The Cycloid The cycloid is the locus of a point fixed to the rim of a circular disk that is rolling along a straight line (see figure). The parametric equations for the cycloid are $latex \displaystyle x = r (\theta - \sin\theta)\,, \qquad y = r (1 - \cos\theta ) \ \ \ \ \ … Continue reading Mamikon’s Theorem and the area under a cycloid arch
Month: February 2021
Machine Learning and Climate Change Prediction
Current climate prediction models are markedly defective, even in reproducing the changes that have already occurred. Given the great importance of climate change, we must identify the causes of model errors and reduce the uncertainty of climate predictions [TM205 or search for “thatsmaths” at irishtimes.com]. The Charney Report In 1979, a study group led by … Continue reading Machine Learning and Climate Change Prediction
Apples and Lemons in a Doughnut
A ring torus (or, simply, torus) is a surface of revolution generated by rotating a circle about a coplanar axis that does not intersect it. We let $latex {r}&fg=000000$ be the radius of the circle and $latex {R}&fg=000000$ the distance from the axis to the centre of the circle, with $latex {R>r}&fg=000000$. If the axis … Continue reading Apples and Lemons in a Doughnut
Complexity: are easily-checked problems also easily solved?
From the name of the Persian polymath Al Khwarizmi, who flourished in the early ninth century, comes the term algorithm. An algorithm is a set of simple steps that lead to the solution of a problem. An everyday example is a baking recipe, with instructions on what to do with ingredients (input) to produce a … Continue reading Complexity: are easily-checked problems also easily solved?
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