Related papers: The Poincar\'e problem in the dicritical case
Earlier, there were defined two generalized (``motivic'') versions of the Poincar\'e series of a collection of plane valuations on the algebra ${\mathcal O}_{{\mathbb C}^2,0}$ of germs of holomorphic functions in two variables. One of them…
We introduce and study the conical curvature-dimension condition, $CCD(K,N)$, for graphs. We show that $CCD(K,N)$ provides necessary and sufficient conditions for the underlying graph to satisfy a sharp global Poincar\'e inequality which in…
We prove a Poincar\'e, and a general Sobolev type inequalities for functions with compact support defined on a $k$-rectifiable varifold $V$ defined on a complete Riemannian manifold with positive injectivity radius and sectional curvature…
We continue the analysis of modular invariant functions, subject to inhomogeneous Laplace eigenvalue equations, that were determined in terms of Poincar\'e series in a companion paper. The source term of the Laplace equation is a product of…
A notion of Poincar\'e series was introduced by A.Campillo et al. It was developed by Delgado and Gusein-Zade for a multi-index filtration corresponding to the sequence of blow-ups. The present paper suggests the way to generalize the…
In this paper, we study the Poincar\'e compactification of the limiting planar semiflow of a coupled PDE-ODE system composed by a reaction-diffusion equation with large diffusion coupled with an ODE by a boundary condition in a heating…
We introduce excess logarithmic residues for one-dimensional holomorphic foliations tangent to a divisor. They arise from the comparison between the logarithmic normal sheaf and the ordinary normal sheaf of the foliation, and measure the…
Braid Floer homology is an invariant of proper relative braid classes. Closed integral curves of 1-periodic Hamiltonian vector fields on the 2-disc may be regarded as braids. If the Braid Floer homology of associated proper relative braid…
Earlier, there was computed the Poincar\'e series of a valuation or of a collection of valuations on the ring of germs of holomorphic functions in two variables. For a collection of several plane curve valuations it appeared to coincide…
We prove a fractional version of Poincar\'e inequalities in the context of $\R^n$ endowed with a fairly general measure. Namely we prove a control of an $L^2$ norm by a non local quantity, which plays the role of the gradient in the…
We discuss the occurrence of Poincar\'e-Andronov-Hopf bifurcations in parameter dependent ordinary differential equations, with no a priori assumptions on special coordinates. The first problem is to determine critical parameter values from…
We approach the problem of uniformization of general Riemann surfaces through consideration of the curvature equation, and in particular the problem of constructing Poincar\'e metrics (i.e., complete metrics of constant negative curvature)…
Let $V$ be a locally bounded measurable function such that $e^{-V}$ is bounded and belongs to $L^1(dx)$, and let $\mu_V(dx):=C_V e^{-V(x)} dx$ be a probability measure. We present the criterion for the weighted Poincar\'{e} inequality of…
We consider a family of holomorphic PDEs whose singular locus is given by the zero set of an analytic map $P$ with $P(0)=0$. Our goal is to establish conditions for the existence and uniqueness of formal power series solutions and to…
Tate's algorithm tells us that for an elliptic curve $E$ over a local field $K$ of residue characteristic $\geq 5$, $E/K$ has potentially good reduction if and only if $\text{ord}(j_E)\geq 0$. It also tells us that when $E/K$ is semistable…
To a plane curve singularity one associates a multi-index filtration on the ring of germs of functions of two variables defined by the orders of a function on irreducible components of the curve. The Poincare series of this filtration…
A domain $S\subset{\mathbb{R}}^d$ is said to fulfill the Poincar\'{e} cone property if any point in the boundary of $S$ is the vertex of a (finite) cone which does not otherwise intersects the closure $\bar{S}$. For more than a century,…
Using recent results on string on $AdS_{3}\times N^d$, where N is a d-dimensional compact manifold, we re-examine the derivation of the non trivial extension of the (1+2) dimensional-Poincar\'e algebra obtained by Rausch de Traubenberg and…
Poincar\'e profiles are a family of analytically defined coarse invariants, which can be used as obstructions to the existence of coarse embeddings between metric spaces. In this paper we calculate the Poincar\'e profiles of all connected…
We establish a strategy for finding sharp upper and lower numerical bounds of the Poincar\'e constant on a class of planar domains with piecewise self-similar boundary. The approach consists of four main components: W1) tight inner-outer…