Calculus: Readings from the Mathematics teacher

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3. We next fix an r > 1 and an offset vector c in Zr, and ask how far the expected value measure p(H(N), offsets c) is from the limit measure p(univ, offsets c). 4. Let r > 1 be an integer, c in Zr an offset vector, and E > 0 a real number. 3, we have discrep(p(H(N), offsets c), p(univ, offsets c)) < NE-1/(2r+4) PROOF. (A, H(N), offsets c). For any point x in R', we denote by R(x) the rectangle (-oo, x]. We have Ip(N)(R(x)) - p(R(x))I = f H (N) f f (u(A, N)(R(x)) - p(R(x)))dHaar Ip(A, N)(R(x)) - p(R(x))IdHaar (N) (Sup, Ip(A, N)(R(y)) - p(R(y))I)dHaar (N) fH I discrep(µ(A, N), µ)IdHaar N) < NE-1/(2r+4) NE-1/(2r+4).

For any integer N > 2, denote by RN(ordered) the dosed subset o f RN consisting o f those points (x(1), ... , x(N)) which satisfy x(1) < x(2) < < x(N). 5. COMBINATORICS OF SPACINGS: THIRD DISCUSSION The "order" map RN order", 51 RN(ordered) C RN, "arrange the coordinates in increasing X F4 order (X), is a continuous map of RN to itself. PROOF. To check that a map X - Y of topological spaces is continuous, it suffices to exhibit a finite covering of its source X by closed sets Zi such that the map restricted to each Zi is continuous.

3. Let a < b < a + 2Tr be real numbers, and denote by Z[a, b] the closed interval in the unit circle which is the isomorphic image of the closed interval [a, b] by the map x'--r exp(ix). Fix N > 1. Denote by [a, b]N(ordered) the subset of [a, b]N consisting of those N-tuples (cpl, ... , (pn) of elements of [a, b] which satisfy a < p(1) 5 cp(2) < ... < ,p(N) < b. 8. APPENDIX 33 Then the map [a, b] N (ordered) ' PN, Z [a, b], a < cp(1) < ci(2) < < cp(N) < b - fl(T - exp(icp(j))) i is bicontinuous.

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