Related papers: Critical structures of inner functions

In this survey paper, we discuss the problem of characterizing the critical sets of bounded analytic functions in the unit disk of the complex plane. This problem is closely related to the Berger-Nirenberg problem in differential geometry…

We establish an extension of Liouville's classical representation theorem for solutions of the partial differential equation $\Delta u=4 e^{2u}$ and combine this result with methods from nonlinear elliptic PDE to construct holomorphic maps…

A classical result due to Blaschke states that for every analytic self-map $f$ of the open unit disk of the complex plane there exists a Blaschke product $B$ such that the zero sets of $f$ and $B$ agree. In this paper we show that there is…

We present a numerical model for determining a finite Blaschke product of degree $n+1$ having $n$ preassigned distinct critical points $z_1,\dots,z_n$ in the complex (open) unit disk $\mathbb{D}$. The Blaschke product is uniquely determined…

Inner functions play a central role in function theory and operator theory on the Hardy space over the unit disk. Motivated by recent works of C. B\'en\'eteau et al. and of D. Seco, we discuss inner functions on more general weighted Hardy…

We consider critical points of a class of functionals on compact four-dimensional manifolds arising from Regularized Determinants for conformally covariant operators, whose explicit form was derived in [10], extending Polyakov's formula.…

We study the invariant subspaces generated by inner functions for a class of $\mathcal{P}^t(\mu)$-spaces which can be identified as spaces of analytic functions in the unit disk $\mathbb{D}$, where $\mu$ is a measure supported in the closed…

We give a generalization of the notion of finite Blaschke products from the perspective of generalized inner functions in various reproducing kernel Hilbert spaces. Further, we study precisely how these functions relate to the so-called…

Relative equilibria of Lagrangian and Hamiltonian systems with symmetry are critical points of appropriate scalar functions parametrized by the Lie algebra (or its dual) of the symmetry group. Setting aside the structures - symplectic,…

Over an algebraically closed field of positive characteristic, there exist rational functions with only one critical point. We give an elementary characterization of these functions in terms of their continued fraction expansions. Then we…

In this paper, we prove new existence and multiplicity results for critical points of lower semicontinuous functionals in Banach spaces, complementing the nonsmooth critical point theory set forth by Szulkin and avoiding the need of the…

In this paper, some existence results for sign-changing critical points of locally Lipschitz functionals in real Banach space are obtained by the method combining the invariant sets of descending ow method with a quantitative deformation.…

We construct a Riemannian metric on the $ 2 $-dimensional torus, such that for infinitely many eigenvalues of the Laplace-Beltrami operator, a corresponding eigenfunction has infinitely many isolated critical points. A minor modification of…

Eigenvalues in the essential spectrum of a weighted Sturm-Liouville operator are studied under the assumption that the weight function has one turning point. An abstract approach to the problem is given via a functional model for indefinite…

Recently the first author studied the bifurcation of critical points of families of functionals on a Hilbert space, which are parametrised by a compact and orientable manifold having a non-vanishing first integral cohomology group. We…

Motivated by the Invariant Subspace Problem, we describe explicitly the closed subspace $H^2$ generated by the limit points in the $H^2$ norm of the orbit of a thin Blaschke product $B$ under composition operators $C_\phi$ induced by…

Let $f=B_1/B_2$ be a ratio of finite Blaschke products having no critical points on $\partial\mathbb{D}$. Then $f$ has finitely many critical level curves (level curves containing critical points of $f$) in the disk, and the non-critical…

In this work, we introduce the notion of a two-Krein space and show that, starting from any classical Krein space, it is possible to construct spaces endowed with an indefinite two-inner product (admitting both positive and negative…

Finite-dimensional model spaces are quotient spaces of the Hardy space on the open unit disc, determined by finite Blaschke products. Composition operators, on the other hand, act by composing Hardy space functions with analytic self-maps…

Recent advancements in finite element methods allows for the implementation of mesh cells with curved edges. In the present work, we develop the tools necessary to employ multiply connected mesh cells, i.e. cells with holes, in planar…