Kähler-Ricci flow on threefolds collapses in finite time.
problem Finite time collapse of Kähler-Ricci flow on threefolds.
method Analysis of solutions to Kähler-Ricci flow on compact threefolds.
result If the flow encounters a finite time collapsing singularity, the manifold admits a Fano fibration.
Finite-time extinction and smoothing effects in fractional fast diffusion on manifolds.
problem Finite-time extinction and smoothing effects in fractional fast diffusion equations.
method Nonlinear semigroups techniques, weighted Lp spaces, fractional Green function. result Sharp extinction rates and pointwise lower bounds for solutions.
Finite time for Ricci flow on certain manifolds.
problem Finite time of Ricci flow on specific manifolds.
method Analysis of homogeneous Ricci flows on noncompact manifolds.
result Finite extinction time for a family of homogeneous Ricci flows.
We investigate the limiting behavior of the unnormalized Kahler-Ricci flow on a Kahler manifold with a polarized initial Kahler metric. We prove that the Kahler-Ricci flow becomes extinct in finite time if and only if the manifold has positive first Chern class and the initial Kahler class is proportional to the first …
Study on spherical surfaces in complex plane evolving under mean curvature flow.
problem Finding conditions for Lagrangian spherical surfaces to become extinct in finite time.
method Evolution under mean curvature flow of embedded Lagrangian spherical surfaces in C2. result Condition for extinction and convergence to Clifford torus after rescaling.
Finite time for subsolutions on Riemannian manifolds proved.
problem Finite extinction time for subsolutions of a specific equation on Riemannian manifolds.
method Proved finite extinction time using weighted Sobolev inequality and assumptions on p, q, and ρ.
result Weak subsolutions to the equation have a finite extinction time.
The paper studies how smooth convex shapes in a ball evolve over time.
problem Evolution of smooth convex shapes in a ball by a specific curvature flow.
method Evolution by α-Gauss curvature flow, using conformal maps and entropy estimates. result The shapes become extinct in finite time and converge to a hemisphere.
Corrects a mistake in Ricci Flow's curve-shrinking flow equations.
problem Incorrect upper bound for total curvature evolution.
method Shows the multiplicative constant depends on initial total curvature and length.
result Finite-time extinction result remains unaffected.
The study examines the long-term behavior of mean curvature flows in closed 3-manifolds.
problem Understanding the long-term behavior of mean curvature flows in closed 3-manifolds.
method The approach involves constructing piecewise almost regular flows and applying perturbative arguments.
result The study constructs minimal surfaces in 3-manifolds via parabolic methods.
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…
This is an expository article with complete proofs intended for a general non-specialist audience. The results are two-fold. First, we discuss a geometric invariant, that we call the width, of a manifold and show how it can be realized as the sum of areas of minimal 2-spheres. For instance, when M is a homotopy 3-sph…
In this paper we will discuss how one may be able to use mean curvature flow to tackle some of the central problems in topology in 4-dimensions. We will be concerned with smooth closed 4-manifolds that can be smoothly embedded as a hypersurface in R^5. We begin with explaining why all closed smooth homotopy spheres can…
Chen's flow leads to finite-time singularities for closed submanifolds.
problem Understanding the finite-time singularities of Chen's fourth-order curvature flow.
method Investigates the flow's behavior, proving finite-time extinction and concentration of curvature.
result Chen's flow leads to finite-time singularities for closed submanifolds, characterized by concentration of curvature in specific dimensions.
New law predicts first extinction in resampling processes.
problem Intractable extinction times in resampling processes.
method Modeling multinomial updates as independent square-root diffusions.
result Closed-form law for first-extinction time with linear cost.
Ricci flows with almost maximal extinction time are nearly round.
problem Understanding the rigidity of Ricci flows with positive curvature.
method Analyzing the relationship between Ricci flows, their extinction times, and curvature properties.
result Ricci flows with almost maximal extinction time are nearly round.
Develops a method to disaggregate aerosol optical depth into vertical extinction profiles.
problem Uncertainty in measuring aerosol vertical distributions due to limited observations.
method Bayesian nonparametric Gaussian process modeling using meteorological predictors.
result Model reconstructs realistic extinction profiles with well-calibrated uncertainty, outperforming idealized baselines.
Study on Ricci flows of awesome homogeneous spaces, proving finite extinction time.
problem Understanding the long-time behavior of Ricci flows on homogeneous spaces.
method Analyzing Ricci flows on non-compact manifolds, focusing on finite extinction time.
result Ricci flows on non-contractible spaces have finite extinction time, confirming conjecture.
Study smooth solutions to fractional mean curvature flow, proving uniqueness and finite extinction time.
problem Understanding evolution of surfaces with fractional mean curvature.
method Established a comparison principle and evolutions equations for fractional geometric quantities.
result Proved uniqueness and finite extinction time for compact solutions.
Proves existence of mean curvature flow with surgery for free boundary surfaces.
problem Existence of mean curvature flow with free boundary.
method New approach for flows with surgery in free boundary setting.
result Flow converges to stable minimal surfaces without surgeries for large t.
We introduce a fractional Yamabe flow involving nonlocal conformally invariant operators on the conformal infinity of asymptotically hyperbolic manifolds, and show that on the conformal spheres $(\Sn, [g_{\Sn}])$, it converges to the standard sphere up to a Möbius diffeomorphism. This result allows us to obtain extinct…
This work addresses the {\em singularity formation} of complete non-compact solutions to the conformally flat Yamabe flow whose conformal factors have {\em cylindrical behavior at infinity}. Their singularity profiles happen to be {\em Yamabe solitons}, which are {\em self-similar solutions} to the fast diffusion equat…
Gaia will obtain astrometry and spectrophotometry for essentially all sources in the sky down to a broad band magnitude limit of G=20, an expected yield of 10^9 stars. Its main scientific objective is to reveal the formation and evolution of our Galaxy through chemo-dynamical analysis. In addition to inferring position…
This paper models stock prices using a Janardan Galton Watson process.
problem Modeling stock price fluctuations and predicting market trends.
method Extends Janardan Galton Watson process to model stock prices, considering initial close price and number of offspring.
result The model predicts return values and probability of market extinction.
Finite extinction time for Hermitian curvature flow on homogeneous spaces.
problem Finite extinction time of Hermitian curvature flow on compact homogeneous spaces.
method Proved finite extinction time and analyzed flow behavior.
result Finite extinction time T>0 for the flow. Ancient solutions found on flag manifolds from invariant Einstein metrics.
problem Understanding the behavior of Ricci flow on flag manifolds.
method Global study of the dynamical system induced by the Ricci flow, using invariant Einstein metrics and Poincaré compactification.
result Non-collapsed ancient solutions emerge from invariant Einstein metrics, with a Type I singularity in finite time.
Both theoretical and applied economics have a great deal to say about many aspects of the firm, but the literature on the extinctions, or demises, of firms is very sparse. We use a publicly available data base covering some 6 million firms in the US and show that the underlying statistical distribution which characteri…
The study examines the long-term behavior of a flow on Lie groups.
problem Understanding the long-time behavior of the pluriclosed flow on Lie groups.
method Analysis of left-invariant Hermitian structures on Lie groups, proving convergence and existence of solutions.
result Solutions on certain Lie groups converge to self-similar solutions, some of which are shrinking solitons.
Study shows flows with critical forcing term satisfy area change formula.
problem Analyzing flows with critical forcing terms and their area change.
method Identifying minimal conditions for Brakke flows to satisfy area change formula.
result Generalized BV flows with critical forcing terms have a lower bound for extinction time.
Essential principal components simplify spectral analysis with minimal training data.
problem Accurate spectral quantification from complex mixtures.
method Identifying essential principal components and using molar extinction coefficients.
result Near one-to-one projection from principal components to mixture constituents.
We consider radial solutions to the fast diffusion equation ut=Δum on the hyperbolic space HN for N≥2, m∈(ms,1), ms=N+2N−2. By radial we mean solutions depending only on the geodesic distance r from a given point o∈HN. We investigate their fine asymptotics near…
New continuous-time optimization algorithms converge in finite time to local minima.
problem Finding local minima in optimization problems.
method Discontinuous dynamical systems with finite-time convergence via Lyapunov-based differential inequality.
result Finite-time convergence to strict local minima with provable settling time.
Study finite time singularities in Ricci flow with bounded scalar curvature.
problem Understanding finite time singularities in Ricci flow with bounded scalar curvature.
method Analyzing blow-up sequences of locally Type I singularities.
result Every blow-up sequence of a locally Type I singularity has a specific property.
Continuity of second derivative in level set flow determined.
problem Regularity of level set flow solutions.
method Analysis of singular times and singular sets.
result Second derivative is continuous if and only if flow has a single singular time.
Finite-time degeneration of harmonic map flows on surfaces is proven under specific conditions.
problem Finite-time degeneration of harmonic map flows on surfaces.
method Proving finite-time degeneration under specific conditions of image stretching rate.
result Sharp conditions for finite-time degeneration of harmonic map flows are established.
Finite-time blow-up in Yang-Mills flow for small energy initial connections.
problem Finite-time blow-up of Yang-Mills flow solutions.
method Analyzing the Yang-Mills flow on Riemannian and Kähler manifolds.
result Finite-time blow-up occurs for small energy initial connections.
Estimates for VPMCF show ancient MCF solutions and finite-time behavior.
problem Volume Preserving Mean Curvature Flow (VPMCF) behavior and singularities.
method Nonlocal estimates and blowup analysis.
result Ancient solutions to MCF and finite-time behavior of VPMCF.
The paper proves a statement about surfaces diffeomorphic to annuli.
problem Proving a statement about surfaces diffeomorphic to annuli in Perelman's paper.
method Uses extrinsic techniques, co-area formula, and is potentially generalizable.
result Potential generalizability to higher dimensions.
The paper analyzes deep neural networks using control theory to set a time limit for their convergence.
problem Understanding the finite-time convergence of deep neural networks.
method Lyapunov based analysis of the loss function, control theory framework, finite-time control of non-linear systems.
result A priori guarantees of finite-time convergence for deep neural networks are provided.
RW-based learning is vulnerable to the Pac-Man attack, which eliminates active RWs.
problem Vulnerability of RW-based learning to malicious behavior.
method Proposed the Average Crossing (AC) algorithm to prevent RW extinction.
result RW-based stochastic gradient descent remains convergent under AC, even in the presence of Pac-Man.
No finite-time singularities in Yang-Mills flow in 4D.
problem Finite-time singularities in Yang-Mills flow.
method Weighted energy identity and sharp decay estimates.
result Long-time existence of Yang-Mills flow in 4D.
We give a bound on the extinction time for a compact, strictly convex hypersurface in R^{n+1} evolving by a geometric flow where the velocity is given in terms of the curvature. This result generalizes a theorem of Colding and Minicozzi for mean curvature flow solutions to a wider class of flows studied by Ben Andrews.…
We give concentration bounds for martingales that are uniform over finite times and extend classical Hoeffding and Bernstein inequalities. We also demonstrate our concentration bounds to be optimal with a matching anti-concentration inequality, proved using the same method. Together these constitute a finite-time versi…
In this paper, a geometric function is introduced to reflect the attenuation speed of impact of one firm's default to its partner. If two firms are competitions (copartners), the default intensity of one firm will decrease (increase) abruptly when the other firm defaults. As time goes on, the impact will decrease gradu…
Study shows Whitney sphere collapses to a point in finite time.
problem Understanding the evolution of Whitney sphere under mean curvature flow.
method Investigated equivariant Lagrangian spheres in \(\mathbb{C}^n\) using mean curvature flow.
result Equivariant Lagrangian spheres collapse to a point in finite time and converge to a plane with multiplicity two.
Study of deep neural networks using finite-time Lyapunov exponents.
problem Understanding the geometric structures in input space formed by deep neural networks.
method Analogy with dynamical systems, computing finite-time Lyapunov exponents.
result Ridges of large positive exponents divide input space into regions associated with different classes.
Study on SA with heavy-tailed and LRD noise, establishing finite-time bounds.
problem Analyzing stochastic approximation under heavy-tailed and LRD noise.
method Noise-averaging argument to regularize impact of non-classical noise.
result Established first finite-time moment bounds for SA under heavy-tailed and LRD noise.
In this paper we discuss a simple relation, which was previously missed, between the high co-dimensional isoperimetric problem of finding a filling with small volume to a given cycle, and extinction estimates for singular, high co-dimensional, mean curvature flow. The utility of this viewpoint is first exemplified by t…
First-order method solves stochastic bilevel optimization with linear constraints.
problem Stochastic bilevel optimization with linear constraints and noise.
method Developed a novel framework using gradient-based techniques and smoothed penalty functions.
result Achieved finite-time convergence guarantees for (δ,ε)-Goldstein stationary points.