By Graham Staplehurst, Peter C. Fenlon, Angus McBride
Minas Tirith, the Guarded urban, rises from the bottom of Mount Mindolluin. Capital of Gondor, it's the preeminent image of the fight opposed to Sauron of Mordor. No different urban in western Middle-earth deals better desire for the security of the loose Peoples. Minas Tirith instructions the huge hole among the White Mountains and the Mountains of Shadow that encircle the darkish Lord's place of birth. Its seven partitions and 7 degrees appear to develop out of the stone, as though carved by way of Giants. every one tier of the Royal urban wraps round the rock and looms above the encircling zone, its ramparts glimmering white with every one dawn. excessive atop the capital, at the uppermost point, the Tower of Ecthelion reaches skyward like a pearl-and-crystal spire. Amidst the typhoon spawned by way of the Lord of the earrings, this fortress is the final carry in Minas Tirith, an unheard of bastion in an ominous time.
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Additional resources for Minas Tirith: Cities of Middle-earth (Middle Earth Role Playing MERP)
A JENSEN'S INEQUALITY. Suppose that f is convex on I. If a I, ... , an E I and AI, ... , An are nonnegative real numbers such that Al + ... + An = I, then Equality occurs if and only if all the aj are equal or f is a linear function. , f is concave downward), as is the case for the sine function in our Problem, then the inequality is reversed. Let's prove Jensen's inequality in the case where n = 4 and A = 1/2. Since f is convex (we'll take the interval I for granted), f (a + b) < f(a) + f(b) 2 ' 2 with equality if and only if a = b (we'll assume that can use this hypothesis twice: f (a l + a2 : a3 + a4 ) = f f is not a linear function).
Aha! solutions are to be found! 1 Algebra Passing Time At some time between 3:00 and 4:00, the minute hand of a clock passes the hour hand. Exactly what time is this? ) Solution Let's solve the problem in a mundane way first (before giving an aha! solution). We reckon time in minutes from the top of the hour (12 on the clock). Suppose that the minute hand is at t minutes and the hour hand is, correspondingly, at 15 + t /12 minutes. ) When the two hands coincide, we have t = 15 + t /12, and hence t = 16 + 4/11.
Now that we know the symmetry group of the cube (S4), we find that the symmetry group of the regular tetrahedron goes along for the ride. Every symmetry of the cube automatically gives a symmetry of the regular tetrahedron (with the tetrahedron inscribed in the cube), or it moves the vertices of the tetrahedron to the other four vertices of the cube. How many symmetries of the regular tetrahedron are there? Since we can put the tetrahedron down on any of its four faces and then rotate the tetrahedron in any of three ways, the regular tetrahedron has 4·3 = 12 symmetries.
Minas Tirith: Cities of Middle-earth (Middle Earth Role Playing MERP) by Graham Staplehurst, Peter C. Fenlon, Angus McBride