Closed Unit Disk Research Articles

Blaschke product for bordered surfaces

Any ramified holomorphic covering of a closed unit disc by another such a disc is given by a finite Blaschke product. The inverse is also true. In this note we give two explicit constructions for a holomorphic ramified covering of a disc by other bordered Riemann surface. The machinery used here strongly resembles the description of magnetic configurations in submicron planar magnets.

Cyclicity in Dirichlet Spaces

AbstractLet $\unicode[STIX]{x1D707}$ be a positive finite Borel measure on the unit circle and ${\mathcal{D}}(\unicode[STIX]{x1D707})$ the associated harmonically weighted Dirichlet space. In this paper we show that for each closed subset $E$ of the unit circle with zero $c_{\unicode[STIX]{x1D707}}$ -capacity, there exists a function $f\in {\mathcal{D}}(\unicode[STIX]{x1D707})$ such that $f$ is cyclic (i.e., $\{pf:p\text{ is a polynomial}\}$ is dense in ${\mathcal{D}}(\unicode[STIX]{x1D707})$ ), $f$ vanishes on $E$ , and $f$ is uniformly continuous. Next, we provide a sufficient condition for a continuous function on the closed unit disk to be cyclic in ${\mathcal{D}}(\unicode[STIX]{x1D707})$ .

A general method for constructing essential uniform algebras

A general method for constructing essential uniform algebras with prescribed properties is presented. Using the method, the following examples are constructed: an essential, natural, regular uniform algebra on the closed unit disc; an essential, natural c

Some results for strong differential subordination of analytic functions defined by differential operator

The purpose of this paper is to introduce some results associated with the strong differential subordinations of analytic functions defined in the open unit disk and closed unit disk of the complex plane.

Semiglobal state synchronization for continuous‐ or discrete‐time multiagent systems subject to actuator saturation

SummaryThis paper studies the semiglobal state synchronization of homogeneous networks with diffusive full‐state coupling or partial‐state coupling in the presence of input saturation. Both continuous‐ and discrete‐time agents are considered. We assume that agents are, at most, critically unstable, that is, the agents either have all its eigenvalues in the closed left‐half complex plane for continuous‐time systems or have all its eigenvalues in the closed unit disc for discrete‐time systems. The communication network is associated with a graph containing a directed spanning tree. In this paper, we propose a linear static protocol for multiagent systems with full‐state coupling and a linear dynamic protocol for multiagent systems with partial‐state coupling based on a low‐gain methodology, such that state synchronization is achieved among agents for any initial conditions in an a priori given compact set.

Fundamental group and analytic disks

Let $W$ be a domain in a connected complex manifold $M$ and $w_0\in W$. Let ${\mathcal A}_{w_0}(W,M)$ be the space of all continuous mappings of a closed unit disk $\overline D$ into $M$ that are holomorphic on the interior of $\overline D$, $f(\partial\mathbb D)\subset W$ and $f(1)=w_0$. On the homotopic equivalence classes $\eta_1(W,M,w_0)$ of ${\mathcal A}_{w_0}(W,M)$ we introduce a binary operation $\star$ so that $\eta_1(W,M,w_0)$ becomes a semigroup and the natural mappings $\iota_1:\,\eta_1(W,M,w_0)\to\pi_1(W,w_0)$ and $\delta_1:\,\eta_1(W,M,w_0)\to\pi_2(M,W,w_0)$ are homomorphisms. \par We show that if $W$ is a complement of an analytic variety in $M$ and if $S=\delta_1(\eta_1(W,M,w_0))$, then $S\cap S^{-1}=\{e\}$ and any element $a\in\pi_2(M,W,w_0)$ can be represented as $a=bc^{-1}=d^{-1}g$, where $b,c,d,g\in S$. \par Let ${\mathcal R}_{w_0}(W,M)$ be the space of all continuous mappings of $\overline D$ into $M$ such that $f(\partial{\mathbb D})\subset W$ and $f(1)=w_0$. We describe its open dense subset ${\mathcal R}^{\pm}_{w_0}(W,M)$ such that any connected component of ${\mathcal R}^{\pm}_{w_0}(W,M)$ contains at most one connected component of ${\mathcal A}_{w_0}(W,M)$.

A non-Archimedean Ohsawa–Takegoshi extension theorem

We prove an Ohsawa–Takegoshi-type extension theorem on the Berkovich closed unit disc over certain non-Archimedean fields. As an application, we establish a non-Archimedean analogue of Demailly’s regularization theorem for quasisubharmonic functions on the Berkovich unit disc.

How Many Eigenvalues of a Product of Truncated Orthogonal Matrices are Real?

ABSTRACTA truncation of a Haar distributed orthogonal random matrix gives rise to a matrix whose eigenvalues are either real or complex conjugate pairs, and are supported within the closed unit disk. This is also true for a product Pm of m independent truncated orthogonal random matrices. One of most basic questions for such asymmetric matrices is to ask for the number of real eigenvalues. In this paper, we will exploit the fact that the eigenvalues of Pm form a Pfaffian point process to obtain an explicit determinant expression for the probability of finding any given number of real eigenvalues. We will see that if the truncation removes an even number of rows and columns from the original Haar distributed orthogonal matrix, then these probabilities will be rational numbers. Finally, based on exact finite formulas, we will provide conjectural expressions for the asymptotic form of the spectral density and the average number of real eigenvalues as the matrix dimension tends to infinity.

Synchronization in a network of identical continuous‐ or discrete‐time agents with unknown nonuniform constant input delay

SummaryThis paper studies the synchronization problem for multiagent systems with identical continuous‐ or discrete‐time agents with unknown nonuniform constant input delays. The agents are connected via full‐ or partial‐state coupling. The agents are assumed to be asymptotically null controllable, ie, all eigenvalues are in the closed left‐half complex plane for continuous‐time agents or in the closed unit disc for discrete‐time agents. We derive an upper bound for the input delay tolerance, which explicitly depends on the agent dynamics. Moreover, for any unknown delay satisfying this upper bound, a low‐gain–based protocol design methodology is proposed without relying on exact knowledge of the network topology such that synchronization is achieved among agents for any network graph in a given set.

Schoenberg’s theorem for real and complex Hilbert spheres revisited

Schoenberg’s theorem for the complex Hilbert sphere proved by Christensen and Ressel in 1982 by Choquet theory is extended to the following result: Let L denote a locally compact group and let D¯ denote the closed unit disc in the complex plane. Continuous functions f:D¯×L→C such that f(ξ⋅η,u−1v) is a positive definite kernel on the product of the unit sphere in ℓ2(C) and L are characterized as the functions with a uniformly convergent expansion f(z,u)=∑m,n=0∞φm,n(u)zmz¯n,where φm,n is a double sequence of continuous positive definite functions on L such that ∑φm,n(eL)<∞ (eL is the neutral element of L). It is shown how the coefficient functions φm,n are obtained as limits from expansions for positive definite functions on finite dimensional complex spheres via a Rodrigues formula for disc polynomials.Similar results are obtained for the real Hilbert sphere.

THE ESSENTIAL NORMS OF COMPOSITION OPERATORS ON WEIGHTED DIRICHLET SPACES

Let $\unicode[STIX]{x1D711}$ be an analytic self-map of the unit disc. If $\unicode[STIX]{x1D711}$ is analytic in a neighbourhood of the closed unit disc, we give a precise formula for the essential norm of the composition operator $C_{\unicode[STIX]{x1D711}}$ on the weighted Dirichlet spaces ${\mathcal{D}}_{\unicode[STIX]{x1D6FC}}$ for $\unicode[STIX]{x1D6FC}&gt;0$. We also show that, for a univalent analytic self-map $\unicode[STIX]{x1D711}$ of $\mathbb{D}$, if $\unicode[STIX]{x1D711}$ has an angular derivative at some point of $\unicode[STIX]{x2202}\mathbb{D}$, then the essential norm of $C_{\unicode[STIX]{x1D711}}$ on the Dirichlet space is equal to one.

Asymptotics for polynomials orthogonal in an indefinite metric

We continue studying polynomials generated by the Szegő recursion when a finite number of Verblunsky coefficients lie outside the closed unit disk. We prove some asymptotic results for the corresponding orthogonal polynomials and then translate them to the real line to obtain the Szegő asymptotics for the resulting polynomials. The latter polynomials give rise to a non-symmetric tridiagonal matrix but it is a finite-rank perturbation of a symmetric Jacobi matrix.

A note on the differential operator on generalized Fock spaces

It has long been known that the differential operator D represents a typical example of unbounded operators on many Banach spaces including the classical Fock spaces, the Fock–Sobolev spaces, and the generalized Fock spaces where the weight decays faster than the Gaussian weight. In this note we identify Fock type spaces where the operator admits boundedness, compactness and membership in the Schatten Sp class spectral structures. We also showed that its nontrivial spectrum while acting on such spaces is precisely the closed unit disk D‾ in the complex plane.

Solvability conditions and design for synchronization of discrete‐time multiagent systems

SummaryThis paper provides solvability conditions for state synchronization with homogeneous discrete‐time multiagent systems with a directed and weighted communication network under partial‐ or full‐state coupling. Our solvability conditions reveal that the synchronization problem is solvable for all possible, a priori given, set of graphs associated with a communication network only under the condition that the agents are at most weakly unstable (ie, agents have all eigenvalues in the closed unit disc). However, if an upper bound on the eigenvalues inside the unit disc of the row stochastic matrices associated with any graph in a given set of graphs is known, then one can achieve synchronization for a class of unstable agents. We provide protocol design for at most weakly unstable agents based on a direct eigenstructure assignment method and a standard H2 discrete‐time algebraic Riccati equation. We also provide protocol design for strictly unstable agents (ie, agents have at least one eigenvalue outside the unit disc) based on the standard H2 discrete‐time algebraic Riccati equation.

Holomorphic isometries between products of complex unit balls

We first give an exposition on holomorphic isometries from the Poincaré disk to polydisks and from the Poincaré disk to the product of the Poincaré disk with a complex unit ball. As an application, we provide an example of proper holomorphic map from the unit disk to the complex unit ball that is irrational, algebraic and holomorphic on a neighborhood of the closed unit disk. We also include some new results on holomorphic isometries.

Combinatorial models for spaces of cubic polynomials

W. Thurston constructed a combinatorial model of the Mandelbrot set M2 such that there is a continuous and monotone projection of M2 to this model. We propose the following related model for the space MD3 of critically marked cubic polynomials with connected Julia set and all cycles repelling. If (P,c1,c2)∈MD3, then every point z in the Julia set of the polynomial P defines a unique maximal finite set Az of angles on the circle corresponding to the rays, whose impressions form a continuum containing z. Let G(z) denote the convex hull of Az. The convex sets G(z) partition the closed unit disk. For (P,c1,c2)∈MD3 let c1⁎ be the co-critical point ofc1. We tag the marked dendritic polynomial (P,c1,c2) with the set G(c1⁎)×G(P(c2))⊂D‾×D‾. Tags are pairwise disjoint; denote by MD3comb their collection, equipped with the quotient topology. We show that tagging defines a continuous map from MD3 to MD3comb so that MD3comb serves as a model for MD3.

Tests for complete K-spectral sets

Let Φ be a family of functions analytic in some neighborhood of a complex domain Ω, and let T be a Hilbert space operator whose spectrum is contained in Ω‾. Our typical result shows that under some extra conditions, if the closed unit disc is complete K′-spectral for φ(T) for every φ∈Φ, then Ω‾ is complete K-spectral for T for some constant K. In particular, we prove that under a geometric transversality condition, the intersection of finitely many K′-spectral sets for T is again K-spectral for some K≥K′. These theorems generalize and complement results by Mascioni, Stessin, Stampfli, Badea–Beckermann–Crouzeix and others. We also extend to non-convex domains a result by Putinar and Sandberg on the existence of a skew dilation of T to a normal operator with spectrum in ∂Ω. As a key tool, we use the results from our previous paper [11] on traces of analytic uniform algebras.

Compact Brownian surfaces I: Brownian disks

We show that, under certain natural assumptions, large random plane bipartite maps with a boundary converge after rescaling to a one-parameter family (BD_L, 0 < L < infinity) of random metric spaces homeomorphic to the closed unit disk of R^2, the space BD_L being called the Brownian disk of perimeter L and unit area. These results can be seen as an extension of the convergence of uniform plane quadrangulations to the Brownian map, which intuitively corresponds to the limit case where L = 0. Similar results are obtained for maps following a Boltzmann distribution, in which the perimeter is fixed but the area is random.

Mean action and the Calabi invariant

Given an area-preserving diffeomorphism of the closed unit disk which is a rotation near the boundary, one can naturally define an ``action'' function on the disk which agrees with the rotation number on the boundary. The Calabi invariant of the diffeomorphism is the average of the action function over the disk. Given a periodic orbit of the diffeomorphism, its ``mean action'' is defined to be the average of the action function over the orbit. We show that if the Calabi invariant is less than the boundary rotation number, then the infimum over periodic orbits of the mean action is less than or equal to the Calabi invariant. The proof uses a new filtration on embedded contact homology determined by a transverse knot, which may be of independent interest. (An analogue of this filtration can be defined for any other version of contact homology in three dimensions that counts holomorphic curves.)

Finite Blaschke products and the construction of rational Γ-inner functions

LetΓ=def{(z+w,zw):|z|≤1,|w|≤1}⊂C2. A Γ-inner function is a holomorphic map h from the unit disc D to Γ whose boundary values at almost all points of the unit circle T belong to the distinguished boundary bΓ of Γ. A rational Γ-inner function h induces a continuous map h|T from T to bΓ. The latter set is topologically a Möbius band and so has fundamental group Z. The degree of h is defined to be the topological degree of h|T. In a previous paper the authors showed that if h=(s,p) is a rational Γ-inner function of degree n then s2−4p has exactly n zeros in the closed unit disc D−, counted with an appropriate notion of multiplicity. In this paper, with the aid of a solution of an interpolation problem for finite Blaschke products, we explicitly construct the rational Γ-inner functions of degree n with the n zeros of s2−4p prescribed.