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Routh-Hurwitz Criterion II

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In the application of the Routh-Hurwitz criterion, two specific scenarios can arise that complicate stability analysis.
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Consider an electrical power grid, where stability is essential to prevent blackouts. The Routh-Hurwitz criterion is a valuable tool for assessing system stability under varying load conditions or faults. By analyzing the closed-loop transfer function, the Routh-Hurwitz criterion helps determine whether the system remains stable.
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Deformation occurs in axial and transverse directions when an axial load is applied to a slender bar. This deformation impacts the cubic element within the bar, transforming it into either a rectangular parallelepiped or a rhombus, contingent on its orientation. This transformation process induces shearing strain. Axial loading elicits both shearing and normal strains. Applying an axial load instigates equal normal and shearing stresses on elements oriented at a 45° angle to the load axis.
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Using Microwave and Macroscopic Samples of Dielectric Solids to Study the Photonic Properties of Disordered Photonic Bandgap Materials
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Borromean hypergraph formation in dense random rectangles.

Alexander R Klotz1

  • 1Department of Physics and Astronomy, <a href="https://ror.org/0080fxk18">California State University, Long Beach</a>, 1250 Bellflower Boulevard, California 90840, USA.

Physical Review. E
|October 19, 2024
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Summary
This summary is machine-generated.

We developed a simple model to understand Borromean links in entangled networks. Dense packing of shapes can form these complex links, potentially creating a connected network in systems like kinetoplast DNA.

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Area of Science:

  • Topology
  • Polymer Physics
  • Biophysics

Background:

  • Borromean links are complex three-body topological structures.
  • Studying their formation in entangled polymer networks is challenging due to computational difficulties with knot invariants.
  • These links are relevant to biological systems like kinetoplast DNA.

Purpose of the Study:

  • To develop a minimal model for studying stochastic Borromean link formation.
  • To investigate Borromean link formation in dense packings of randomly oriented rectangles.
  • To explore the potential for percolating hypergraphs formed by Hopf and Borromean linking.

Main Methods:

  • A minimal model was developed to simulate link formation.
  • Rectangles were randomly oriented and densely packed in a volume.
  • The formation of Hopf and Borromean links was evaluated.
  • Percolation thresholds for Borromean hypergraphs were analyzed.

Main Results:

  • Dense packings of rectangles can form Borromean triplets and larger clusters.
  • High densities of linked rectangles can create a percolating hypergraph through the network.
  • The study provides data on the percolation threshold of Borromean hypergraphs.

Conclusions:

  • Borromean connectivity can arise from dense packing of simple geometric objects.
  • The findings have implications for understanding topological structures in kinetoplast DNA and similar biological systems.
  • This model offers a computationally tractable approach to studying complex topological phenomena.