Related papers: Parametric Coarea Inequality for 1-cycles

A famous result by R\"odl, Ruci\'nski, and Szemer\'edi guarantees a (tight) Hamilton cycle in $k$-uniform hypergraphs $H$ on $n$ vertices with minimum $(k-1)$-degree $\delta_{k-1}(H)\geq (1/2+o(1))n$, thereby extending Dirac's result from…

In \cite{reed97}, Reed conjectures that the inequality $\chi (G) \leq \left \lceil \textstyle {1/2} (\omega (G) + \Delta (G) + 1) \right \rceil$ holds for any graph $G$. We prove this holds for a graph $G$ if $\bar{G}$ is disconnected. From…

We prove a restriction isomorphism for Chow groups of zero-cycles with coefficients in Milnor K-theory for smooth projective schemes over excellent henselian discrete valuation rings. Furthermore, we study torsion subgroups of these groups…

In this article, we compute the precise value of the maximum likelihood degree of the Gaussian graphical model of a cycle, confirming a conjecture due to Drton, Sturmfels and Sullivant.

We give a short proof of a reverse isoperimetric inequality due to Y. Groman and J. P. Solomon.

By means of the mathematical analysis theory, inequality theory, mathematical induction and the dimension reduction method, under the proper hypotheses, we establish the following cyclic inequalities: \[\sum_{i=1}^{n}…

Recently, Ni and Pan proved a $q$-congruence on certain sums involving central $q$-binomial coefficients, which was conjectured by Guo. In this paper, we give a generalization of this $q$-congruence and confirm another $q$-congruence, also…

We prove a geometric criterion on a $\SL$-invariant ergodic probability measure on the moduli space of holomorphic abelian differentials on Riemann surfaces for the non-uniform hyperbolicity of the Kontsevich--Zorich cocycle on the real…

We establish a vanishing result for the $L_{q,p}$-cohomology ($q\ge p$) of a twisted cylinder, which is a generalization of a warped cylinder. The result is new even for warped cylinders. We base on the methods for proving the $(p,q)$…

In this paper we prove the following. Let $\Sigma$ be an $n$--dimensional closed hyperbolic manifold and let $g$ be a Riemannian metric on $\Sigma \times \mathbb{S}^1$. Given an upper bound on the volumes of unit balls in the Riemannian…

We study the geometry of the action of SL(2,R) on the projective line in order to present a new and simpler proof of the Herman-Avila-Bochi formula. This formula gives the average Lyapunov exponent of a class of 1-families of…

In an algebraic family of rational maps of $\mathbb{P}^1$, we show that, for almost every parameter for the trace of the bifurcation current of a marked critical value, the critical value is Collet-Eckmann. This extends previous results of…

We prove some new results about the spacing between neighboring zeros of paraorthogonal polynomials on the unit circle. Our methods also provide new proofs of some existing results. The main tool we will use is a formula for the phase of…

P\'{o}lya's conjecture on the eigenvalues of the Laplacian has been one of the core problems in spectral geometry. Building upon the recent breakthrough works on P\'{o}lya's conjecture for balls and annuli by Filonov, Levitin, Polterovich…

Let $X$ be the product of a surface satisfying $b_2=\rho$ and of a curve over a finite field. We study a strong form of the integral Tate conjecture for $1$-cycles on $X$. We generalize and give unconditional proofs of several results of…

The dual Kontsevich cycles in the double dual of associative graph homology correspond to polynomials in the Miller-Morita-Mumford classes in the integral cohomology of mapping class groups. I explain how the coefficients of these…

We prove the interior null-controllability of one-dimensional parabolic equations with time independent measurable coefficients.

Avila and Viana exhibit an explicit sufficient condition for the Lyapunov exponents of a linear cocycle over a Markov map to have multiplicity 1. Here, in terms of geometric perturbations, we prove that this sufficient criterion is generic…

We prove three inequalities relating some invariants of sets of matrices, such as the joint spectral radius. One of the inequalities, in which proof we use geometric invariant theory, has the generalized spectral radius theorem of Berger…

In this note we show the convergence of the fundamental solutions of the parabolic equations assuming the Cheeger-Gromov convergence of the underlying manifolds and the uniform $L^1$-bound of the solutions. We also prove a local integral…