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‘For the case of a finite-dimensional real linear space these are infinite systems whose adjoint cone is topologically closed.’
‘According to embedding theory, the Poincare map is topologically equivalent to a Poincare surface of section that intersects the system's trajectory in its characteristic state-space.’
‘A Mobius Strip made of paper: if you consider the thickness of the paper, a ring that makes a mobius strip is topologically the same as solid tori.’
‘In particular, he conjectured that any smooth 3-surface that has no edges, no corkscrew-like twists, and no doughnut-like holes must be topologically equivalent to a 3-sphere.’
‘For example, tethering of nucleosomes to the surface may at least transiently strengthen their ability to create a local topologically closed domain, leading to enhanced torsional stress within that region when hSwi-Snf is active.’
‘On the other hand, information is preserved in topologically trivial metrics.’
‘He was able to show that any 2-dimensional surface having the same fundamental group as the 2-dimensional surface of a sphere is topologically equivalent to a sphere.’
‘It implies that contact may obtain between A and B only if either A or B is topologically closed while the other is topologically open in the relevant contact area; but the appeal to indeterminacy allows one to leave the matter unsettled.’
‘In effect, any object that mathematically behaves like a 3-sphere is topologically a 3-sphere, no matter how distorted or twisted its shape may be.’
‘This is because every organism is a closed surface, topologically speaking, and it has been known for many years what kinds of closed surfaces can and cannot be continuously mapped to each other.’
‘These two-dimensional examples give us an idea of how to build the more difficult three-dimensional, topologically compact spaces.’
‘For a space that is topologically equivalent to the real line and for metrics that are consistent with that topology, each coordinate system determines a metric and each metric determines a coordinate system.’
‘I've heard you refer to the inverse as being topologically finite.’
‘In addition, the measurement results can be displayed dynamically, topologically, statistically, or in a tabular format on a monitor.’
‘Apart from the importance in understanding the mechanisms involved in vivo, the interplay between conformation and interactions in topologically constrained polymers is of interest in its own right from a biophysical point of view.’
‘Unless matter is to be created or destroyed, we seem to have to say that the break is half-open, with one new matter-surface being topologically closed and the other being topologically open.’
‘We then propose a classification of bacterial phenotypes based on the topology of the metabolic phase diagram, and enumerate the possible topologically distinct phenotypes that can be achieved through mutations of the PTS.’
‘Highly significant correlations are found between the most topologically relevant sites in our analysis and the single point mutations known to be associated with the arousal of the genetic forms of prion disease.’
‘We show that this small and topologically simple protein is remarkably mechanically resistant, reinforcing the view that extension of hydrogen-bonded parallel -strands is key to defining mechanical strength.’
‘Thus, the situation in much of the remainder of the dermal skeleton, where individual elements are indistinguishable morphologically or topologically, ensures that discussion of patterns such as fusion or loss are entirely vacuous.’