We completely solve the family of relative Thue equations x − (t − 1)xy − (t+ 2)xy − y = μ, where the parameter t, the root of unity μ and the solutions x and y are integers in the same imaginary quadratic number field. This is achieved using the hypergeometric method for |t| ≥ 53 and Baker’s method combined with a computer search using continued fractions for the remaining values of t.

We concentrate on automatic addition of untimed and real-time UNITY properties to programs by local redesign. The main focus of this paper is to identify instances where addition of UNITY properties can be achieved efficiently (in polynomial time) and where the problem of adding UNITY properties is difficult (NP-complete). Regarding addition of UNITY properties in polynomial time, we present a ...

lanczos-type algorithms are well known for their inherent instability. they typically breakdown occurs when relevant orthogonal polynomials do not exist. current approaches to curing breakdown rely on jumping over the non-existent polynomials to resume computation. this may have to be used many times during the solution process. we suggest an alternative to jumping, which consists of restarting...

The Volume Conjecture loosely states that the limit of the n-th colored Jones polynomial of a hyperbolic knot, evaluated at the primitive complex n-th root of unity is a sequence of complex numbers that grows exponentially. Moreover, the exponential growth rate is proportional to the hyperbolic volume of the knot. We provide an efficient formula for the colored Jones function of the simplest hy...

We evaluate multiple polylogarithm values at sixth roots of unity up to weight six, i.e. of the form G(a1, . . . , aw; 1) where the indices ai are equal to zero or a sixth root of unity, with a1 6= 1. For w ≤ 6, we construct bases of the linear spaces generated by the real and imaginary parts of G(a1, . . . , aw; 1) and obtain a table for expressing them as linear combinations of the elements o...

An exact S-matrix is conjectured for the imaginary coupled d (3) 4 affine Toda field theory, using the U q (g (1) 2) symmetry. It is shown that this S-matrix is consistent with the results for the case of real coupling using the breather-particle correspondence. For q a root of unity it is argued that the theory can be restricted to yield Φ(11|14) perturbations of W A 2 minimal models and the r...