Hyperbolic spaces and quadratic forms

Applications of quadratic D-forms to generalized quadratic forms

In this paper, we study generalized quadratic forms over a division algebra with involution of the first kind in characteristic two. For this, we associate to every generalized quadratic from a quadratic form on its underlying vector space. It is shown that this form determines the isotropy behavior and the isometry class of generalized quadratic forms.

Decomposable quadratic forms in Banach spaces

A continuous quadratic form on a real Banach space X is called decomposable if it is the difference of two nonnegative (i.e., positively semidefinite) continuous quadratic forms. We prove that if X belongs to a certain class of superreflexive Banach spaces, including all Lp(μ) spaces with 2 ≤ p < ∞, then each continuous quadratic form on X is decomposable. On the other hand, on each infinite-di...

Hyperbolic uniformizations through computations on ternary quadratic forms

Orders in indefinite quaternion algebras provide Fuchsian groups acting on the Poincare half-plane, used to construct the associated Shimura curves. We explain how, by using embedding theory, the elements of those Fuchsian groups depend on representations of integers by suitable ternary quadratic forms. Thus the explicit computation of those representations leads to explicit presentations and f...

Minimization of Constrained Quadratic Forms in Hilbert Spaces

A common optimization problem is the minimization of a symmetric positive definite quadratic form 〈x, Tx〉 under linear constraints. The solution to this problem may be given using the Moore–Penrose inverse matrix. In this work at first we extend this result to infinite dimensional complex Hilbert spaces, where a generalization is given for positive operators not necessarily invertible, consider...

Quadratic Functions, Optimization, and Quadratic Forms