Related papers: Weighted Lojasiewicz inequalities and regularity o…

In real algebraic geometry, Lojasiewicz's theorem asserts that any integral curve of the gradient flow of an analytic function that has an accumulation point has a unique limit. Lojasiewicz proved this result in the early 1960s as a…

This paper gives an essentially self-contained exposition (except for an appeal to the Lojasiewicz gradient inequality) of geometric invariant theory from a differential geometric viewpoint. Central ingredients are the moment-weight…

The classical Lojasiewicz inequality and its extensions for partial differential equation problems (Simon) and to o-minimal structures (Kurdyka) have a considerable impact on the analysis of gradient-like methods and related problems:…

Let $\Sigma$ be a compact oriented surface and $N$ a compact K\"ahler manifold with nonnegative holomorphic bisectional curvature. For a solution of harmonic map flow starting from an almost-holomorphic map $\Sigma \to N$ (in the energy…

The harmonic map energy of a map from a closed, constant-curvature surface to a closed target manifold can be seen as a functional on the space of maps and domain metrics. We consider the gradient flow for this energy. In the absence of…

The recent study by Waclawczyk et al. [J. Phys. A: Math. Theor. 50, 175501 (2017)] possesses three shortcomings: (i) The analysis misses a key aspect of the LMN equations which makes their Lie-group symmetry results incomplete. In…

We generalize the definition of topological entropy given by Adler, Konheim, and McAndrew (AKM) for piecewise continuous self-maps defined on a compact interval (pc-maps). For this notion of entropy, we prove that the properties of the…

We develop a dynamical method for proving the sharp Berezin-Li-Yau inequality. The approach is based on the volume-preserving mean curvature flow and a new monotonicity principle for the Riesz mean $R_\Lambda(\Omega_t)$. For convex domains…

We rigorously show that a large family of monotone quantities along the weak inverse mean curvature flow is the limit case of the corresponding ones along the level sets of $p$-capacitary potentials. Such monotone quantities include…

We show the equivalences of several notions of entropy, like a version of the topological entropy of the geodesic flow and the Minkowski dimension of the boundary, in metric spaces with convex geodesic bicombings satisfying a uniform…

For a real analytic complex vector field $L$ in an open set of $\mathbb{R}^2$, with local first integrals that are open maps, we attach a number $\mu \ge 1$ (obtained through Lojasiewicz inequalities) and show that the equation $Lu=f$ has…

In this paper, we give some {\L}ojasiewicz-type inequalities and a nonsmooth slope inequality on non-compact domains for continuous definable functions in an o-minimal structure. We also give a necessary and sufficicent condition for which…

We prove that the log-Brunn-Minkowski inequality (log-BMI) for the Lebesque measure in dimension $n$ would imply the log-BMI and, therefore, the B-conjecture for any log-concave density in dimension $n$. As a consequence, we prove the…

We study harmonic map sequences from surfaces to compact homogeneous spaces. For sequences developing a single bubble, we derive refined asymptotic expansions in the neck region and prove new obstruction relations among the leading…

Stochastic monotonicity is a well known partial order relation between probability measures defined on the same partially ordered set. Strassen Theorem establishes equivalence between stochastic monotonicity and the existence of a coupling…

We consider the inverse problem of determining the Lam\'{e} parameters and the density of a three-dimensional elastic body from the local time-harmonic Dirichlet-to-Neumann map. We prove uniqueness and Lipschitz stability of this inverse…

We prove a generalization of Dunham Jackson's famous approximation inequality to the case of compact sets in the complex plane admitting both upper and lower bounds for their Green's functions, i.e. the well known Holder Continuity Property…

We introduce a map which reproduces qualitatively many fundamental properties of the dynamics of heavy particles in fluid flows. These include a uniform rate of decrease of volume in phase space, a slow-manifold effective dynamics when the…

In this paper, we prove the uniqueness of solutions to the logarithmic Minkowski problem in $\mathbb{R}^3$ without symmetry condition, provided the density of the measure is close to $1$ in $C^{\alpha}$ norm. This result also implies the…

We prove the equivalence of ensembles for Bernoulli measures on $\mathbb{Z}$ conditioned on two conserved quantities under the situation that one of them is spatially inhomogeneous. For the proof, we extend the classical local limit theorem…