An elementary treatise on curve tracing by Percival Frost

By Percival Frost

This obtainable remedy covers orders of small amounts, types of parabolic curves at an unlimited distance, kinds of curves locally of the starting place, and different types of branches whose tangents on the starting place are the coordinate axes. extra subject matters contain asymptotes, analytical triangle, singular issues, extra. 1960 variation.

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An elementary treatise on curve tracing

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Auerbach investigates convex sets K which have the property that every chord of some given length, a say, determines an arc of fixed length. He first shows that, if there are such examples, then these same chords determine sectors of fixed area - hence his interest in floating bodies and Ulam's problem. He considers bodies with smooth boundaries and investigates the angle a{9) between the area (and perimeter) bisecting chord in direction 9 and the tangent at the point from which the chord emanates.

In particular, this shows that K i-» F(iir*) is an SL(n) invariant and homogeneous valuation. The next result shows that there are no further examples. 52 Theorem 2 ( 4 6 ). A functional $ : VQ —» R is a measurable, SL(n) invariant valuation which is homogeneous of degree q if and only if there is a constant c € R suc/i t/iat $(P) = < c for q = 0 cV(P) /or q = n cV(P*) for q = —n 0 otherwise 1 /or every P G PQ . It is not known if there are additional examples if $ is not homogeneous. We conjecture that every SL(n) invariant and continuous valuation is a linear combination of a constant, the volume of the body and the volume of the polar body.

Averaging over G, since

Val G Val G , of degree n, such that for

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