https://ogma.newcastle.edu.au/vital/access/ /manager/Index ${session.getAttribute("locale")} 5 Ramanujan-type supercongruences https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:11918 Wed 11 Apr 2018 16:53:56 AEST ]]> Linear independence of values of Tschakaloff functions with different parameters https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:11878 qt(z)= ∞ / ∑ / ν=0 q-tν(ν+1)/2zν and Θ(q-t,z)= ∞ / ∑ / ν=-∞ q-tν²zν at different rational points z ≠ 0 and with different positive integer parameters t, where q ∈ ℤ {0, ± 1}.]]> Wed 11 Apr 2018 15:16:43 AEST ]]> Rational tetrahedra with edges in geometric progression https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:4995 Wed 11 Apr 2018 14:45:34 AEST ]]> Cyclic q-MZSV sum https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:11894 Wed 11 Apr 2018 12:36:01 AEST ]]> Euler's factorial series and global relations https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:44631 ∞k=0 K! (—z)k, we address arithmetic and analytical questions related to its values in both p-adic and Archimedean valuations.]]> Tue 18 Oct 2022 13:31:37 AEDT ]]> Explicit Drinfeld moduli schemes and Abhyankar’s generalized iteration conjecture https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:40248 k be a field containing Fq. Let ψ be a rank r Drinfeld Fq[t]-module determined by ψt(X) = tX + a₁Xq+···+ar−1Xqr−1+Xqr, where t, a₁,...,ar−1 are algebraically independent over k. Let n ∈ Fq[t] be a monic polynomial. We show that the Galois group of ψₙ(X) over k(t, a₁,...,ar−1) is isomorphic to GLr(Fq[t]/nFq[t]), settling a conjecture of Abhyankar. Along the way we obtain an explicit construction of Drinfeld moduli schemes of level tn. Video. For a video summary of this paper, please visit https://youtu.be/TInrNq02-UA.]]> Thu 28 Jul 2022 11:09:26 AEST ]]> On the generating function of the integer part: [nα + γ] https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:14077 n and qn are the numerators and denominators of the convergents of the continued fraction expansion of α and t**n and s**n are particular algorithmically generated sequences of best approximates for the non-homogeneous diophantine approximation problem of minimizing |nα + γ − m|. This generalizes results of Böhmer and Mahler, who considered the special case where γ = 0. This representation allows us to easily derive various transcendence results. For example, ∑∞n=1 [ne +1/2 ]/2n is a Liouville number. Indeed the first series is Liouville for rational z, w∈ [−1, 1] with |zw| ≠ 1 provided α has unbounded continued fraction expansion. A second application, which generalizes a theorem originally due to Lord Raleigh, is to give a new proof of a theorem of Fraenkel, namely [nα + γ]∞n=1 and [nα′ + γ′]∞n=1 partition the non-negative integers if and only if 1/α + 1/α′ = 1 and γ/α + γ′/α′ = 0 (provided some sign and integer independence conditions are placed on α, β, γ, γ′). The analysis which leads to the results is quite delicate and rests heavily on a functional equation for G. For this a natural generalization of the simple continued fraction to Kronecker′s forms |nα + γ − m| is required.]]> Sat 24 Mar 2018 08:22:34 AEDT ]]> On simultaneous diophantine approximations to ζ(2) and ζ(3) https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:19583 Sat 24 Mar 2018 07:58:20 AEDT ]]> Cyclic polygons with rational sides and area https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:4922 Sat 24 Mar 2018 07:21:12 AEDT ]]> Drinfeld modular polynomials in higher rank II: Kronecker congruences https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:40246 Fri 08 Jul 2022 13:23:56 AEST ]]>