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Spin systems where the difference in chemical shifts of the coupled nuclei is greater than ten times J are called first-order spin systems. These nuclei are weakly coupled, and their chemical shifts and coupling constant can generally be estimated from the well-separated signals in the spectrum.
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¹H NMR: Long-Range Coupling01:27

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The coupling interactions of nuclei across four or more bonds are usually weak, with J values less than 1 Hz. While these are usually not observed in spectra, the presence of multiple bonds along the coupling pathway can result in observable long-range coupling.
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The probability of having two carbon-13 atoms next to each other is negligible because of the low natural abundance of carbon-13. Consequently, peak splitting due to carbon-carbon spin-spin coupling is not observed in spectra. However, protons up to three sigma bonds away split the carbon signal according to the n+1 rule, resulting in complicated spectra.
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Quorum sensing is a mechanism of bacterial communication that enables coordinated gene expression in response to changes in population density. This facilitates collective behaviors that enhance survival, resource acquisition, and ecological adaptation. This process relies on small signaling molecules called autoinducers that accumulate as bacterial populations grow. When a critical threshold concentration of autoinducers is reached, bacterial cells collectively modify gene expression,...
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Second Order systems II01:18

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In an underdamped second-order system, where the damping ratio ζ is between 0 and 1, a unit-step input results in a transfer function that, when transformed using the inverse Laplace method, reveals the output response. The output exhibits a damped sinusoidal oscillation, and the difference between the input and output is termed the error signal. This error signal also demonstrates damped oscillatory behavior. Eventually, as the system reaches a steady state, the error diminishes to zero.
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Periodic coupling inhibits second-order consensus on networks.

Fabian Baumann1, Igor M Sokolov1,2, Melvyn Tyloo3

  • 1Institut für Physik, Humboldt-Universität zu Berlin, Newtonstraße 15, 12489 Berlin, Germany.

Physical Review. E
|December 17, 2020
PubMed
Summary
This summary is machine-generated.

Time-periodic coupling modulations can unexpectedly prevent consensus in second-order systems. This study identifies specific frequencies that induce parametric resonance, hindering collective agreement in networks.

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Area of Science:

  • Network science
  • Control theory
  • Dynamical systems

Background:

  • Consensus algorithms are crucial for distributed decision-making and multi-agent coordination.
  • Second-order consensus models incorporate Newtonian dynamics of physical agents.

Purpose of the Study:

  • To investigate the impact of time-periodic coupling modulations on second-order consensus models.
  • To identify mechanisms that inhibit the formation of collective consensus states.

Main Methods:

  • Spectral decomposition of the consensus model.
  • Analytical derivation of conditions for parametric resonance.
  • Extension of parametric resonance theory to networks with time-periodic couplings.

Main Results:

  • Small time-periodic coupling modulations can inhibit consensus formation.
  • Parametric resonance is induced at specific intermediate coupling frequencies.
  • Resonance frequencies are precisely predicted and linked to the network's Laplacian spectrum.

Conclusions:

  • Time-periodic coupling modulations can disrupt consensus in second-order systems.
  • The study provides a precise formalism to predict and quantify resonance phenomena in such networks.
  • Findings are crucial for designing robust coordination strategies in networked systems.