Making Concrete Garden Ornaments by Sherri Warner Hunter

By Sherri Warner Hunter

A celebrated instructor has accrued the tried-and-true suggestions utilized by sculptors and folks artists, and adapted them for low-tech atmosphere. paintings with molds; shape over armatures; version or carve with concrete; and deal with concrete surfaces with mosaics, embedded gadgets, incising, and coloring. For magnificent open air elaborations, craft a Birdbath, Little woman Fountain, and extra. Twenty-five initiatives in all.

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5 propositionally as a family of tautologies K K;:'. Let X be a family of subsets of In]. We encode X with the underlying variables Pij, i :::; m and j :::; n, where Pij has the value True or False depending on whether a 1 or a 0 is in the (i,j) entry of the incidence matrix of X. The propositional formula, K K;:', states that either the set of subsets (described by the Pij'S) is not hereditary, or two subsets are the same, or for all k:::; n, IXI~k ~ I{O, ... ,m-l}l~k. We can express the fact that the Pij 's represent a hereditary family by the following formula: 1\ 1\ (Pij --+ V (-'P£j 1\ 1\ 1<£<111.

X n , such that when we evaluate C on II, ... , In" it does not output a satisfying truth assignment. Because the total number of 3CNF formulas on n variables is roughly 2 n3 , this takes about 20(n 3 ) symbols; thus the entire tautology is expressible in 20(n 3 ) symbols. Let us call the above family of tautologies NOTPOLYn , where n is the number of underlying variables. p;, The obvious way to prove this tautology is to go-through all possible circuits of size m, and for each of them, check all possible 3CN F formulas on n variables, and exhaustively check that for each one, the circuit errs on some input.

The first-order theory I ~o + ~o- PH P is defined to be the theory I ~o plus P HP( h) for every ~o -defined function h. 2 The theories equivalent. L. Bonet, S. Buss, and T. 6. 6. o-Bondy (this was first noted by Krajicek [16]). 6. o-formula which maps [a] one-to-one into [a - 1]. Define an a x a matrix A by letting its (i,j)-entry equal 1 if and only if h(j + 1) = i + 1. ) Then A violates Bondy's principle. Thus we have shown that if the pigeonhole principle fails, then Bondy's theorem fails. 6.

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