Search results for "Poincaré conjecture"
showing 10 items of 33 documents
Principal Poincar\'e Pontryagin Function associated to some families of Morse real polynomials
2014
It is known that the Principal Poincar\'e Pontryagin Function is generically an Abelian integral. We give a sufficient condition on monodromy to ensure that it is an Abelian integral also in non generic cases. In non generic cases it is an iterated integral. Uribe [17, 18] gives in a special case a precise description of the Principal Poincar\'e Pontryagin Function, an iterated integral of length at most 2, involving logarithmic functions with only one ramification at a point at infinity. We extend this result to some non isodromic families of real Morse polynomials.
Poincaré Week in Göttingen, 22–28 April 1909
2018
When Paul Wolfskehl died in 1906, his will established a prize for the first mathematician who could supply a proof of Fermat’s Last Theorem, or give a counterexample refuting it. The interest from this prize money was later used to bring world-renowned mathematicians to Gottingen to deliver a series of lectures. Hilbert was apparently very pleased with this arrangement, and once jested that the only thing that kept him from proving Fermat’s famous conjecture was the thought of killing the goose that laid these golden eggs.
Multiplicity of fixed points and growth of ε-neighborhoods of orbits
2012
We study the relationship between the multiplicity of a fixed point of a function g, and the dependence on epsilon of the length of epsilon-neighborhood of any orbit of g, tending to the fixed point. The relationship between these two notions was discovered before (Elezovic, Zubrinic, Zupanovic) in the differentiable case, and related to the box dimension of the orbit. Here, we generalize these results to non-differentiable cases introducing a new notion of critical Minkowski order. We study the space of functions having a development in a Chebyshev scale and use multiplicity with respect to this space of functions. With the new definition, we recover the relationship between multiplicity o…
Characterizations of Orlicz-Sobolev Spaces by Means of Generalized Orlicz-Poincaré Inequalities
2012
Let Φ be anN-function. We show that a functionu∈LΦ(ℝn)belongs to the Orlicz-Sobolev spaceW1,Φ(ℝn)if and only if it satisfies the (generalized) Φ-Poincaré inequality. Under more restrictive assumptions on Φ, an analog of the result holds in a general metric measure space setting.
Maximal function estimates and self-improvement results for Poincaré inequalities
2018
Our main result is an estimate for a sharp maximal function, which implies a Keith–Zhong type self-improvement property of Poincaré inequalities related to differentiable structures on metric measure spaces. As an application, we give structure independent representation for Sobolev norms and universality results for Sobolev spaces. peerReviewed
Quasihyperbolic boundary conditions and Poincaré domains
2002
We prove that a domain in ${\Bbb R}^n$ whose quasihyperbolic metric satisfies a logarithmic growth condition with coefficient $\beta\le 1$ is a (q,p)-\Poincare domain for all p and q satisfying $p\in[1,\infty)\cap(n-n\beta,n)$ and $q\in[p,\beta p^*)$ , where $p^*=np/(n-p)$ denotes the Sobolev conjugate exponent. An elementary example shows that the given ranges for p and q are sharp. The proof makes use of estimates for a variational capacity. When p=2 we give an application to the solvability of the Neumann problem on domains with irregular boundaries. We also discuss the relationship between this growth condition on the quasihyperbolic metric and the s-John condition.
Poincaré inequalities and Steiner symmetrization
1996
A complete geometric characterization for a general Steiner symmetric domain Ω ⊂ Rn to satisfy the Poincare inequality with exponent p > n−1 is obtained and it is shown that this range of exponents is best possible. In the case where the Steiner symmetric domain is determined by revolving the graph of a Lipschitz continuous function, it is shown that the preceding characterization works for all p > 1 and furthermore for such domains a geometric characterization for a more general Sobolev–Poincare inequality to hold is given. Although the operation of Steiner symmetrization need not always preserve a Poincare inequality, a general class of domains is given for which Poincare inequalities are…
Klein, Mittag-Leffler, and the Klein-Poincaré Correspondence of 1881–1882
2018
If a modern-day Plutarch were to set out to write the “Parallel lives” of some famous modern-day mathematicians, he could hardly do better than to begin with the German, Felix Klein (1849–1925), and the Swede, Gosta Mittag-Leffler (1846–1927). Both lived in an age ripe with possibilities for the mathematics profession and, like few of their contemporaries, they seized upon these new opportunities whenever and however they arose. Even when their chances for success looked dismal, they forged ahead, winning over the skeptics as they did so. Although accomplished and prolific researchers (Klein’s work has even enjoyed the appellation “great”), they owed much of their success to their talents a…
The Poincar\'e-Cartan Form in Superfield Theory
2018
An intrinsic description of the Hamilton-Cartan formalism for first-order Berezinian variational problems determined by a submersion of supermanifolds is given. This is achieved by studying the associated higher-order graded variational problem through the Poincar\'e-Cartan form. Noether theorem and examples from superfield theory and supermechanics are also discussed.
The equality case in a Poincaré–Wirtinger type inequality
2016
It is known that, for any convex planar set W, the first non-trivial Neumann eigenvalue μ1 (Ω) of the Hermite operator is greater than or equal to 1. Under the additional assumption that Ω is contained in a strip, we show that β1 (Ω) = 1 if and only if Ω is any strip. The study of the equality case requires, among other things, an asymptotic analysis of the eigenvalues of the Hermite operator in thin domains.