Product formulas for the 5-division points on the Tate normal form and the Rogers–Ramanujan continued fraction

Explicit formulas are proved for the 5-torsion points on the Tate normal form E5 of an elliptic curve having (X,Y)=(0,0) as a point of order 5. These formulas express the coordinates of points in E5[5]−⟨(0,0)⟩ as products of linear fractional quantities in terms of 5-th roots of unity and a parameter u, where the parameter b which defines the curve E5 is given as b=(ε5u5−ε−5)/(u5+1) and ε=(−1+5–√)/2. If r(τ) is the Rogers-Ramanujan continued fraction and b=r5(τ), then the coordinates of points of order 5 in E5[5]−⟨(0,0)⟩ are shown to be products of linear fractional expressions in r(5τ) with coefficients in Q(ζ5)

Solutions of diophantine equations as periodic points of pp-adic algebraic functions, III

All the periodic points of a certain algebraic function related to the Rogers-Ramanujan continued fraction $r(\tau)$ are determined. They turn out to be $0, \frac{-1 \pm \sqrt{5}}{2}$, and the conjugates over $\mathbb{Q}$ of the values $r(w_d/5)$, where $w_d$ is one of a specific set of algebraic integers, divisible by the square of a prime divisor of 5, in the field $K_d=\mathbb{Q}(\sqrt{-d})$, as $-d$ ranges over all negative quadratic discriminants for which $\left(\frac{-d}{5}\right) = +1$. This yields new insights on class numbers of orders in the fields $K_d$. Conjecture 1 of Part I is proved for the prime $p=5$, showing that the ring class fields over fields of type $K_d$ whose conductors are relatively prime to $5$ coincide with the fields generated over $\mathbb{Q}$ by the periodic points (excluding -1) of a fixed $5$-adic algebraic function.Comment: 38 page

Synthetic foundations of cevian geometry, III: The generalized orthocenter

In this paper, the third in the series, we define the generalized orthocenter $H$ corresponding to a point $P$, with respect to triangle $ABC$, as the unique point for which the lines $HA, HB, HC$ are parallel, respectively, to $QD, QE, QF$, where $DEF$ is the cevian triangle of $P$ and $Q=K \circ \iota(P)$ is the $isotomcomplement$ of $P$, both with respect to $ABC$. We prove a generalized Feuerbach Theorem, and characterize the center $Z$ of the cevian conic $\mathcal{C}_P$, defined in Part II, as the center of the affine map $\Phi_P = T_P \circ K^{-1} \circ T_{P'} \circ K^{-1}$, where $T_P$ is the unique affine map for which $T_P(ABC)=DEF$; $T_{P'}$ is defined similarly for the isotomic conjugate $P'=\iota(P)$ of $P$; and $K$ is the complement map. The affine map $\Phi_P$ fixes $Z$ and takes the nine-point conic $\mathcal{N}_H$ for the quadrangle $ABCH$ (with respect to the line at infinity) to the inconic $\mathcal{I}$, defined to be the unique conic which is tangent to the sides of $ABC$ at the points $D, E, F$. The point $Z$ is therefore the point where the nine-point conic $\mathcal{N}_H$ and the inconic $\mathcal{I}$ touch. This theorem generalizes the usual Feuerbach theorem and holds in all cases where the point $P$ is not on a median, whether the conics involved are ellipses, parabolas, or hyperbolas, and also holds when $Z$ is an infinite point. We also determine the locus of points $P$ for which the generalized orthocenter $H$ coincides with a vertex of $ABC$; this locus turns out to be the union of three conics minus six points. All our proofs are synthetic, and combine affine and projective arguments.Comment: 34 pages, 7 figure

Synthetic foundations of cevian geometry, I: Fixed points of affine maps in triangle geometry

We give synthetic proofs of many new results in triangle geometry, focusing especially on fixed points of certain affine maps which are defined in terms of the cevian triangle $DEF$ of a point $P$ with respect to a given triangle $ABC$, as well as the cevian triangle of the isotomic conjugate $P'$ of $P$ with respect to $ABC$. We prove a formula for the cyclocevian map in terms of the isotomic and isogonal maps using an entirely synthetic argument, and show that the complement $Q$ of the isotomic conjugate $P'$ has many interesting properties. If $T_P$ is the affine map taking $ABC$ to $DEF$, we show synthetically that $Q$ is the unique ordinary fixed point of $T_P$ when $P$ is any point not lying on the sides of triangle $ABC$, its anti-complementary triangle, or the Steiner circumellipse of $ABC$. We also show that $T_P(Q')=P$ if $Q'$ is the complement of $P$, and that the affine map $T_P T_{P'}$ is either a homothety or a translation which always has the $P$-ceva conjugate of $Q$ as a fixed point. Finally, we show that $P$ lies on the Steiner circumellipse if and only if $T_PT_{P'}=K^{-1}$, where $K$ is the complement map for $ABC$. This paper forms the foundation for several more papers to follow, in which the conic on the 5 points $A,B,C,P,Q$ is studied and its center is characterized as a fixed point of the map $\lambda=T_{P'} T_P^{-1}$.Comment: 25 pages, 6 figure

On sums of Rudin-Shapiro coefficients II

Let {a(n)} be the Rudin-Shapiro sequence, and let s(n) = ∑a(k) and t(n) = ∑(-1)k a(k). In this paper we show that the sequences {s(n)/√n} and {t(n)/√n} do not have cumulative distribution functions, but do have logarithmic distribution functions (given by a specific Lebesgue integral) at each point of the respective intervals [√3/5, √6] and [0, √3]. The functions a(x) and s(x) are also defined for real x ≥ 0, and the function [s(x) – a(x)]/√x is shown to have a Fourier expansion whose coefficients are related to the poles of the Dirichlet series ∑a(n)/n, where Re τ > ½

The Homflypt skein module of a connected sum of 3-manifolds

If M is an oriented 3-manifold, let S(M) denote the Homflypt skein module of M. We show that S(M_1 connect sum M_2) is isomorphic to S(M_1) tensor S(M_2) modulo torsion. In fact, we show that S(M_1 connect sum M_2) is isomorphic to S(M_1) tensot S(M_2) if we are working over a certain localized ring. We show the similar result holds for relative skein modules. If M contains a separating 2-sphere, we give conditions under which certain relative skein modules of M vanish over specified localized rings.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-31.abs.htm

Towards an Adaptive Skeleton Framework for Performance Portability

The proliferation of widely available, but very different, parallel architectures makes the ability to deliver good parallel performance on a range of architectures, or performance portability, highly desirable. Irregularly-parallel problems, where the number and size of tasks is unpredictable, are particularly challenging and require dynamic coordination. The paper outlines a novel approach to delivering portable parallel performance for irregularly parallel programs. The approach combines declarative parallelism with JIT technology, dynamic scheduling, and dynamic transformation. We present the design of an adaptive skeleton library, with a task graph implementation, JIT trace costing, and adaptive transformations. We outline the architecture of the protoype adaptive skeleton execution framework in Pycket, describing tasks, serialisation, and the current scheduler.We report a preliminary evaluation of the prototype framework using 4 micro-benchmarks and a small case study on two NUMA servers (24 and 96 cores) and a small cluster (17 hosts, 272 cores). Key results include Pycket delivering good sequential performance e.g. almost as fast as C for some benchmarks; good absolute speedups on all architectures (up to 120 on 128 cores for sumEuler); and that the adaptive transformations do improve performance

Periodic points of algebraic functions and Deuring’s class number formula

The exact set of periodic points in Q of the algebraic function ˆ F(z) = (−1±p1 − z4)/z2 is shown to consist of the coordinates of certain solutions (x, y) = ( , ) of the Fermat equation x4+y4 = 1 in ring class fields f over imaginary quadratic fields K = Q(p−d) of odd conductor f, where −d = dKf2 1 (mod 8). This is shown to result from the fact that the 2-adic function F(z) = (−1 + p1 − z4)/z2 is a lift of the Frobenius automorphism on the coordinates for which | |2 < 1, for any d 7 (mod 8), when considered as elements of the maximal unramified extension K2 of the 2-adic field Q2. This gives an interpretation of the case p = 2 of a class number formula of Deuring. An algebraic method of computing these periodic points and the corresponding class equations H−d(x) is given that is applicable for small periods. The pre-periodic points of ˆ F(z) in Q are also determined