Lossless image compression has often employed techniques quite separate from those used for text compression or lossy image compression; most standards today employ modelling followed by coding (e.g.â€¦ (More)

Given a large positive integer N , how quickly can one construct a prime number larger than N (or between N and 2N)? Using probabilistic methods, one can obtain a prime number in time at most log Nâ€¦ (More)

OPTIMAL CONTEXT TREES HARALD HELFGOTT Abstract. We show how to construct an optimal context tree for a given plaintext and context order. Our algorithm runs in time linear in the size of theâ€¦ (More)

For a global field K and an elliptic curve EÎ· over K(T ), Silvermanâ€™s specialization theorem implies rank(EÎ·(K(T ))) â‰¤ rank(Et(K)) for all but finitely many t âˆˆ P(K). If this inequality is strict forâ€¦ (More)

Let A be a subset of the primes. Let Î´P (N) = |{n âˆˆ A : n â‰¤ N}| |{n prime : n â‰¤ N}| . We prove that, if Î´P (N) â‰¥ C log log logN (log logN)1/3 for N â‰¥ N0, where C and N0 are absolute constants, then Aâ€¦ (More)

We the compare the number of phrases created by Ziv/Lempel '78 parsing of a binary sequence and of its reversal. We show that the two parsings can vary by a factor that grows at least as fast as theâ€¦ (More)

This submission to arXiv is the report of a panel session at the 2018 International Congress of Mathematicians (Rio de Janeiro, August). It is intended that, while v1 is that report, this stays aâ€¦ (More)