By Cao F.

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**Example text**

The Sicilian city of Syracuse allied itself with Carthage during the second Punic war, and thus was attacked by Rome, portending what would ultimately happen to the entire Hellenic world. During a long siege, soldiers of the Roman general Marcellus were terriﬁed by the ingenious war machines defending the city, invented by the Syracusan Archimedes (c. ). These included catapults to hurl great stones, as well as ropes, pulleys, and hooks to raise and smash Marcellus’s ships, and perhaps even burning mirrors setting ﬁre to their sails.

39. Prove the inequality i=1 ik < (n + 1)k+1 / (k + 1) of Roberval using Pascal’s equation. Can you also use it to prove his second inequality n nk+1 / (k + 1) < i=1 ik , which is harder to show? 40. Compare and contrast the work of Archimedes and Pascal toward ﬁnding the area of a bounded curvilinear region in the plane. Cite speciﬁc contributions of each author toward the solution of this problem. 41. 38 to prove that lim n→∞ n k i=1 i k+1 n = 1 , k+1 a result known as Wallis’s theorem (discussed further in the next section).

Thus, [Proposition 10, Corollary 1] n · OA2 > 3 OP 2 + OQ2 + · · · + OY 2 , whence C1 > 3f , or f < 13 C1 , which is impossible, since f > R1 ≯ 13 C1 . Since then R1 is neither greater nor less than 13 C1 , R1 = 1 3 C1 . Therefore 1 C1 . 25), but only on the inequalities from Corollary 1. 26). 19. Compare the translations given by Heath and Dijksterhuis. Do they say the same thing? What are the advantages and disadvantages of the verbal versus the modern symbolic approaches? 20. 7. Also show that they are equiv3 3 n alent to n3 < i=1 i2 < (n+1) .

### Curve evolution and image processing by Cao F.

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