We present a systematic technique to find explicit solutions of birational maps, provided that these solutions are given in terms of elliptic functions. The two main ingredients are the following: (i) application of classical addition theorems for elliptic functions and (ii) experimental technique to detect an algebraic curve containing a given sequence of points in a plane. These methods are applied to Kahan–Hirota–Kimura discretizations of the periodic Volterra chains with three and four particles.
This is an Accepted Manuscript of an article published by Taylor & Francis in “Experimental mathematics” on 24th August 2016, available online: http://www.tandfonline.com/doi/full/10.1080/10586458.2016.1166354
Let q = 3 be a period. There are at least two (1, q)-periodic trajectories inside any smooth strictly convex billiard table. We quantify the chaotic dynamics of axisymmetric billiard tables close to their boundaries by studying the asymptotic behavior of the differences of the lengths of their axisymmetric (1, q)-periodic trajectories as q ¿ +8. Based on numerical experiments, we conjecture that, if the billiard table is a generic axisymmetric analytic strictly convex curve, then these differences behave asymptotically like an exponentially small factor q-3e-rq times either a constant or an oscillating function, and the exponent r is half of the radius of convergence of the Borel transform of the well-known asymptotic series for the lengths of the (1, q)-periodic trajectories. Our experiments are focused on some perturbed ellipses and circles, so we can compare the numerical results with some analytical predictions obtained by Melnikov methods. We also detect some non-generic behaviors due to the presence of extra symmetries. Our computations require a multiple-precision arithmetic and have been programmed in PARI/GP.
ATR points were introduced by Darmon as a conjectural con-
struction of algebraic points on certain elliptic curves for which
the Heegner-point method is not in general available. So far,
the only numerical evidence, provided by Darmon–Logan and
In those special cases, the ATR points can be obtained from the
already existing Heegner points, thanks to results from Zhang
In this paper, we compute for the first time an algebraic ATR point
on a curve that is not uniformizable by any Shimura curve, thus
providing the first piece of numerical evidence that Darmon’s
construction works beyond geometric modularity. To this pur-
pose, we improve the method proposed by Darmon and Logan
by removing the requirement that the real quadratic base field
be norm-Euclidean and accelerating the numerical integration
of Hilbert modular forms.
The Monotone Upper Bound Problem asks for the maximal number M(d,n) of vertices on a strictly-increasing edge-path on a simple d-polytope with n facets. More specifically, it asks whether the upper bound
M(d,n) ≤ Mubt(d,n)
provided by McMullen’s (1970) Upper Bound Theorem is tight, where Mubt(d,n) is the number of vertices of a dual-to-cyclic d-polytope with n facets.
It was recently shown that the upper bound M(d,n) ≤ Mubt(d,n) holds with equality for small dimensions (d ≤ 4: Pfeifle, 2003) and for small corank (n ≤ d + 2: Gärtner et al., 2001). Here we prove that it is not tight in general: In dimension d=6 a polytope with n=9 facets can have Mubt(6,9)=30 vertices, but not more than 27 ≤ M(6,9) ≤ 29 vertices can lie on a strictly-increasing edge-path.
The proof involves classification results about neighborly polytopes, Kalai’s (1988)
concept of abstract objective functions, the Holt-Klee conditions (1998), explicit
enumeration, Welzl’s (2001) extended Gale diagrams, randomized generation of instances,
as well as non-realizability proofs via a version of the Farkas lemma.