The first quantitatively correct pictorial atlas of the cut locus of a nontrivially deformed standard torus in $\mathbb{R}^3$ given a non-symmetrically placed starting point is presented along with a description of the software tool Loki used to generate it. Loki can compute the cut locus from a point on a genus-1 two-dimensional Riemannian manifold defined either by a parametrization or its metric, these to be given in closed form. The algorithm computes a piecewise polynomial approximation to the exponential map and inverts this numerically, thus correctly taking into account the global nature of the problem. As an example of its use in motivating and guiding traditional mathematical research, we provide a preliminary conjecture based upon the output of this software and both a counterexample and a proof motivated by the conjecture.

It has been conjectured for some time that the set of poles of a rotationally symmetric two-sheeted hyperboloid breaks into two disjoint sets if symmetry is broken by contraction perpendicular to the original axis of symmetry. We provide the first reliable visualizations of this process, confirming previous conjectures and motivating new ones.

The article Darmon (2002) proposes a conjectural p-adic analytic construction of points on (modular) elliptic curves, points which are defined over ring class fields of real quadratic fields. These points are related to classical Heegner points in the same way as Stark units to circular or elliptic units.

If K is a real quadratic field, the Stark-Heegner points attached to K are conjectured to satisfy an analogue of the Shimura reciprocity law, so that they can in principle be used to find explicit generators for the ring class fields of K. It is also expected that their heights can be expressed in terms of derivatives of the Rankin L-series attached to E and K, in analogy with the Gross-Zagier formula.

The main goal of this paper is to describe algorithms for calculating Stark-Heegner points and supply numerical evidence for the Shimura reciprocity and Gross-Zagier conjectures, focussing primarily on elliptic curves of prime conductor.

Let G be a simple algebraic group and P a parabolic subgroup of G. The group P acts on the Lie algebra $\mathfrak p_u$ of its unipotent radical $ P_u$ via the adjoint action. The modality of this action, mod ($P : \mathfrak p_u)$, is the maximal number of parameters upon which a family of P-orbits on $\mathfrak p_u$ depends. More generally, we also consider the modality of the action of P on an invariant subspace $\mathfrak n$ of $\mathfrak p_u$, that is mod ($P :\mathfrak n)$. In this note we describe an algorithmic procedure, called MOP, which allows one to determine upper bounds for mod ($P :\mathfrak n)$.

The classification of the parabolic subgroups P of exceptional groups with a finite number of orbits on $\mathfrak p_u$ was achieved with the aid of MOP. We describe the results of this classification in detail in this paper. In view of the results from Hille and Röhrle (1999), this completes the classification of parabolic subgroups of all reductive algebraic groups with this finiteness property.

Besides this result we present other applications of MOP, and illustrate an example.

Cubic forms in three variables are parametrised by points of a projective space $\P^9$. We study the subvarieties in this space defined by decomposable forms. Specifically, we calculate their equivariant minimal resolutions and describe their ideals invariant-theoretically.

In this paper we develop a method for an estimation of the degree of $C^0$-sufficiency of an analytic function germ $f:(\C^n,0)\to (\C,0)$. The difference between the estimation we give and the degree of $C^0$-sufficiency of $f$ will be $\leq 1$, by virtue of the results of Chang-Lu and Bochnak-Kucharz.

We consider the existence of power integral bases in composites of polynomial orders of number fields. We prove that if the degree of the composite field equals the product of the degrees of its subfields and the minimal polynomials of the generating elements of the polynomial orders have a multiple linear factor in their factorization modulo q, then the composite order admits no power integral bases. As an application we provide several examples including a parametric family of "simplest sextic fields.''

What's the best way to represent an isometry of hyperbolic 3-space $\bH^3$? Geometers traditionally worked in $\text{SL}(2,\bC)$, but for software development many now prefer the Minkowski space model of $\bH^3$ and the orthogonal group {O}(3,1). One powerful advantage is that ideas and computations in $S^3$ using matrices in {O}(4) carry over directly to $\bH^3$ and {O}(3, 1). Furthermore, {O}(3,1) handles orientation reversing isometries exactly as it handles orientation preserving ones. Unfortunately in computations one encounters a nagging dissimilarity between {O}(4) and {O}(3,1): while numerical errors in {O}(4) are negligible, numerical errors in {O}(3,1) tend to spiral out of control. The question we ask (and answer) in this article is, "Are exponentially compounded errors simply a fact of life in hyperbolic space, no matter what model we use? Or would they be less severe in $\text{SL}(2,\bC)$?" In other words, is numerical instability the Achilles' heel of {O}(3,1)?

Distinguished involutions in the affine Weyl groups, defined by G. Lusztig, play an essential role in the Kazhdan-Lusztig combinatorics of these groups. A distinguished involution is called canonical if it is the shortest element in its double coset with respect to the finite Weyl group. Each two-sided cell in the affine Weyl group contains precisely one canonical distinguished involution. We calculate the canonical distinguished involutions in the affine Weyl groups of rank ≤ 7. We also prove some partial results relating canonical distinguished involutions and Dynkin's diagrams of the nilpotent orbits in the Langlands dual group.

We discuss a method of producing computer assisted proofs of almost everywhere strong convergence of the d-dimensional Gauss algorithm. This algorithm is equivalent to Brun's algorithm and to the modified Jacobi-Perron algorithm considered by Podsypanin and Schweiger. In this paper we focus on the reduction of the problem to a finite number of calculations. These calculations have been carried out for the three-dimensional algorithm and the results, which prove almost everywhere strong convergence, will be published separately.

A proof that the three-dimensional Gauss algorithm is strongly convergent almost everywhere is given. This algorithm is equivalent to Brun's algorithm and to the modified Jacobi-Perron algorithm considered by Podsypanin and Schweiger. The proof involves the rigorous computer assisted estimation of the largest Lyapunov exponent of a cocycle associated to the algorithm. To the best of my knowledge, this is the first proof of almost everywhere strong convergence of a Jacobi-Perron type algorithm in dimension greater than two.

In a previous article, the authors described an algorithm to determine whether a finite metric polyhedral complex satisfied various local curvature conditions such as being locally CAT(0). The proof made use of Tarski's theorem about the decidability of first order sentences over the reals in an essential way, and thus it was not immediately applicable to a specific finite complex. In this article, we describe an algorithm restricted to 3-dimensional complexes which uses only elementary 3-dimensional geometry. After describing the procedure, we include several examples involving Euclidean tetrahedra which were run using an implementation of the algorithm in GAP.