Jove
Visualize
Contact Us
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Concept Videos

Block Diagram Reduction01:22

Block Diagram Reduction

The process of deriving the transfer function of a control system often involves reducing its block diagram to a single block. This simplification can be achieved through a series of strategic operations, including relocating branch points and comparators. These operations preserve the overall function of the system while allowing for easier manipulation and combination of blocks.
The first step in this process is the identification and relocation of a branch point. A branch point, where a...
Fundamental Theorem of Algebra01:30

Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra is central to the study of polynomial equations, asserting that every non-constant polynomial with complex coefficients has at least one complex zero. This means that a polynomial of degree n ≥ 1, written as:  with an ≠ 0, has at least one solution in the complex number system. Since the set of real numbers is a subset of complex numbers, this theorem applies equally to polynomials with real coefficients.Building on this result, the Complete Factorization...
Elements of Block Diagrams01:25

Elements of Block Diagrams

Block diagrams serve as a visual representation of the input-output relationships within a system. An illustrative example is a heating system, where the set temperature activates the furnace to warm the room to the desired level. Block diagrams are versatile, modeling linear systems through Laplace transform variables and nonlinear systems using time domain variables.
A block diagram typically includes essential elements such as comparators, blocks, and feedback loops. Each of these elements...
Theorems of Pappus and Guldinus: Problem Solving01:12

Theorems of Pappus and Guldinus: Problem Solving

Pappus and Guldinus's theorems are powerful mathematical principles that are used for finding the surface area and volume of composite shapes. For example, consider a cylindrical storage tank with a conical top. Finding the surface area or volume can be challenging for such complex shapes. These theorems are particularly useful in calculating the volume and surface area of such systems. Here, the cylindrical storage tank with a conical top can be broken down into two simple shapes: a cylinder...
Rationalizing Substitutions01:29

Rationalizing Substitutions

Integrals involving non-rational functions are often difficult to evaluate using standard techniques, especially when radicals appear in the integrand. Rationalizing substitution provides a systematic method for simplifying such integrals by converting them into rational forms that are easier to handle.Consider a rod whose linear mass density depends on a constant linear density, a characteristic length, and the distance from the left end of the rod. Determining the total mass requires...
Periodic Classification of the Elements04:00

Periodic Classification of the Elements

The periodic table arranges atoms based on increasing atomic number so that elements with the same chemical properties recur periodically. When their electron configurations are added to the table, a periodic recurrence of similar electron configurations in the outer shells of these elements is observed. Because they are in the outer shells of an atom, valence electrons play the most important role in chemical reactions. The outer electrons have the highest energy of the electrons in an atom...

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by
Same journal

Quadratic Sparse Domination and Weighted Estimates for Non-integral Square Functions.

Journal of geometric analysis·2026
Same journal

Flows of Conformally Coclosed <math><msub><mi>G</mi> <mn>2</mn></msub></math> -Structures with Dilaton.

Journal of geometric analysis·2026
Same journal

Kähler-Einstein Metrics.

Journal of geometric analysis·2026
Same journal

On Shape Optimization with Large Magnetic Fields in Two Dimensions.

Journal of geometric analysis·2026
Same journal

Families of proper holomorphic maps.

Journal of geometric analysis·2026
Same journal

Distributional Sectional Curvature Bounds for Riemannian Metrics of Low Regularity.

Journal of geometric analysis·2026

Related Experiment Video

Updated: May 23, 2026

Swin-PSAxialNet: An Efficient Multi-Organ Segmentation Technique
04:48

Swin-PSAxialNet: An Efficient Multi-Organ Segmentation Technique

Published on: July 5, 2024

A Schwarz Lemma for the Pentablock.

Nujood M Alshehri1, Zinaida A Lykova1

  • 1School of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon Tyne, NE1 7RU UK.

Journal of Geometric Analysis
|May 22, 2026
PubMed
Summary
This summary is machine-generated.

This study introduces a Schwarz lemma for the pentablock, a complex domain relevant to μ-synthesis. Researchers developed a construction theory for rational maps, proving a Schwarz lemma for this domain.

Keywords:
Distinguished boundaryInner functionsPentablockSchwarz lemma

More Related Videos

Quadruple-Checkerboard: A Modification of the Three-Dimensional Checkerboard for Studying Drug Combinations
11:15

Quadruple-Checkerboard: A Modification of the Three-Dimensional Checkerboard for Studying Drug Combinations

Published on: July 24, 2021

Related Experiment Videos

Last Updated: May 23, 2026

Swin-PSAxialNet: An Efficient Multi-Organ Segmentation Technique
04:48

Swin-PSAxialNet: An Efficient Multi-Organ Segmentation Technique

Published on: July 5, 2024

Quadruple-Checkerboard: A Modification of the Three-Dimensional Checkerboard for Studying Drug Combinations
11:15

Quadruple-Checkerboard: A Modification of the Three-Dimensional Checkerboard for Studying Drug Combinations

Published on: July 24, 2021

Area of Science:

  • Complex Analysis
  • Operator Theory
  • Control Theory

Background:

  • The pentablock is a bounded, non-convex domain in C^3, arising from μ-synthesis problems.
  • Rational maps and inner functions are crucial in complex analysis and systems theory.
  • Schwarz lemmas provide fundamental inequalities for analytic maps between domains.

Purpose of the Study:

  • To prove a Schwarz lemma for the pentablock domain.
  • To develop a structure theory for rational maps into the pentablock.
  • To establish connections between pentablock-inner functions and inner functions of the symmetrized bidisc.

Main Methods:

  • Definition of the pentablock P as the image of 2x2 complex matrices in the unit ball under mapping to (a21, trA, detA).
  • Development of a structure theory for rational maps from the unit disc D to the closed pentablock P̄.
  • Construction of rational P̄-inner functions using zeroes and Fejér-Riesz factorizations.

Main Results:

  • A concrete structure theory for rational maps from D to P̄ that map the unit circle T to the boundary bP̄ is established.
  • Relationships between P̄-inner functions and inner functions of the symmetrized bidisc are identified.
  • A constructive method for generating rational P̄-inner functions of prescribed degree is presented.

Conclusions:

  • The study successfully proves a Schwarz lemma for the pentablock.
  • The developed constructive theory provides an algorithmic approach for creating specific rational inner functions.
  • These findings advance the understanding of complex domains and their associated mappings in mathematical analysis and engineering.