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Hitherto, understanding had been limited to the cases where algebraic numbers are tied to the rational numbers by a commutative group (called a Galois group).
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World cities such as London have tended to defy understanding hitherto because simple hierarchical subdivision has ignored the fact that their polycentricity subsumes a pattern of nested urban movements.
This is shown in Figure 1a, where equations 3 & 4 added give equation 5. Figure 1b depicts the overall mechanistic flowchart for the hitherto held understanding of coupled and uncoupled processes.
By doing so, it broadens the hitherto existing understanding of a static context by defining it as a dynamic factor that needs to be systematically considered and that can be transformed during the intervention.
Most effort hitherto had been devoted to understanding the movements of the Moon through space, but now astronomers began to focus their attention on the character of the Moon itself.
Their venoms contain a diversity of neurotoxins, which represent a tremendous hitherto partially unexplored resource not only for understanding ion channels but also for use in drug design and development.
The differential susceptibility of cell lines and different neuronal populations to prion infection has hitherto been unexplained, and these findings may be critical to understanding prion pathogenesis and selective vulnerability of different cell types to prion infection.
Whereas knowledge derives from virtues that aim at truth, understanding derives at least partly from different virtues, special ones hitherto "unanalyzed, even unrecognized" (Zagzebski 2001, 248).
Scorpion venom represents a tremendous, hitherto partially explored peptide library that has been proven to be useful not only for understanding ion channels but also for drug design.
Tracking cell lines, crucial to understanding why some cells respond to drugs and others do not, is something researchers have hitherto had to do by hand, in a time-consuming and labourious process.The next step will be to use motion capture to see what happens inside a cell.
Seeing the molecular biology of the 1960s as an information science, Mr Agar overlooks Peter Mitchell, whose breakthrough in understanding that cells store energy, not in molecules, but in potential differences revealed an aspect of life hitherto undreamed of.
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com