We prove an analogue of the Oppenheim conjecture for a system comprising inhomogeneous quadratic form and linear in 3 variables using dynamics on space affine lattices.

Abstract— A linear space-time block code (STBC) is a vector space spanned by its defining weight matrices over the real number field. We define a Quadratic Form (QF), called the Hurwitz-Radon QF (HRQF), on this vector space and give a QF interpretation of the ML decoding complexity of a STBC. It is shown that the ML decoding complexity is only a function of the weight matrices defining the code...

Quadratic forms capture multivariate information in a single number, making them useful, for example, in hypothesis testing. When a quadratic form is large and hence interesting, it might be informative to partition the quadratic form into contributions of individual variables. In this paper it is argued that meaningful partitions can be formed, though the precise partition that is determined w...

We explore methods for determining the underlying structure of certain classes of continued fractions . The goal is to develop closed form expressions for the continued fractions of many quadratic irrationals. Consider a finite difference equation satisfying: • Gn+1 = anGn + bnGn−1. • an = m, and bn = l for all n, where m, l ∈ N note: m = l = 1 gives the Fibonacci numbers. Let Gn denote the n t...

An n-ary integral quadratic form is a formal expression Q(x1, · · · , xn) = ∑ 1≤i,j≤n aijxixj in nvariables x1, · · · , xn, where aij = aji ∈ Z. We present a randomized polynomial time algorithm that given a quadratic form Q(x1, · · · , xn), a prime p, and a positive integer k outputs a U ∈ GLn(Z/p Z) such that U transforms Q to its p-adic canonical form.