW complex bifurcates from and persists outside a secondary toral CW complex.
A flow-spine of a 3-manifold is a spine admitting a flow that is transverse to the spine, where the flow in the complement of the spine is diffeomorphic to a constant flow in an open ball. We say that a contact structure on a closed, connected, oriented 3-manifold is supported by a flow-spine if it has a contact form w…
We propose GraphNVP, the first invertible, normalizing flow-based molecular graph generation model. We decompose the generation of a graph into two steps: generation of (i) an adjacency tensor and (ii) node attributes. This decomposition yields the exact likelihood maximization on graph-structured data, combined with t…
Analysis of Vlasov plasma dynamics using matched pair Lie-Poisson formulation.
problem Understanding the dynamics of Vlasov plasma and its kinetic moments.
method Hamiltonian (Lie-Poisson) analysis and matched pair decomposition.
result Observation of mutual interactions between subdynamics in Vlasov plasma.
We prove that the geodesic flow on the unit tangent bundle to a hyperbolic 2-orbifold is left-handed if and only if the orbifold is a sphere with three conic points. As a consequence, on the unit tangent bundle to a 3-conic sphere, the lift of every finite collection of closed geodesics that is zero in integral homolog…
New criteria for ideal circle patterns on surfaces.
problem Determining when a surface supports ideal circle patterns.
method Introducing a character L(D,Φ) and using combinatorial Ricci flows. result Simpler and more easily verifiable criteria for ideal circle patterns.