Explanipedia

Function theory on the annulus in the dp-norm Open

Jim Agler, Zinaida A. Lykova, N. J. Young · 2025

Function theory in the bfd-norm on an elliptical region Open

Jim Agler, Zinaida A. Lykova, N. J. Young · 2024

Let E be the open region in the complex plane bounded by an ellipse. The B. and F. Delyon norm ‖⋅‖bfd on the space Hol(E) of holomorphic functions on E is defined by‖f‖bfd=defsupT∈Fbfd(E)⁡‖f(T)‖, where Fbfd(E) is the class of operators T s…

Function theory in the bfd-norm on an elliptical region Open

Jim Agler, Zinaida A. Lykova, N. J. Young · 2024

Let $E$ be the open region in the complex plane bounded by an ellipse. The B. and F. Delyon norm $\|\cdot\|_{\mathrm{bfd}}$ on the space $\mathrm{Hol}(E)$ of holomorphic functions on $E$ is defined by $$ \|f\|_{\mathrm{bfd}} \stackrel{\rm …

On the operators with numerical range in an ellipse Open

Jim Agler, Zinaida A. Lykova, N. J. Young · 2024

We give new necessary and sufficient conditions for the numerical range W(T) of an operator T∈B(H) to be a subset of the closed elliptical set Kδ⊆C given byKδ=def{x+iy:x2(1+δ)2+y2(1−δ)2≤1}, where 0<δ<1. Here B(H) denotes the collection of …

A Hilbert space approach to singularities of functions Open

Jim Agler, Zinaida A. Lykova, N. J. Young · 2022

We introduce the notion of a pseudomultiplier of a Hilbert space H of functions on a set Ω. Roughly, a pseudomultiplier of H is a function which multiplies a finite-codimensional subspace of H into H, where we allow the possibility that a …

Why physics instructors choose to include computation in their courses Open

N. J. Young · 2022

Exterior products of operators and superoptimal analytic approximation Open

Dimitrios Chiotis, Zinaida A. Lykova, N. J. Young · 2021

We give a new algorithm for the construction of the unique superoptimal analytic approximant of a given continuous matrix-valued function on the unit circle, making use of exterior powers of operators in preference to spectral or {\em Wien…

Exterior Powers and Pointwise Creation Operators Open

Dimitrios Chiotis, Zinaida A. Lykova, N. J. Young · 2021

We develop a theory of pointwise wedge products of vector-valued functions on the circle and the disc, and obtain results which give rise to a new approach to the analysis of the matricial Nehari problem. We investigate properties of point…

Intrinsic Directions, Orthogonality, and Distinguished Geodesics in the Symmetrized Bidisc Open

Jim Agler, Zinaida A. Lykova, N. J. Young · 2021

The symmetrized bidisc $$\begin{aligned} G {\mathop {=}\limits ^\mathrm{{def}}}\{(z+w,zw):|z|<1,\quad |w|<1\}, \end{aligned}$$ under the Carathéodory metric, is a complex Finsler space of cohomogeneity 1 …

Intrinsic Directions, Orthogonality and Distinguished Geodesics in the Symmetrized Bidisc Open

Jim Agler, Zinaida A. Lykova, N. J. Young · 2020

The symmetrized bidisc \[ G \stackrel{\rm{def}}{=}\{(z+w,zw):|z|<1,\ |w|<1\}, \] under the Carathéodory metric, is a complex Finsler space of cohomogeneity $1$ in which the geodesics, both real and complex, enjoy a rich geometry. As a Fins…

Exterior powers and pointwise creation operators Open

Dimitrios Chiotis, Zinaida A. Lykova, N. J. Young · 2020

We develop a theory of pointwise wedge products of vector-valued functions on the circle and the disc, and obtain results which give rise to a new approach to the analysis of the matricial Nehari problem. We investigate properties of point…

Geodesics, Retracts, and the Norm-Preserving Extension Property in the Symmetrized Bidisc Open

Jim Agler, Zinaida A. Lykova, N. J. Young · 2019

A set $V$ in a domain $U$ in $\mathbb{C}^n$ has the {\em norm-preserving extension property} if every bounded holomorphic function on $V$ has a holomorphic extension to $U$ with the same supremum norm. We prove that an algebraic subset of …

A geometric characterization of the symmetrized bidisc Open

Jim Agler, Zinaida A. Lykova, N. J. Young · 2019

Non‐commutative manifolds, the free square root and symmetric functions in two non‐commuting variables Open

Jim Agler, John E. McCarthy, N. J. Young · 2018

The richly developed theory of complex manifolds plays important roles in our\nunderstanding of holomorphic functions in several complex variables. It is\nnatural to consider manifolds that will play similar roles in the theory of\nholomor…

Characterizations of Some Domains via Carathéodory Extremals Open

Jim Agler, Zinaida A. Lykova, N. J. Young · 2018

In this paper we characterize the unit disc, the bidisc and the symmetrized bidisc $$\begin{aligned} G =\{(z+w,zw):|z|<1,\ |w|<1\} \end{aligned}$$in terms of the possession of small classes of analytic maps into the unit disc that suffice …

Analytic interpolation into the tetrablock and a $\mu$-synthesis problem Open

Zinaida A. Lykova, N. J. Young, A. Ajibo · 2018

We give a solvability criterion for a special case of the $\\mu$-synthesis\nproblem. That is, we prove the necessity and sufficiency of a condition for the\nexistence of an analytic $2 \\times 2$ matrix-valued function on the disc\nsubject…

Analytic interpolation into the tetrablock and a $μ$-synthesis problem Open

Zinaida A. Lykova, N. J. Young, A. Ajibo · 2018

We give a solvability criterion for a special case of the $μ$-synthesis problem. That is, we prove the necessity and sufficiency of a condition for the existence of an analytic $2 \times 2$ matrix-valued function on the disc subject to a b…

Characterizations of some domains via Carathéodory extremals Open

Jim Agler, Zinaida A. Lykova, N. J. Young · 2018

In this paper we characterize the unit disc, the bidisc and the symmetrized bidisc \[ G =\{(z+w,zw):|z|<1,\ |w|<1\} \] in terms of the possession of small classes of analytic maps into the unit disc that suffice to solve all Carathéodory e…

Algebraic and geometric aspects of rational Γ-inner functions Open

Jim Agler, Zinaida A. Lykova, N. J. Young · 2018

The setG=def{(z+w,zw):|z|<1,|w|<1}⊂C2 has intriguing complex-geometric properties; it has a 3-parameter group of automorphisms, its distinguished boundary is a ruled surface homeomorphic to the Möbius band and it has a special subvariety w…

Possibilistic Fuzzy Local Information C-Means for Sonar Image\n Segmentation Open

Alina Zare, N. J. Young, Daniel Suen, Thomas Nabelek, Aquila Galusha , et al. · 2017

Side-look synthetic aperture sonar (SAS) can produce very high quality images\nof the sea-floor. When viewing this imagery, a human observer can often easily\nidentify various sea-floor textures such as sand ripple, hard-packed sand, sea\n…

Realization of functions on the symmetrized bidisc Open

Jim Agler, N. J. Young · 2017

We prove a realization formula and a model formula for analytic functions with modulus bounded by 1 on the symmetrized bidiscG=def{(z+w,zw):|z|<1,|w|<1}. As an application we prove a Pick-type theorem giving a criterion for the existence o…

Realization of functions on the symmetrized bidisc Open

Jim Agler, N. J. Young · 2017

We prove a realization formula and a model formula for analytic functions with modulus bounded by $1$ on the symmetrized bidisc \[ G\stackrel{\rm def}{=} \{(z+w,zw): |z|<1, \, |w| < 1\}. \] As an application we prove a Pick-type theorem gi…

A rich structure related to the construction of analytic matrix functions Open

Daniel C. Brown, Zinaida A. Lykova, N. J. Young · 2016

We study certain interpolation problems for analytic 2×2 matrix-valued functions on the unit disc. We obtain a new solvability criterion for one such problem, a special case of the μ-synthesis problem from robust control theory. For certai…

Systematic review and meta-analysis of mesh implantation during primary stoma formation to prevent parastomal hernia Open

Stephen J. Chapman, Bradford J. Wood, Thomas M Drake, N. J. Young, David Jayne · 2016

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

Jim Agler, Zinaida A. Lykova, N. J. Young · 2016

A rich structure related to the construction of analytic matrix functions Open

David Colin Brown, Zinaida A. Lykova, N. J. Young · 2016

We analyse two special cases of $μ$-synthesis problems which can be reduced to interpolation problems in the set of analytic functions from the disc into the symmetrised bidisc and into the tetrablock. For these inhomogeneous domains we st…

Geodesics, retracts, and the norm-preserving extension property in the\n symmetrized bidisc Open

Jim Agler, Zinaida A. Lykova, N. J. Young · 2016

A set $V$ in a domain $U$ in $\\mathbb{C}^n$ has the {\\em norm-preserving\nextension property} if every bounded holomorphic function on $V$ has a\nholomorphic extension to $U$ with the same supremum norm. We prove that an\nalgebraic subse…

A Nagy-Foias model for commuting pairs of contractions Open

N. J. Young · 2016

The starting point for the Nagy-Foias model for a contractive operator $T$ on Hilbert space is Sz.-Nagy's observation that $T$ has a canonical minimal unitary dilation to a larger Hilbert space. For a {\em pair} $T=(T_1,T_2)$ of commuting …

Algebraic and geometric aspects of rational $\Gamma$-inner functions Open

Jim Agler, Zinaida A. Lykova, N. J. Young · 2015

The set \\[ \\Gamma {\\stackrel{\\rm def}{=}} \\{(z+w,zw):|z|\\leq 1,|w|\\leq 1\\}\n\\subset {\\mathbb{C}}^2 \\] has intriguing complex-geometric properties; it has a\n3-parameter group of automorphisms, its distinguished boundary is a rul…

Algebraic and geometric aspects of rational $Γ$-inner functions Open

Jim Agler, Zinaida A. Lykova, N. J. Young · 2015

The set \[ Γ{\stackrel{\rm def}{=}} \{(z+w,zw):|z|\leq 1,|w|\leq 1\} \subset {\mathbb{C}}^2 \] has intriguing complex-geometric properties; it has a 3-parameter group of automorphisms, its distinguished boundary is a ruled surface homeomor…

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