arXiv · 0709.1826
Some results on singularities of plurisubharmonic functions are put into the context of tropical mathematics.
Explore connections, maps & timelines
Alexander Rashkovskii. 2008-12-05. Tropical analysis of plurisubharmonic singularities. https://arxiv.org/abs/0709.1826
Cite the original work for its findings. Save a collection to share your selection of sources.
KEEP EXPLORING
The Chebyshev polynomials are utilized in this study to define the subclass of the bi-univalent function. Also, Chebyshev polynomial bounds and Fekete-Szego inequalities for functions defined in the classes are established.
A maximal plurisubharmonic function on an infinite-dimensional Hilbert space can have an injective, noncompact Levi operator at every point. This raises the question of what replaces the finite-dimensional determinant criterion for maximality. We prove that a twice continuously differentiable plurisubharmonic function is maximal if its Levi operator is compact at every point, without any boundedness assumption on the function. Using the normalized determinants of Fujii and Seo, we also characterize local maximality by the condition that the infimum of the spectrum of the Levi operator is zero. The same condition characterizes maximality when the function is bounded on the closure of every bounded open set whose closure lies in the domain. The compactness result follows from a more general spectral criterion. Its proof combines a bounded transformation with the smooth variational principle of Borwein and Preiss to establish comparison on bounded open sets whose closures need not be compact.
We study the vector Dyson equation $$-\frac{1}{m(z)}=z\mathbf{1}-\mathbf{a}+Sm(z),$$ with parameter $z$ in the complex upper half-plane $\mathbb{C}_+$, where $\mathbf{a}\in\mathbb R^d$ and $S$ is a nonnegative matrix, not necessarily symmetric. This equation has a unique vector solution $m(z)\in\mathbb{C}_+^d$, for which we establish a complete measure decomposition and prove regularity. We then develop a graph-theoretic approach to the singularity and stability problem for non-symmetric matrices $S$. The graph structure of $S$ identifies the possible degeneracies of the stability operator as $z$ approaches the real axis. In particular, for non-backtracking matrices, we prove square-root growth at regular edges, cubic-root growth at regular cusps, and complete stability estimates. We also obtain the corresponding estimates for symmetric matrices in the periodic setting.