19 December, 2024 in expository, math.MG | Tags: cosmic distance ladder, quaternions, spherical trigonometry | by Terence Tao | 21 comments Hamilton’s quaternion number system is a non-commutative extension of the complex numbers, consisting of numbers of the form where are real numbers, and...
8 September, 2024 in math.AP, math.MP | Tags: finite time blowup, incompressible Euler equations | by Terence Tao | 7 comments Given a smooth compact Riemannian manifold , the incompressible Euler equations can be written in abstract index notation as where is a time-dependent scalar field...
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First, the visual assessment of MR images is greatly dependent on the high-level expertise of radiologists, leading to discrepancies in the assessment results. Second, some features of MR images reflecting tumor heterogeneities cannot be observed on the visual assessment. Third, the visual assessment...
On the other hand, semantic relationships are tied to the underlying concepts being expressed, and they are consistent across languages and speaking style. For instance, in “The dog chased the cat” and “The cat was chased by the dog,” the same semantic content is expressed with different...
[146], a boost in citations can be seen in late 2018 and 2019. On the left, we display a sample problem solved in that year by one of the articles, specifically a type of time-dependent equation. Some of the addressed problems to be solved in the first vanilla PINN, for example, ...
Wherever a loose bed of sand is subject to sufficiently strong winds, aeolian dunes form at predictable wavelengths and growth rates. As dunes mature and coarsen, however, their growth trajectories become more idiosyncratic; nonlinear effects, sediment s
Downhill running (DR) is a whole-body exercise model that is used to investigate the physiological consequences of eccentric muscle actions and/or exercise-induced muscle damage (EIMD). In a sporting context, DR sections can be part of running disciplines (off-road and road running) and can...