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Related Concept Videos

¹H NMR: Long-Range Coupling01:27

¹H NMR: Long-Range Coupling

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.
In alkenes, spin information is communicated via σ–π overlap, as seen in allylic (four-bond) and homoallylic (five-bond) couplings. These coupling interactions are stronger when the σ bond is parallel to the alkene π orbitals.
Spin–Spin Coupling: Two-Bond Coupling (Geminal Coupling)01:20

Spin–Spin Coupling: Two-Bond Coupling (Geminal Coupling)

Two NMR-active nuclei bonded to a central atom can be involved in geminal or two-bond coupling. Geminal coupling is commonly seen between diastereotopic protons in chiral molecules and unsymmetrical alkenes, among others.
The central atom need not be NMR-active because its electrons are affected by the electron polarization of the spin-active atoms. However, spin information is transmitted less effectively than in one-bond coupling, and 2J values are usually weaker than 1J values. The energy of...
NMR Spectroscopy: Spin–Spin Coupling01:08

NMR Spectroscopy: Spin–Spin Coupling

The spin state of an NMR-active nucleus can have a slight effect on its immediate electronic environment. This effect propagates through the intervening bonds and affects the electronic environments of NMR-active nuclei up to three bonds away; occasionally, even farther. This phenomenon is called spin–spin coupling or J-coupling. Coupling interactions are mutual and result in small changes in the absorption frequencies of both nuclei involved. While nuclei of the same element are involved in...
¹H NMR Signal Multiplicity: Splitting Patterns01:13

¹H NMR Signal Multiplicity: Splitting Patterns

When protons A and X are coupled, their nuclear spin energy levels are slightly modified. This is because the energy required to excite proton A to a spin state parallel to proton X is slightly different from the energy required for it to become anti-parallel to spin X. Consequently, there are two possible excitation frequencies for A (A1 and A2), depending on the spin state of X, and vice versa. The mutual nature of coupling implies that the difference between frequencies A1 and A2, indicated...
Spin–Spin Coupling: One-Bond Coupling01:17

Spin–Spin Coupling: One-Bond Coupling

Coupling interactions are strongest between NMR-active nuclei bonded to each other, where spin information can be transmitted directly through the pair of bonding electrons. While nuclei polarize their electrons to the opposite spins, the bonding electron pair has opposite spins. Configurations with antiparallel nuclear spins are expected to be lower in energy. When coupling makes antiparallel states more favorable, J is considered to have a positive value. The one-bond coupling constant, 1J,...
Network Function of a Circuit01:25

Network Function of a Circuit

Frequency response analysis in electrical circuits provides vital insights into a circuit's behavior as the frequency of the input signal changes. The transfer function, a mathematical tool, is instrumental in understanding this behavior. It defines the relationship between phasor output and input and comes in four types: voltage gain, current gain, transfer impedance, and transfer admittance. The critical components of the transfer function are the poles and zeros.

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New Framework for Understanding Cross-Brain Coherence in Functional Near-Infrared Spectroscopy (fNIRS) Hyperscanning Studies
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Outer synchronization of coupled networks using arbitrary coupling strength.

Zhuchun Li1, Xiaoping Xue

  • 1Department of Mathematics, Harbin Institute of Technology, 150001 Harbin, China. lizhuchun@gmail.com

Chaos (Woodbury, N.Y.)
|July 2, 2010
PubMed
Summary
This summary is machine-generated.

This study demonstrates that outer synchronization between two coupled networks is achievable with any coupling strength for balanced network topologies. Numerical examples confirm this, showing coupling strength impacts short-term performance but not the transverse Lyapunov exponent.

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Area of Science:

  • Complex Networks
  • Systems Science
  • Control Theory

Background:

  • Coupled network synchronization is crucial in various scientific domains.
  • Outer synchronization, a specific type of synchronization, remains an area of active research.
  • Understanding synchronization dynamics in networks with balanced structures is essential.

Purpose of the Study:

  • To investigate and prove the conditions for achieving outer synchronization between two coupled networks.
  • To analyze the influence of coupling strength on outer synchronization performance.
  • To explore the relationship between outer synchronization and the transverse Lyapunov exponent.

Main Methods:

  • Utilizing the Lyapunov function approach to establish theoretical guarantees for synchronization.
  • Applying concepts from the average-consensus problem to network analysis.
  • Conducting numerical simulations to validate theoretical findings.

Main Results:

  • Outer synchronization can be asymptotically reached for networks with balanced structure topology, irrespective of coupling strength.
  • Numerical examples illustrate that coupling strength affects the transient behavior of outer synchronization.
  • The transverse Lyapunov exponent is demonstrated to be independent of the coupling strength value.

Conclusions:

  • The theoretical framework confirms the robust achievement of outer synchronization in balanced networks.
  • While coupling strength influences initial convergence, it does not affect the long-term synchronization state or the transverse Lyapunov exponent.
  • The findings provide valuable insights into the control and design of synchronized network systems.