We propose a construction of lattices from (skew-) polynomial codes, by endowing quotients some ideals in both number fields and cyclic algebras with suitable trace form. give criteria for unimodularity. This yields integral unimodular multiplicative structure. Examples are provided.

Ramanujans Ternary Quadratic Form represents a series of numbers that satisfy a tripartite quadratic relation. In the present work, we examine the sequence of numbers generated by such forms and other related forms obtained by varying the coefficients and exponents to other values. Chaotic characterization using standard techniques such as Lyapunov Exponents, Kolmogorov Entropy, Fractal Dimensi...

It is a well-known theorem in linear models that the idempotency of a matrix is a sufficient and necessary condition for a quadratic form in normal variates to have a chi-square distribution, but its proofs in the early literature were incorrect or incomplete. Driscoll (1999) provided an improved proof, and this article presents a simple proof. More importantly, we establish and prove a general...

The systematic study of quadratic forms over an arbitrary field of characteristic = 2 was initiated by Witt [W]. Two milestone papers in the area are a paper of Pfister [P0] relating quadratic forms and orderings and the paper of Milnor [M] pointing out a possible relationship between the Witt ring of quadratic forms and the Galois cohomology of the field (a relationship eventually verified by ...

Let Q be a nondegenerate quadratic form, and L is a nonzero linear form of dimension d > 3. As a generalization of the Oppenheim conjecture, we prove that the set {(Q(x), L(x)) : x ∈ Z} is dense in R provided that Q and L satisfy some natural conditions. The proof uses dynamics on homogeneous spaces of Lie groups.

     The quadratic loss function has been used by decision-theoretic statisticians and economists for many years.  In this paper  the estimation of scale parameter under a bounded loss function, which is adequate for assessing quality and quality improvement, is considered with restriction to the principles of invariance and risk unbiasedness. An implicit form of minimum risk scale equivariant ...