https://nova.newcastle.edu.au/vital/access/ /manager/Index en-au 5 https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:14231 n-1 ≡ -1 mod n, then n must be a prime. We survey what is known about this interesting and now fairly old conjecture. Giuga proved that n is a counterexample to his conjecture if and only if each prime divisor p of n satisfies (p - 1) | (n/p - 1) and p | (n/p - 1). Using this characterization, he proved computationally that any counterexample has at least 1,000 digits; equipped with more computing power, E. Bedocchi later raised this bound to 1,700 digits. By improving on their method, we determine that any counterexample has at least 13,800 digits. We also give some new results on the second of the above conditions. This leads, in our opinion, to some interesting questions about what we call Giuga numbers and Giuga sequences.]]> Wed 11 Apr 2018 15:58:03 AEST ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:13055 0 they are s-Hölder, and so proximally, subdifferentiable only on dyadic rationals and nowhere else. As applications we construct Lipschitz functions with prescribed Hölder and approximate subderivatives.]]> Sat 24 Mar 2018 08:15:39 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:13075 p version of the classical Blaschke-Santalo inequality for polar volumes as a consequence of more subtle convexity estimates for the volume of the p -ball in Euclidean space. We also give analogs for the (p, q) -substitution norm.]]> Sat 24 Mar 2018 08:15:38 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:13123 Sat 24 Mar 2018 08:15:04 AEDT ]]>