Existence of translating solutions shown for curve diffusion flow.
problem Existence of translating solutions for curve diffusion flow.
method Higher order curve shortening flow approach.
result Properly immersed translating solutions exist.
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48 results for “curved diffusions”
Smooth convergence shown for curve diffusion flows.
problem Embeddedness and global existence of curves.
method Exponentially fast convergence established.
result Smooth convergence for curve diffusion flows.
In this article we investigate the dynamics of special solutions to the surface diffusion flow of idealised ribbons. This equation reduces to studying the curve diffusion flow for the profile curve of the ribbon. We provide: (1) a complete classification of stationary solutions; (2) qualitative results on shrinkers, tr…
Length-constrained curve diffusion flow for open curves with endpoints on two intersecting linesmath.DG
Curve diffusion flow straightens curves with endpoints on intersecting lines.
problem Straightening open curves with endpoints on intersecting lines.
method Curve diffusion flow with mixed boundary conditions.
result The curve converges to a circular arc of the same length.
The paper shows that energy futures yield curves have an affine geometry.
problem Estimating dynamic behavior of yield curves from data while avoiding arbitrage.
method Finite dimensional models for yield curves, diffusion coefficients, and compatibility conditions.
result The compatibility of yield curves with diffusion coefficients forces an affine geometry.
Identifies smooth curves for financial models.
problem Consistent term structures with flexible diffusion.
method Analyzes manifolds of curves for Heath-Jarrow-Morton models.
result Term structures cannot be affine but must be linear-rational.
We prove a blow-up criterion in terms of an -bound of the curvature for solutions to the curve diffusion flow if the maximal time of existence is finite. In our setting, we consider an evolving family of curves driven by curve diffusion flow, which has free boundary points supported on a line. The evolving curve h…
We show that any initial closed curve suitably close to a circle flows under length-constrained curve diffusion to a round circle in infinite time with exponential convergence. We provide an estimate on the total length of time for which such curves are not strictly convex. We further show that there are no closed tran…
Study on curve diffusion flows with scale-critical curvature term.
problem Analyzing stability of curve diffusion flows with scale-critical curvature.
method Introduced and studied a one-parameter family of curve diffusion flows with a scale-critical cubic curvature term. Analyzed dynamical stability of homothetic circles using variational methods.
result Established that any small perturbation of an -fold circle monotonically approaches the unit -circle after rescaling, translation, and reparametrisation.
Jellyfish existmath.DG
New jellyfish found in various flows.
problem Existence of geometrically distinct shapes in flows.
method Analyzing elastic, curve diffusion, and ideal flows.
result Infinitely many distinct shapes discovered.
We study in details the long-time asymptotic behavior of a relativistic diffusion taking values in the unitary tangent bundle of a curved Lorentzian manifold, namely a spatially flat and fast expanding Robertson-Walker space-time. We prove in particular that the Poisson boundary of the diffusion can be identified with …
In this paper we establish a general form of the isoperimetric inequality for immersed closed curves (possibly non-convex) in the plane under rotational symmetry. As an application we obtain a global existence result for the surface diffusion flow, providing that an initial curve is -close to a multiply covered ci…
High-dimensional curved diffusions show abrupt convergence at a critical time.
problem Understanding abrupt convergence in high-dimensional curved diffusions.
method Functional inequalities and spectral rigidity.
result Abrupt convergence (cutoff) occurs in high dimensions, linked to spectral rigidity.
Global minimizers for fast diffusion versus nonlocal interactions on negatively curved manifoldsmath.AP
Study finds conditions for global minimizers on curved manifolds with fast diffusion and nonlocal interactions.
problem Existence of global minimizers for a free energy functional on negatively curved manifolds.
method Investigation of Carlson-Levin type inequalities for Cartan-Hadamard manifolds.
result Establishes necessary and sufficient conditions for the existence of global energy minimizers.
A new method uses string method to explore diffusion models.
problem Understanding the geometry of learned distributions in diffusion models.
method String method to compute continuous paths between samples.
result The string method identifies realistic morphing sequences and transition pathways.
In this paper we construct a parametrization-free embedding technique for numerically evolving reaction-diffusion PDEs defined on algebraic curves that possess an isolated singularity. In our approach, we first desingularize the curve by appealing to techniques from algebraic geometry. We create a family of smooth curv…
New method synthesizes data on curved spaces for better interpolation.
problem Synthesizing data on curved spaces for better interpolation.
method Riemannian Diffusion Schrödinger Bridge
result Generalizes Diffusion Schrödinger Bridge to curved spaces for better interpolation.
We show short-time existence for curves driven by curve diffusion flow with a prescribed contact angle : The evolving curve has free boundary points, which are supported on a line and it satisfies a no-flux condition. The initial data are suitable curves of class with . For …
In this paper we consider the steepest descent -gradient flow of the length functional for immersed plane curves, known as the curve diffusion flow. It is known that under this flow there exist both initially immersed curves which develop at least one singularity in finite time and initially embedded curves whi…
The latter author, together with collaborators, proposed a numerical scheme to calculate the price of barrier options. The scheme is based on a symmetrization of diffusion process. The present paper aims to give a mathematical credit to the use of the numerical scheme for Heston or SABR type stochastic volatility model…
Defending Diffusion Models Against Membership Inference Attacks via Higher-Order Langevin Dynamicscs.LG
Paper defends diffusion models from membership inference attacks using Langevin dynamics.
problem Defending diffusion models against membership inference attacks.
method Uses critically-damped higher-order Langevin dynamics with auxiliary variables.
result Demonstrates improved resistance to membership inference attacks through theoretical investigation and validation.
Simulates financial market orders using anomalous diffusion models.
problem Anomalous diffusion in financial market order dynamics.
method Discrete Time Random Walk with Sibuya waiting times, non-uniform sampling, and cubic spline interpolation.
result Demonstrates price impact for different forcing functions and model parameters.
A coupling method and an analytic one allow us to prove new lower bounds for the spectral gap of reversible diffusions on compact manifolds. Those bounds are based on the a notion of curvature of the diffusion, like the coarse Ricci curvature or the Bakry--Emery curvature-dimension inequalities. We show that when this …
The paper studies stochastic optimization on matrices and its limits as dimensions grow.
problem Optimizing functions on large symmetric matrices using stochastic gradient descent.
method Deterministic limits of random curves on matrices, using graphons and stochastic differential equations.
result The limit is a gradient flow on graphons, extending classical McKean-Vlasov limits.
The study examines how the number of noise samples affects diffusion models' performance.
problem Understanding the balance between generalization and memorization in diffusion models.
method Theoretical analysis and empirical experiments with Denoising Score Matching (DSM) using random features.
result Precise expressions for test and train errors under specific conditions reveal the mechanisms of generalization and memorization.
DiTSNe-Ia model accurately reconstructs supernovae spectra from light curves.
problem Difficult identification and interpretation of diverse sub-populations of supernovae.
method Variational diffusion-based generative model conditioned on light curves.
result DiTSNe-Ia achieves significantly more accurate reconstructions than SALT3 across all phases.
Study shows how diffusion models learn on low-dimensional manifolds.
problem Learning efficiency of diffusion models on manifolds.
method Analyzes denoising score matching with random feature neural networks.
result Sample complexity scales linearly with intrinsic dimension, not ambient dimension.
Develops polynomial diffusion models for multi-factor commodity futures dynamics.
problem Modeling futures prices using latent state variables for short and long-term stochastic factors.
method Polynomial diffusion models to incorporate non-linear effects, two filtering methods for estimation.
result Accurate estimation of futures prices despite parameter identification issues in polynomial diffusion models.
New method shortens and straightens curves, proving convergence and well-posedness.
problem Shortening and straightening of curves.
method Conceptual shift in curve shortening to tangent aligning, variational study of geometric flows.
result Proves convergence to a straight line and global well-posedness for various geometric flows.
At the heart of technology transitions lie complex processes of social and industrial dynamics. The quantitative study of sustainability transitions requires modelling work, which necessitates a theory of technology substitution. Many, if not most, contemporary modelling approaches for future technology pathways overlo…
Construct intrinsic Langevin dynamics for rigid inclusions on curved surfaces.
problem Stochastic dynamics of rigid inclusions on curved surfaces.
method Cartan's method of moving frames, Hamiltonian equations, intrinsic Langevin equations, Fokker-Planck equation.
result Extracted overdamped equations for accurate simulations of diffusion processes.
Curves become nearly circular over time without initial assumptions.
problem Understanding the asymptotic behavior of area-preserving flows.
method Proving asymptotic circularity without additional assumptions.
result Immortal solutions become asymptotically circular without initial assumptions.
We study the curve diffusion flow for closed curves immersed in the Minkowski plane , which is equivalent to the Euclidean plane endowed with a closed, symmetric, convex curve called an indicatrix that scales the length of a vector in depending on its length. The indiactrix $\partial\mathcal{…
Adaptive sampling improves graph diffusion models by maintaining uniform information speed.
problem Standard diffusion models overlook non-homogeneous dynamics on complex manifolds.
method Information-geometric framework using Fisher-Rao metric and Drift Variation Score (DVS).
result DVS solver ensures uniform rate of distributional change, improving structural fidelity and efficiency.
Study on existence of ground states on curved spaces with conditions on potential growth.
problem Existence of ground states for aggregation-diffusion models on Cartan-Hadamard manifolds.
method Investigation of a free energy functional on Cartan-Hadamard manifolds, considering entropy and interaction energies.
result Necessary and sufficient conditions for existence of ground states are found, depending on the growth of the attractive potential.
We introduce variational approximations for curve evolutions in two-dimensional Riemannian manifolds that are conformally flat, i.e.\ conformally equivalent to the Euclidean space. Examples include the hyperbolic plane, the hyperbolic disk, the elliptic plane as well as any conformal parameterization of a two-dimension…
Diffusion models don't overfit, contrary to expectations.
problem Understanding generalization in diffusion models.
method Fundamental impossibility results and analysis of score matching.
result Diffusion models exhibit classical U-shaped loss curve, not double descent.
Capital distribution curve is defined as log-log plot of normalized stock capitalizations ranked in descending order. The curve displays remarkable stability over periods of time. Theory of exchangeable distributions on set partitions, developed for purposes of mathematical genetics and recently applied in non-parametr…
We study a class of nonlocal, energy-driven dynamical models that govern the motion of closed, embedded curves from both an energetic and dynamical perspective. Our energetic results provide a variety of ways to understand physically motivated energetic models in terms of more classical, combinatorial measures of compl…
A new method de-randomizes MCMC dynamics using the Stein operator.
problem Estimating complex target distributions in Bayesian inference.
method De-randomized kernel-based particle samplers that discretize the fiber-gradient Hamiltonian flow.
result GSVGD de-randomizes complex MCMC dynamics, maintaining high sample quality.
Non-negative curvature affects Markov chains' mixing and expansion properties.
problem Understanding the behavior of Markov chains with non-negative curvature.
method Analyzing conductance, displacement, and cutoff phenomenon in sparse Markov chains.
result Non-negatively curved Markov chains exhibit specific, non-standard behavior in terms of mixing and expansion.
Computing the SSRq-fin.MF
Model-free expression for SSR derived in terms of characteristic function.
problem Calculating the skew-stickiness-ratio (SSR) in financial markets.
method Model-free expression using characteristic function, focusing on diffusion and affine forward variance cases.
result General formula for SSR simplifies and becomes particularly tractable in affine forward variance cases, with a limit of for short-term limit.
We analyze DMs using spectral methods to design effective noise schedules.
problem Lack of theoretical foundation for synthesis process decisions in DMs.
method Introduced a frequency response perspective based on Gaussianity assumption.
result Proposed a spectral transfer function to understand DM inference process.
Study Chen's flow of curves in two settings: closed circles and lines, identifying geometric conditions for global behavior.
problem Understanding the global behavior of Chen's flow of curves in two settings.
method Investigated two settings: closed immersed ω-circles and immersed lines with a cocompactness condition. Analyzed geometric conditions and curvature effects.
result Identified conditions ensuring the flow shrinks every initial curve to a point, including a rescaling method.
Hydrodynamic Coupling of Particle Inclusions Embedded in Curved Lipid Bilayer Membranescond-mat.soft
We develop theory and computational methods to investigate particle inclusions embedded within curved lipid bilayer membranes. We consider the case of spherical lipid vesicles where inclusion particles are coupled through (i) intramembrane hydrodynamics, (ii) traction stresses with the external and trapped solvent flui…
We perform a geometric study of the equilibrium locus of the flow that models the diffusion process over a circular network of cells. We prove that when considering the set of all possible values of the parameters, the equilibrium locus is a smooth manifold with corners, while for a given value of the parameters, it is…
Light curve completion and forecasting using fast and scalable Gaussian processes (MuyGPs)astro-ph.IM
Predict missing and future data points in light curves using scalable Gaussian Processes.
problem Gappy time-series data from commercial cameras confound light curve prediction.
method MuyGPs, a scalable framework for hyperparameter estimation of Gaussian Processes using nearest neighbors sparsification and local cross-validation.
result MuyGPs enable accurate prediction of missing and future data points in light curves.
Graphs with bounded degrees and non-negative Ollivier-Ricci curvature have subexponential growth and diffusive random walk.
problem Understanding geometric properties of graphs with non-negative Ollivier-Ricci curvature.
method Analyzing the geometric properties of graphs with non-negative Ollivier-Ricci curvature, proving subexponential growth and diffusive random walk.
result For graphs with bounded degrees and non-negative Ollivier-Ricci curvature, the average log-volume growth and random walk displacement are subexponential.