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Resistance Conditions and Applications

Juha Kinnunen, Pilar Silvestre (2013)

Analysis and Geometry in Metric Spaces

This paper studies analytic aspects of so-called resistance conditions on metric measure spaces with a doubling measure. These conditions are weaker than the usually assumed Poincaré inequality, but however, they are sufficiently strong to imply several useful results in analysis on metric measure spaces. We show that under a perimeter resistance condition, the capacity of order one and the Hausdorff content of codimension one are comparable. Moreover, we have connections to the Sobolev inequality...

Resolvent conditions and powers of operators

Olavi Nevanlinna (2001)

Studia Mathematica

We discuss the relation between the growth of the resolvent near the unit circle and bounds for the powers of the operator. Resolvent conditions like those of Ritt and Kreiss are combined with growth conditions measuring the resolvent as a meromorphic function.

Restricted interpolation by meromorphic inner functions

Alexei Poltoratski, Rishika Rupam (2016)

Concrete Operators

Meromorphic Inner Functions (MIFs) on the upper half plane play an important role in applications to spectral problems for differential operators. In this paper, we survey some recent results concerning function theoretic properties of MIFs and show their connections with spectral problems for the Schrödinger operator.

Results on the deficiencies of some differential-difference polynomials of meromorphic functions

Xiu-Min Zheng, Hong-Yan Xu (2016)

Open Mathematics

In this paper, we study the relation between the deficiencies concerning a meromorphic function f(z), its derivative f′(z) and differential-difference monomials f(z)mf(z+c)f′(z), f(z+c)nf′(z), f(z)mf(z+c). The main results of this paper are listed as follows: Let f(z) be a meromorphic function of finite order satisfying lim sup r→+∞ T(r, f) T(r,  f ′ ) <+∞, lim sup r → + ∞ T ( r , f ) T ( r , f ' ) < + ∞ , and c be a non-zero complex constant, then δ(∞, f(z)m f(z+c)f′(z))≥δ(∞, f′) and δ(∞,f(z+c)nf′(z))≥ δ(∞, f′). We also investigate the value...

Résurgence paramétrique et exponentielle petitesse de l'écart des séparatrices du pendule rapidement forcé

David Sauzin (1995)

Annales de l'institut Fourier

Henri Poincaré avait déjà remarqué que les variétés stable et instable du pendule perturbé, défini par l’hamiltonien H ( q , p , t ) = p 2 / 2 + ( - 1 + cos q ) ( 1 - μ sin ( t / ϵ ) ) , ne coïncident pas lorsque que le paramètre μ n’est pas nul, mais qu’on peut leur associer un même développement formel divergent en puissance de ϵ . Cette divergence est ici analysée au moyen de la récente théorie de la résurgence, et du calcul étranger qui permet de trouver un équivalent asymptotique de l’écart des deux variétés pour ϵ tendant vers zéro - du moins cela est-il montré...

Riemann and Klein surfaces with nodes viewed as quotients.

Ignacio C. Garijo (2006)

Revista Matemática Complutense

If G is a group of automorphisms that acts properly discontinuously on a Riemann or Klein surface X, then there exists a unique structure of Riemann or Klein surface on X/G such that the projection π: X → X/G is a morphism. The analogous result is not true when we deal with surfaces with nodes. In this paper we give a new definition of a group that acts properly discontinuously on a surface with nodes in order to obtain a similar theorem.

Riemann problem on the double of a multiply connected circular region

V. V. Mityushev (1997)

Annales Polonici Mathematici

The Riemann problem has been solved in [9] for an arbitrary closed Riemann surface in terms of the principal functionals. This paper is devoted to solution of the problem only for the double of a multiply connected region and can be treated as complementary to [9,1]. We obtain a complete solution of the Riemann problem in that particular case. The solution is given in analytic form by a Poincaré series.

Riemann surfaces with boundary and natural triangulations of the Teichmüller space

Gabriele Mondello (2011)

Journal of the European Mathematical Society

We compare some natural triangulations of the Teichmüller space of hyperbolic surfaces with geodesic boundary and of some bordifications. We adapt Scannell–Wolf’s proof to show that grafting semi-infinite cylinders at the ends of hyperbolic surfaces with fixed boundary lengths is a homeomorphism. This way, we construct a family of equivariant triangulations of the Teichmüller space of punctured surfaces that interpolates between Bowditch–Epstein–Penner’s (using the spine construction) and Harer–Mumford–Thurston’s...

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