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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.
The first scenario occurs when a singular zero appears in the first column of the Routh table. This situation creates a division by zero issues. To resolve this, a small positive or negative number, denoted as epsilon (∈), is substituted for the zero. The stability analysis proceeds by assuming a sign for ∈. If ∈ is positive, any sign change in the first...
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Routh-Hurwitz Criterion I01:15

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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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In this lesson, we delve into the role of ring conformation and its stability, which determines the spatial arrangement and, consequently, the molecular symmetry and stereoisomerism of cyclic compounds. 1,2-Dimethylcyclohexane is used as a case study to evaluate the possible number of stereoisomers. Here, given the multiple (n = 2) chiral centers, there are 2n = 4 possible configurations that lack a plane of symmetry, as the ring skeleton exists in a non-planar chair conformation. In addition,...
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Angle of Twist - Elastic Range01:13

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Consider a cylindrical shaft with a length denoted by L and a consistent cross-sectional radius referred to as r. This shaft undergoes a torque at the free end. The highest shearing strain within the shaft is directly proportional to the twist angle and the radial distance from the shaft axis. When the shaft behaves elastically, this shearing strain can be articulated using variables such as the applied torque, radial distance, the polar moment of inertia, and the modulus of rigidity. By...
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¹H NMR of Conformationally Flexible Molecules: Temporal Resolution00:52

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At room temperature, the chair conformer of cyclohexane undergoes rapid ring flipping between two equivalent chair conformers at a rate of approximately 105 times per second. These two chair conformers are in equilibrium. The rapid ring flipping results in the interconversion of the axial proton to an equatorial proton and an equatorial to the axial proton. Such interconversions are too rapid and cannot be detected on the NMR timescale. Hence, the NMR spectrometer cannot distinguish between the...
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The chair conformation is the most stable form of cyclohexane due to the absence of angle and torsional strain. The absence of angle strain is a result of cyclohexane’s bond angle being very close to the ideal tetrahedral bond angle of 109.5° in its chair conformer. Similarly, the torsional strain is also absent owing to the perfectly staggered arrangement of bonds.
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Related Experiment Video

Updated: Jun 5, 2025

Magnetic Tweezers for the Measurement of Twist and Torque
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Tensor ring rank determination using odd-dimensional unfolding.

Yichun Qiu1, Guoxu Zhou2, Chao Li3

  • 1School of Automation, Guangdong University of Technology, Guangzhou 510006, China; Center for Advanced Intelligence Project (AIP), RIKEN, Tokyo 103-0027, Japan.

Neural Networks : the Official Journal of the International Neural Network Society
|December 5, 2024
PubMed
Summary
This summary is machine-generated.

Determining tensor ring (TR) ranks is challenging. This study introduces an odd-dimensional unfolding method, enhancing TR-rank determination for noisy data and high dimensions.

Keywords:
Low-rank approximationRank determinationSingular value thresholdingTensor networkTensor ring decomposition

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Area of Science:

  • Multilinear algebra
  • Data analysis
  • Numerical methods

Background:

  • Tensor ring (TR) decomposition is vital for high-dimensional data analysis.
  • Determining TR-ranks is difficult, with existing methods sensitive to initial ranks and lacking adaptivity for noisy or incomplete data.
  • Computational inefficiency is a concern for high-dimensional TR decomposition.

Purpose of the Study:

  • To propose an effective method for determining tensor ring ranks.
  • To address the limitations of existing TR-rank determination techniques, including sensitivity, adaptivity, and computational efficiency.
  • To enhance the applicability of TR decomposition in real-world scenarios.

Main Methods:

  • An odd-dimensional unfolding method leveraging TR model symmetry and bound rank relationships.
  • Singular value thresholding for adaptive TR-rank determination.
  • Randomized sketching techniques for improved computational efficiency and scalability.

Main Results:

  • The proposed method effectively determines TR-ranks.
  • Demonstrated adaptivity in the presence of noisy and incomplete data.
  • Enhanced computational efficiency and scalability for high-dimensional data.

Conclusions:

  • The novel odd-dimensional unfolding method offers a robust solution for TR-rank determination.
  • The approach shows significant potential for applications in rank identification, data denoising, and data completion.
  • This work advances the practical utility of tensor ring decomposition.