https://ogma.newcastle.edu.au/vital/access/ /manager/Index en-au 5 https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:11249 Wed 11 Apr 2018 16:29:26 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:7552 Wed 11 Apr 2018 16:00:04 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:12924 n(r₁,...,r_n) = 1/π²[formula could not be replicated]. By virtue of striking connections with Jacobi ϑ-function values, we are able to develop new closed forms for certain values of the coordinates r_k, and extend such analysis to similar lattice sums. A primary result is that for rational x, y, the natural potential ⏀²(x, y) is 1/π log A where A is an algebraic number. Various extensions and explicit evaluations are given. Such work is made possible by number-theoretical analysis, symbolic computation and experimental mathematics, including extensive numerical computations using up to 20,000-digit arithmetic.]]> Sat 24 Mar 2018 10:37:10 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:13095 where →/r runs over the unit n-cube, with →/q and s fixed, explicitly: ∫₀¹...∫₀¹((r₁-q₁)²+...+(r_n-q_n)²)^s/2dr₁...dr_n. The study of box integrals leads one naturally into several disparate fields of analysis. While previous studies have focused upon symbolic evaluation and asymptotic analysis of special cases (notably s = 1), we work herein more generally—in interdisciplinary fashion—developing results such as: (1) analytic continuation (in complex s), (2) relevant combinatorial identities, (3) rapidly converging series, (4) statistical inferences, (5) connections to mathematical physics, and (6) extreme-precision quadrature techniques appropriate for these integrals. These intuitions and results open up avenues of experimental mathematics, with a view to new conjectures and theorems on integrals of this type.]]> Sat 24 Mar 2018 10:35:17 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:10012 Sat 24 Mar 2018 10:33:33 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:10329 Sat 24 Mar 2018 10:31:09 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:6447 Sat 24 Mar 2018 10:23:28 AEDT ]]>