1. Four equivalent definitions of a quadratic form 2 2. Action of Mn(K) on n-ary quadratic forms 4 3. The category of quadratic spaces 7 4. Orthogonality in quadratic spaces 9 5. Diagonalizability of Quadratic Forms 11 6. Isotropic and hyperbolic spaces 13 7. Witt’s theorems: statements and consequences 15 8. Orthogonal groups, reflections and the proof of Witt Cancellation 17 8.1. The orthogon...

We consider convex relaxations for the problem of minimizing a (possibly nonconvex) quadratic objective subject to linear and (possibly nonconvex) quadratic constraints. Let F denote the feasible region for the linear constraints. We first show that replacing the quadratic objective and constraint functions with their convex lower envelopes on F is dominated by an alternative methodology based ...

This paper concerns binary quadratic forms over F[T ]. It develops theory analogous to the theory of binary quadratic forms over Z. Most although not all of the results are almost identical, while some of the proofs require different techniques. In particular, the form class group is determined when the form takes values in a principal ideal domain, and the ideal class group (and class group is...

Abstract: This paper discusses the quadratic stability and quadratic stabilization problem for a class of nonlinear perturbed discrete time-delay systems. Necessary and sufficient conditions for quadratic stability are presented via S-procedure technique and linear matrix inequality (LMI). Both static and dynamic output feedback controllers are constructed respectively. Furthermore, necessary a...

In his book on compositions of quadratic forms, Shapiro asks whether a quadratic form decomposes as a tensor product of quadratic forms when its adjoint involution decomposes as a tensor product of involutions on central simple algebras. We give a positive answer for quadratic forms defined over local or global fields and produce counterexamples over fields of rational fractions in two variable...

We give some new canonical representations for forms over C. For example, a general binary quartic form can be written as the square of a quadratic form plus the fourth power of a linear form. A general cubic form in (x1, . . . , xn) can be written uniquely as a sum of the cubes of linear forms `ij(xi, . . . , xj), 1 ≤ i ≤ j ≤ n. A general ternary quartic form is the sum of the square of a quad...

In a paper of Kim, Chan, and Rhagavan, the universal ternary classical quadratic forms over quadratic fields of positive discriminant were discovered. Here a proof of the universality of some of these quadratic forms is given using a technique of Liouville. Another quadratic form over the field of discriminant 8 is shown universal by a different elementary approach. Mathematics Subject Classifi...

Several common flexible functional forms are evaluated for Texas agricultural production utilizing three procedures. Nested hypothesis tests indicate that the normalized quadratic is the marginally-preferred functional form followed by the generalized Leontief. Predictive accuracy results are ambiguous between the generalized Leontief and the normalized quadratic. Statistical performance favors...

The two main features of the commented paper are: the class of systems considered and the approach since theproblemsare rather standard – belongingtowhatwe would like to call linear quadratic theory – any theory dealing with a couple ‘‘linear equationþ ‘integral’ quadratic index’’. The most common case of such a couple is represented by ‘‘linear differential equationþ integral quadratic index’’...