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...

The lattice Boussinesq equation (BSQ) is a three-component difference-difference equation defined on an elementary square of the 2D lattice, having 3D consistency. We write the equations in the Hirota bilinear form and construct their multisoliton solutions in terms of Casoratians, following the methodology in our previous papers. In the construction it turns out that instead of the usual discr...

Let F q be the finite field of q elements with characteristic p and F q m its extension of degree m. Fix a nontrivial additive character ψ of F p. For any Laurent polynomial −1 n ], we form the exponential sum S * m (f) := The corresponding L-function L * (f, t) is defined by L * (f, t) := exp (∞ ∑ m=0 S * m (f) t m m). The corresponding L-function L(f, t) is defined as follows L(f, t) := exp (...

Let f1, f2 : CN −→ CN be affine maps fi(x) := Aix + yi (where each Ai is an N -by-N matrix and yi ∈ CN ), and let x1,x2 ∈ AN (C) such that xi is not preperiodic under the action of fi for i = 1, 2. If none of the eigenvalues of the matrices Ai is a root of unity, then we prove that the set {(n1, n2) ∈ N0 : f n1 1 (x1) = f n2 2 (x2)} is a finite union of sets of the form {(m1k + `1,m2k + `2) : k...

We consider 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. We prove that there are only trivial solutions (with |x|, |y| ≤ 1), if |t| is large enough or if the discriminant of the quadratic number field is large enough or if Re t = −1/2 (there a...

We present a new construction for two-dimensional, perfect autocorrelation arrays over roots of unity. These perfect arrays are constructed from a block of perfect column sequences. Other blocks are constructed from the first block, to generate a block-circulant structure. The columns are then multiplied by a perfect sequence over roots of unity, which, when folded into an array commensurate wi...

sends the point z ∈ C to its inversion image about the circle of radius r ∈ R centered at the point a ∈ C. Dynamically, this map is not interesting since iterating twice yields the identity mapping. Using three circles we invert a point z about each circle and form a new map by sending z to the arithmetic mean of the three inversion images. We refer to this as inverting z about the three circle...