arXiv research

Study on Ricci flows of awesome homogeneous spaces, proving finite extinction time.

Study smooth solutions to fractional mean curvature flow, proving uniqueness and finite extinction time.

Finite time for subsolutions on Riemannian manifolds proved.

Kähler-Ricci flow on threefolds collapses in finite time.

Finite time for Ricci flow on certain manifolds.

Finite extinction time for Hermitian curvature flow on homogeneous spaces.

Finite-time extinction and smoothing effects in fractional fast diffusion on manifolds.

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 …

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…

Study on spherical surfaces in complex plane evolving under mean curvature flow.

The paper studies how smooth convex shapes in a ball evolve over time.

New law predicts first extinction in resampling processes.

Ricci flows with almost maximal extinction time are nearly round.

The study examines the long-term behavior of a flow on Lie groups.

Corrects a mistake in Ricci Flow's curve-shrinking flow equations.

The study examines the long-term behavior of mean curvature flows in closed 3-manifolds.

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…

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…

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 paper models stock prices using a Janardan Galton Watson process.

The paper proves a statement about surfaces diffeomorphic to annuli.

Develops a method to disaggregate aerosol optical depth into vertical extinction profiles.

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…

Bing and Moise proved, independently, that any Peano continuum admits a length metric d. We treat non-degenerate Peano continua with a length metric as evolution systems instead of stationary objects. For any compact length space (X, d) we consider a semiflow in the hyperspace $2^X$ of all non-empty closed sets in X. T…

Study shows flows with critical forcing term satisfy area change formula.

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…

Proves existence of mean curvature flow with surgery for free boundary surfaces.

Let $g=(g_{ij})$ be a complete Riemmanian metric on $\R^2$ with finite total area and $I_g=\inf_γI(γ)$ with $I(γ)=L(γ)(A_{in}(γ)^{-1}+A_{out}(γ)^{-1})$ where $γ$ is any closed simple curve in $\R^2$, $L(γ)$ is the length of $γ$, $A_{in}(γ)$ and $A_{out}(γ)$ are the area of the regions inside and outside $γ$ respectivel…

Chen's flow leads to finite-time singularities for closed submanifolds.

We consider radial solutions to the fast diffusion equation $u_t=Δu^m$ on the hyperbolic space $\mathbb{H}^{N}$ for $N \ge 2$, $m\in(m_s,1)$, $m_s=\frac{N-2}{N+2}$. By radial we mean solutions depending only on the geodesic distance $r$ from a given point $o \in \mathbb{H}^N$. We investigate their fine asymptotics near…

Continuity of second derivative in level set flow determined.

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…

Uniform curvature estimates for homogeneous Ricci flows proved.

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…

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.…

Ancient solutions found on flag manifolds from invariant Einstein metrics.

Essential principal components simplify spectral analysis with minimal training data.

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…

The World Trade Web (WTW) is a weighted network whose nodes correspond to countries with edge weights reflecting the value of imports and/or exports between countries. In this paper we introduce to this macroeconomic system the notion of extinction analysis, a technique often used in the analysis of ecosystems, for the…

We consider the initial value problem $u_t = Δ\log u$, $u(x,0) = u_0(x)\ge 0$ in $\R^2$, corresponding to the Ricci flow, namely conformal evolution of the metric $u (dx_1^2 + dx_2^2)$ by Ricci curvature. It is well known that the maximal (complete) solution $u$ vanishes identically after time $T= \frac 1{4π} \int_{\R^…

RW-based learning is vulnerable to the Pac-Man attack, which eliminates active RWs.

Motivated by the scarcity of accurate payoff feedback in practical applications of game theory, we examine a class of learning dynamics where players adjust their choices based on past payoff observations that are subject to noise and random disturbances. First, in the single-player case (corresponding to an agent tryi…

This paper analyzes complex equilibria in a networked bivirus epidemic model.

Target Date Funds may not adapt well, leading to poor performance.

I introduce an algorithm for estimating parameters from multidimensional data based on forward modelling. In contrast to many machine learning approaches it avoids fitting an inverse model and the problems associated with this. The algorithm makes explicit use of the sensitivities of the data to the parameters, with th…

We investigate an economic system in which one large agent - the Japan government changes the environment of numerous smaller agents - the Japan agriculture producers by indirect regulation of prices of agriculture goods. The reason for this intervention was that before the oil crisis in 1974 Japan agriculture producti…

Systems with long-range persistence and memory are shown to exhibit different precursory as well as recovery patterns in response to shocks of exogeneous versus endogeneous origins. By endogeneous, we envision either fluctuations resulting from an underlying chaotic dynamics or from a stochastic forcing origin which ma…

Given a Riemannian metric on a homotopy $n$-sphere, sweep it out by a continuous one-parameter family of closed curves starting and ending at point curves. Pull the sweepout tight by, in a continuous way, pulling each curve as tight as possible yet preserving the sweepout. We show: Each curve in the tightened sweepout …