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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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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...
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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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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.
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A proton M that is coupled to a proton X results in doublet signals for M. However, NMR-active nuclei can be simultaneously coupled to more than one nonequivalent nucleus. When M is coupled to a second proton A, such as in styrene oxide, each peak in the doublet is split into another doublet.
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Network constraints on the mixing patterns of binary node metadata.

Matteo Cinelli1,2, Leto Peel3,4, Antonio Iovanella5

  • 1Ca' Foscari University of Venice, Department of Environmental Sciences, Informatics and Statistics, 30172 Mestre (VE), Italy.

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Network topology significantly constrains assortativity coefficient bounds. Understanding these network constraints is crucial for accurate interpretation of node attribute correlations, preventing misinterpretations in network analysis.

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Area of Science:

  • Network Science
  • Graph Theory
  • Data Analysis

Background:

  • Assortativity coefficient measures the tendency of nodes with similar attributes to connect.
  • It is analogous to Pearson's correlation coefficient for node metadata.
  • Network topology and metadata distribution inherently limit achievable assortativity values.

Purpose of the Study:

  • To quantify the impact of network topology on assortativity coefficient bounds.
  • To investigate how degree distribution and metadata proportions affect extremal assortativity.
  • To provide methods for calculating these bounds for binary node metadata.

Main Methods:

  • Analysis of network constraints on assortativity.
  • Mathematical derivation of bounds for extremal assortativity values.
  • Consideration of binary node metadata and its distribution.

Main Results:

  • Network topology, not just node attributes, dictates assortativity limits.
  • Degree distribution and metadata proportions significantly constrain maximum and minimum assortativity.
  • Identified specific conditions where assortativity bounds are severely limited.

Conclusions:

  • Assortativity interpretation must account for topological constraints.
  • Overlooking these bounds can lead to misinterpretations of network structure and attribute linking.
  • The study provides a framework for more accurate network analysis considering inherent constraints.