恒隆數學獎

8. 香港培正中學 The Johnson-Leader-Russell Question on Square Posets We study the problem of finding the maximum number of maximal chains in a given size-k subset of a square poset [n] x [n]. This was proposed by Johnson, Leader, and Russell but not yet solved. Kittipassorn had given a conjectural solution to the problem. We verify Kittipassorn’s conjecture for 0 ≤ k ≤ 3n-2 and solve a variant problem for the case 3n-1 ≤ k ≤ 4n-4, which also supports the conjecture. For general k, we find that the optimal configuration is given by a 1-Lipschitz function. We also generalize the problem to rectangle posets and give a solution to one particular poset. 9. 聖公會林護紀念中學 FromHappy Function to Linear Happy Functions Given p, q ∈ N and b ∈ N/{1}, the linear happy function is defined as H(x) = ph(x) – q, where h(x) is the ‘happy function’ that maps a positive integer to the squares of its base-b digits. When there exists a positive integer m such that the m-th iterate of H(x) on x is 1, i.e. Hm(x) = 1, then x is called a ’happy number’ in the linear happy function. In this paper, we found the necessary and sufficient conditions for p and q so that happy number exists. We have also investigated the existence of different forms of happy numbers, and found a formula to find L such that H(x) < x for all x ≥ L.

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