What if archimedes had met taylor

MacDougall Every integer larger than 1 is uniquely a product of powers of primes, and we will let P n denote the set of integers with exactly n distinct prime divisors. In this article we study runs of consecutive integers within any of the sets P n. We also derive an upper bound for the length of a run in P n for any n.

The longest run we discovered is in P 3 and has 16 numbers beginning at We conjecture that there is a run of length at least 2 for every n and give a procedure for constructing some such examples. Riley, Jr Suppose you are shooting a billiard ball on elliptical table with no pockets. In what ways is it possible to make your shot return to its starting point? For example, is it possible to specify the number of times the ball must hit the edge before returning to its starting point?

We show that the answers to these and other questions depend first on whether the path of the ball must go around or between the foci and second on the shape eccentricity of the ellipse.

All of this serves to illustrate the subtlety of dynamical systems. Probabilistic Reasoning is not Logical Ruma Falk Most students are pretty well versed in the ground rules of logical deduction. Occurrence of an event might probabilistically support another given uncertain event, namely, enhance its chances, or probabilistically confirm the event in question, that is, render it highly probable.

Six basic rules of logical and probabilistic inferences of the two types are compared with each other. It turns out that probabilistic inferences are sometimes subject to the same rules as logical involvement and some other times they diverge from them. Though applying the logical rules to inferences under uncertainty may often be heuristically expedient, it is wrong in principle and it may incur faulty conclusions in important situations.

Increased sensitivity to such possibilities should caution students, educators, and researchers against perfunctorily applying the habitual syllogistic rules to reasoning under uncertainty. The method of exhaustion is so called because one thinks of the areas measured expanding so that they account for more and more of the required area.

However Archimedes , around BC, made one of the most significant of the Greek contributions. References show. Histoire Sci. J O Fleckenstein, The line of descent of the infinitesimal calculus in the history of ideas, Arch. Torino 46 1 , 1 - T Guitard, On an episode in the history of the integral calculus, Historia Mathematica 14 2 , - Explorer The Category of Matroids.

The structure of the category of matroids and strong maps is investigated: it has coproducts and equalizers, but not products or coequalizers; there are functors from the categories of graphs and … Expand.

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The Rules of Sum and Product.