arXiv research

Framework generates realistic crop images for growth modeling.

Minimal volume entropy vanishes or is positive under certain fiber growth conditions.

Paper improves Stein property for certain Kahler manifolds.

Sharp time analyticity proved for heat equation on Ricci solitons.

In this paper we study volume growth of gradient steady Ricci solitons. We show that if the potential function satisfies a uniform condition, then the soliton has at most Euclidean volume growth.

Climate volatility reduces economic growth, especially in poorer countries.

We prove that any gradient shrinking Ricci soliton has at most Euclidean volume growth. This improves a recent result of H.-D. Cao and D. Zhou by removing a condition on the growth of scalar curvature.

Study volume growth and asymptotic cones of nonnegative Ricci curvature manifolds.

Study Liouville theorems for harmonic maps along ancient super Ricci flows.

We prove flatness of complete Riemannian planes and cylinders without conjugate points under optimal conditions on the area growth.

This paper investigates the relevance of the No-Ponzi game condition for public debt (i.e. the public debt growth rate has to be lower than the real interest rate, a necessary assumption for Ricardian equivalence) and of the transversality condition for the GDP growth rate (i.e. the GDP growth rate has to be lower than…

Study shows fundamental groups of certain curved spaces are simple.

The paper improves quantization error estimates on Riemannian manifolds.

Simplifies analysis of hyperbolic distributions in demographic and economic research.

Ancient caloric functions on manifolds with polynomial growth are studied under volume doubling barrier.

Finite distortion maps cannot have compact branch sets under growth conditions.

For a complete Riemannian metric, a pointwise conformal transformation may lead to a complete or incomplete transformed Riemannian metric, depending on the behavior of the conformal factor. We establish conditions on the growth of the conformal factor towards the infinity of the Riemannian metric, such that the conform…

Bubbles are essential in certain economic models with high growth and low interest rates.

We investigate complete noncompact Ricci-flat manifolds which are not of maximal volume growth. We show that the manifolds with a curvature decay condition and a holonomy decay condition are asymptotic to torus fibrations over ALE spaces. In particular, we classify complete noncompact 4-dimensional hyperkäler manifold…

The study examines polynomial growth functions on gradient shrinking Ricci solitons.

Zipf's law states that the number of firms with size greater than S is inversely proportional to S. Most explanations start with Gibrat's rule of proportional growth but require additional constraints. We show that Gibrat's rule, at all firm levels, yields Zipf's law under a balance condition between the effective grow…

New insights into groups with uniform exponential growth.

Ricci flow shows PIC1 manifolds with maximal volume growth are like Euclidean space.

A general nonlinear logistic equation has been proposed to model long-time saturation in industrial growth. An integral solution of this equation has been derived for any arbitrary degree of nonlinearity. A time scale for the onset of nonlinear saturation in industrial growth can be estimated from an equipartition cond…

Since non-compact RCD(0, N) spaces have at least linear volume growth, we study noncompact RCD(0, N) spaces with linear volume growth in this paper. One of the main results is that the diameter of level sets of a Busemann function grow at most linearly on a noncompact RCD(0, N) space satisfying the linear volume growth…

This article proposes a new approximation scheme for quadratic-growth BSDEs in a Markovian setting by connecting a series of semi-analytic asymptotic expansions applied to short-time intervals. Although there remains a condition which needs to be checked a posteriori, one can avoid altogether time-consuming Monte Carlo…

Ancient solutions to heat equations on graphs are shown to be analytic in time.

In this paper, we study the topology of complete noncompact Riemannian manifolds with asymptotically nonnegative Ricci curvature and large volume growth. We prove that they have finite topological types under some curvature decay and volume growth conditions. We also generize it to the manifolds with $k$-th asymptotica…

New Morse lemma handles functions going to negative infinity.

The paper proves conditions for isoperimetric regions in curved spaces.

We prove that a complete noncompact Kähler manifold $M^{n}$of positive bisectional curvature satisfying suitable growth conditions is biholomorphic to a pseudoconvex domain of {\bf C}$^{n}$ and we show that the manifold is topologically {\bf R}$^{2n}$. In particular, when $M^{n}$ is a Kähler surface of positive bisecti…

Paper establishes a universal growth rate for smooth surrogate losses in classification.

New algorithms optimize private convex optimization with faster rates for functions with κ-growth.

Given that the terminal condition is of at most linear growth, it is well known that a Cauchy problem admits a unique classical solution when the coefficient multiplying the second derivative (i.e., the volatility) is also a function of at most linear growth. In this note, we give a condition on the volatility that is …

Mathematical properties of the historical GDP/cap distributions are discussed and explained. These distributions are frequently incorrectly interpreted and the Unified Growth Theory is an outstanding example of such common misconceptions. It is shown here that the fundamental postulates of this theory are contradicted …

Warning signs about the developing economic crisis in Greece were present in the growth rate of the Gross Domestic Product (GDP) and in the growth of the GDP well before the economic collapse. The growth rate was strongly unstable. On average, in less than 50 years, it decreased 10-folds but after reaching a low minimu…

Sharp Liouville theorem for minimal graphs on manifolds with nonnegative Ricci curvature.

Local smoothing of metrics with small curvature, removing Ricci curvature condition.

Ancient solutions of heat equation with exponential growth are analytic in time.

The paper proves manifold isometries for certain gradient Ricci solitons.

We survey all results concerning the topology of complete noncompact Riemannian manifolds with nonnegative Ricci curvature that have no additional conditions other than restrictions to the dimension, volume growth or diameter growth of the manifold. We will also present relevant examples and list open problems.

For transversely homogeneous foliations on compact manifolds whose global holonomy group has connected closure, it is shown that either all holonomy covers of the leaves have polynomial growth with degree bounded by a common constant, or all holonomy covers of the leaves have exponential growth. This is an extension of…

New approach uses satellite imagery to estimate economic growth.

A singularity theorem based on asymptotic volume growth

Investigates how growth and risk balance in populations, revealing a trade-off.

The paper explores geometry and positive scalar curvature on non-compact manifolds.

Study on existence of ground states on curved spaces with conditions on potential growth.

Analyticity of heat equation extended to Bakry-Émery Ricci curvature manifolds.