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Such complete intersections have important applications in geometry and number theory, because they typically admit rational points, an elementary case of which is the Chevalley–Warning theorem. Fano varieties provide an abstract generalization of these basic examples for which rationality questions are often still tractable.
At the triple junction each of the three boundaries will be one of three types – a ridge (R), trench (T) or transform fault (F) – and triple junctions can be described according to the types of plate margin that meet at them (e.g. fault–fault–trench, ridge–ridge–ridge, or abbreviated F-F-T, R-R-R).
Some troughs look similar to oceanic trenches but possess other tectonic structures. One example is the Lesser Antilles Trough, which is the forearc basin of the Lesser Antilles subduction zone. [8] Also not a trench is the New Caledonia trough, which is an extensional sedimentary basin related to the Tonga-Kermadec subduction zone. [9]
[11] [12] Marianas Trench is an example of a deep slab, thereby creating the deepest trench in the world established by a steep slab angle. [13] Slab breakoff occurs during a collision between oceanic and continental lithosphere, [14] allowing for a slab tear; an example of slab breakoff occurs within the Himalayan subduction zone. [4]
Using geometric language, as is done in incidence geometry, shapes the topics and examples that are normally presented. It is, however, possible to translate the results from one discipline into the terminology of another, but this often leads to awkward and convoluted statements that do not appear to be natural outgrowths of the topics.
Japan Trench: Pacific Ocean 9,000 29,527 5.59 8 Puerto Rico Trench: Atlantic Ocean 8,605 28,232 5.35 9 Yap Trench: Pacific Ocean 8,527 27,976 5.30 10 Richards Deep: Peru–Chile Trench, Pacific Ocean 8,065 26,456 5.01 11 Diamantina Deep: Diamantina fracture zone, Indian Ocean: 8,047 26,401 5.00 12 Romanche Trench: Atlantic Ocean 7,760 25,460 4. ...
In Euclidean and projective geometry, five points determine a conic (a degree-2 plane curve), just as two (distinct) points determine a line (a degree-1 plane curve).There are additional subtleties for conics that do not exist for lines, and thus the statement and its proof for conics are both more technical than for lines.
For example, one can ask if there is a Banach–Tarski paradox in the hyperbolic plane H 2. This was shown by Jan Mycielski and Grzegorz Tomkowicz. [22] [23] Tomkowicz [24] proved also that most of the classical paradoxes are an easy consequence of a graph theoretical result and the fact that the groups in question are rich enough.