the mordell-weil theorem states that the group of rational points‎ ‎on an elliptic curve over the rational numbers is a finitely‎ ‎generated abelian group‎. ‎in our previous paper, h‎. ‎daghigh‎, ‎and s‎. ‎didari‎, on the elliptic curves of the form $ y^2=x^3-3px$‎, ‎‎bull‎. ‎iranian math‎. ‎soc‎.‎‎ 40 (2014)‎, no‎. ‎5‎, ‎1119--1133‎.‎, ‎using selmer groups‎, ‎we have shown that for a prime $p$...

We first define the height function on a domino tiling (as done in [1]) and state some of its basic properties. We then revisit the coupling function and relate it to Green’s function, which allows us to conclude that the coupling function converges in the limit to an analytic function with a pole. Using this, we do a general second moment calculation via the proof found in [1] which writes the...

The interest is in models of integer-valued height functions on shift-invariant planar graphs whose maximum degree three. We prove delocalisation for induced by convex nearest-neighbour potentials, under the condition that each potential function an excited potential, is, a symmetric $V$ with property $V(\pm1)\leq V(0)+\log2$. Examples such include discrete Gaussian and solid-on-solid at invers...

We prove a function field version of the Bounded Height Conjecture formulated by Chatzidakis, Ghioca, Masser and Maurin in 2013.