TAILIEUCHUNG - Báo cáo toán học: "Limit Probabilities for Random Sparse Bit Strings"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Limit Probabilities for Random Sparse Bit Strings. | Limit Probabilities for Random Sparse Bit Strings Katherine St. John Department of Mathematics University of Pennsylvania Philadelphia Pennsylvania 19104 stjohn@ Submitted May 12 1997 Accepted October 14 1997 Abstract Let n be a positive integer c a real positive constant and p n c n. Let Un p be the random unary predicate under the linear order and Sc the almost sure theory of Un c. We show that for every first-order sentence Ộ fộ c lim Pr UJ c has property Ộ is an infinitely differentiable function. Further let S Tc Sc be the set of all sentences that are true in every almost sure theory. Then for every c 0 Sc S. Mathematical Reviews Subject Classification 03C13 60F20 68Q05 1 Introduction Let n be a positive integer and 0 p n 1. The random unary predicate Un p is a probability space over predicates U on n 1 . n with the probabilities determined by Pr U x p n for 1 x n and the events U x are mutually independent over 1 x n. Un p is also called the random bit string. Let Ộ be a first-order sentence in the language with linear order and the unary predicate. In 7 Shelah and Spencer showed that for every such sentence Ộ and for p n n-1 or n-1 k p n n-1 k 1 there exists a constant a such that Jim Pr Un p ộ aệ 1 Current Address Department of Mathematics Santa Clara University Santa Clara CA 950530290 kstjohn@. 1 THE ELECTRONIC .JOURNAL OF COMBINATORICS 4 1997 R23 2 Note that Un p 1 0 means that Un p has property 0. See Section 2 for this and other dehnitions. For each real constant c let Sc be the almost sure theory of the linear ordering with p n n. That is Sc 0 I Ji Pr Un n 0 1 Let To be the almost sure theory for p n n-1 and T1 be the almost sure theory for n-1 p n n-1 2. By the work of Dolan 2 Un p satishes the 0-1 law for p n n-1 and n-1 p n n-1 2 that is for every 0 a 0 or 1 in Equation 1 . This gives that To and T1 are complete theories. Dolan also showed that the 0-1 Law does not hold for n-1 k p n n-1 k 1 k 1. In this paper we will .

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