We are interested in the maximum value achieved by the systole function over all complete finite area hyperbolic surfaces of a given signature (g, n). This maximum is shown to be strictly increasing in terms of the number of cusps for small values of n.We also show that this function is greater than a function that grows logarithmically in function of the ratio g/n.Date: November 2, 2018.2010 Mathematics Subject Classification. Primary: 30F10. Secondary: 32G15, 53C22.
Brooks and Makover introduced an approach to random Riemann surfaces based on associating a dense set of them -Belyi surfaces -with random cubic graphs. In this paper, using Bollobas model for random regular graphs, we examine the topological structure of these surfaces, obtaining in particular an estimate for the expected value of their genus.
We consider random Fibonacci sequences given by x n+1 = ±βx n + x n−1 . Viswanath [Divakar Viswanath, Random Fibonacci sequences and the number 1.13198824 . )] showed that when β = 1, lim n→∞ |x n | 1/n = 1.13 . . . , but his proof involves the use of floating point computer calculations. We give a completely elementary proof that 1.23375 (E(|x n |)) 1/n 1.12095 where E(|x n |) is the expected value for the absolute value of the nth term in a random Fibonacci sequence. We compute this expected value using recurrence relations which bound the sum of all possible nth terms for such sequences. In addition, we give upper and lower bounds for the second moment of the |x n |. Finally, we consider the conjecture of Embree and Trefethen [Mark Embree, Lloyd N. Trefethen, Growth and decay of random Fibonacci sequences, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455 (1987) (1999) 2471-2485], derived using computational calculations, that for values of β < 0.702585 such sequences decay. We show that as β decreases, the critical value where growth can change to decay is in fact 1/ √ 2.
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