We show that the approximation numbers of a compact composition operator on the weighted Bergman spaces Bα of the unit disk can tend to 0 arbitrarily slowly, but that they never tend quickly to 0: they grow at least exponentially, and this speed of convergence is only obtained for symbols which do not approach the unit circle. We also give an upper bounds and explicit an example. Mathematics Su...

It is proved that on any bounded domain in the complex Euclidean space C(n) the Bergman metric is always greater than or equal to the Carathéodory distance. This leads to a number of interesting consequences. Here two such consequences are given. (i) The Bergman metric is complete whenever the Carathéodory distance is complete on a bounded domain. (ii) The Weil-Petersson metric is not uniformly...

A property of the Bergman projection associated to a bounded circular domain containing the origin in C^ is proved: Functions which extend to be holomorphic in large neighborhoods of the origin are characterized as Bergman projections of smooth functions with small support near the origin. For certain circular domains D, it is also shown that functions which extend holomorphically to a neighbor...

Tropical varieties play an important role in algebraic geometry. The Bergman complex B(M) and the positive Bergman complex B+(M) of an oriented matroid M generalize to matroids the notions of the tropical variety and positive tropical variety associated to a linear ideal. Our main result is that if A is a Coxeter arrangement of type Φ with corresponding oriented matroid MΦ, then B (MΦ) is dual ...

This paper surveys a large class of nonlinear extremal problems in Hardy and Bergman spaces. We discuss the general approach to such problems in Hardy spaces developed by S. Ya. Khavinson in the 1960s, but not well known in the West. We also discuss the major difficulties distinguishing the Bergman space setting and formulate some open problems.

Thermal Bergman cyclization of Pt(II) dialkynylporphyrins reveals a marked reduction in the cyclization temperature relative to the free base and Zn(II) derivatives. In contrast, photogenerated (3)ππ* population produces no detectable Bergman photocyclization, suggesting that the photoreactivities of the related free base and Zn(II) derivatives occurs via the (1)ππ* state.

The boundary behavior of the Bergman Kernel function of some Reinhardt domains is studied. Upper and lower bounds for the Bergman kernel function are found at the diagonal points (z, z̄). Let D be the Reinhardt domain D = { z ∈ C | ‖z‖α = n ∑

We prove an exponential estimate for the asymptotics of Bergman kernels of a positive line bundle under hypotheses of bounded geometry. We give further Bergman kernel proofs of complex geometry results, such as separation of points, existence of local coordinates and holomorphic convexity by sections of positive line bundles.