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By Hathaway A.S.

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Let p0 , p1 , . . be non∞ negative, with pk = 1 and μ = kpk , and suppose that P (z) = 0 pk z k t is not a power series in z for any integer t > 1. Then |P (z)| < 1 for |z| < 1; in particular, (1 − P (z))−l is analytic in |z| < 1, say 1 = 1 − P (z) ∞ uk z k . k=0 22 B´ela Bollob´ as The Erd˝os-Feller-Pollard Theorem states that limk→∞ uk = 1/μ if μ < ∞ and uk → ∞ if μ = ∞. The theorem has important consequences in probability theory, and in 1951 de Bruijn and Erd˝ os also used it to study recursion formulae.

Nevertheless, for years no progress was made with the problem so that, eventually, Erd˝ os was tempted to offer $1,000 for a proof or disproof of this assertion. In 1984, Frankl and R¨ odl won the coveted prize when they showed that 1 − l−(r−1) is not a jump-value for r-graphs if r ≥ 3 and l > 2r. In spite of this beautiful result, we are very far from a complete characterization of jump-values. The important topic of Δ-systems was also initiated by Erd˝os. A family of sets {Aγ }γ∈Γ is called a Δ-system if any two sets have precisely the same intersection, that is if the intersection of any two of them is γ∈Γ Aγ .

Erd˝os and Kac proved in 1947 that, in this case, 2 arcsin x1/2 π for all x, 0 ≤ x ≤ 1. Thus Nn /n tends in distribution to the arc sin distribution. What Paul L´evy had proved in 1939 is that this arcsin law holds in the binomial case P(Xk = 1) = P(Xk = −1) = 1/2. In 1953 Erd˝os returned to this theme. In a joint paper with Hunt he proved that if X1 , X2 , . . are independent zero-mean random variables with the same continuous distribution which is symmetric about 0 then, almost surely, lim P(Nn /n < x) = n→∞ 18 B´ela Bollob´ as 1 n→∞ log n n lim k=1 sin Sk = 0.

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Analytical dynamics: being a synopsis of leading topics by Hathaway A.S.


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