Minima Of Indefinite Quadratic Forms Research Articles

Symmetry of topographs of Markoff forms

The Markoff spectrum is defined as the set of normalized values of arithmetic minima of indefinite quadratic forms. In the theory of the Markoff spectrum we observe various kinds of symmetry. Each of Conway’s topographs of quadratic forms which give values in the discrete part of the Markoff spectrum has a special infinite path consisting of edges. It has symmetry with respect to a translation along the path and countable central symmetries by which the path is invariant. We prove that these properties are obtained from the fact that the path is a discretization of a geodesic in the upper half-plane which corresponds to a value of the discrete part of the Markoff spectrum and projects to a simple closed geodesic on the once punctured torus with the highest degree of symmetry.

On palindromes with three or four letters associated to the Markoff spectrum

Let N(2) and N(5) be the sets of integer solutions of 2x2+y12+y22=4xy1y2 and 5x2+y12+y22=5xy1y2, respectively. The elements of these sets give sequences in the non-discrete part of the Markoff spectrum consisting of the normalized values of arithmetic minima of indefinite quadratic forms. Each of these sets can be represented by a bipartite graph. Using this we can construct quadratic forms giving the values of the sequences. We show that, for a quadratic form f which gives a value of the Markoff spectrum corresponding to an integer solution x of N(2) and N(5), the roots of f(ξ,1)=0 have a periodic continued fraction expansion and the period is a palindrome with fixed prefix and suffix on {1,2,3} and {1,2,3,4}, respectively.

Krein space-based H∞ adaptive smoother design for a class of Lipschitz nonlinear discrete-time systems

In this paper, the problem of H∞ adaptive smoother design is addressed for a class of Lipschitz nonlinear discrete-time systems with l2 bounded disturbance input. By comprehensively analyzing the H∞ performance, Lipschitz conditions and unknown parameter’s bounded condition, a positive minimum problem for an indefinite quadratic form is introduced such that the H∞ adaptive smoothing problem is achieved. A Krein space stochastic system with multiple fictitious outputs is constructed by associating with the minimum problem of the introduced indefinite quadratic form. The minimum of indefinite quadratic form is derived in the form of innovations through utilizing Krein space orthogonal projection and innovation analysis approach. Via choosing the suitable fictitious outputs to guarantee the minimum of indefinite quadratic form is positive, the existence condition of the adaptive smoother and its analytical solutions are obtained in virtue of nonstandard Riccati difference equations. The quality of the estimator is checked on an example.

Formula omitted] fixed-lag smoothing for linear discrete time-varying systems with uncertain observations

This paper deals with the problem of H∞ fixed-lag smoothing for linear discrete time-varying systems with uncertain observations and l2-norm bounded noise. First, the design of H∞ smoother is equivalent to the problem that a certain indefinite quadratic form with respect to a new state-space constraints has a minimum and the smoother is such that the minimum is positive. Then, by introducing a Krein space stochastic system and applying re-organized innovation analysis approach, the minimum of indefinite quadratic form and its existence condition are derived. Finally, through analyzing the existence condition of the minimum and guaranteeing the positivity of the minimum, a sufficient and necessary condition for the existence of an H∞ smoother is proposed and the smoother is obtained in terms of two standard Riccati difference equations. A numerical example is given to show the effectiveness of the proposed method.

Inhomogeneous minimum of indefinite quadratic forms in six variables: A conjecture of Watson

The famous conjecture of Watson(11) on the minima of indefinite quadratic forms in n variables has been proved for n ≤ 5, n ≥ 21 and for signatures 0 and ± 1. For the details and history of the conjecture the reader is referred to the author's paper(8). In the succeeding paper (9), we prove Watson's conjecture for signature ± 2 and ± 3 and for all n. Thus only one case for n = 6 (i.e. forms of type (1, 5) or (5, 1)) remains to he proved which we do here; thereby completing the case n = 6. This result is also used in (9) for proving the conjecture for all quadratic forms of signature ± 4. More precisely, here we prove:Theorem 1. Let Q6(x1, …, x6) be a real indefinite quadratic form in six variables of determinant D ( < 0) and of type (5, 1) or (1, 5). Then given any real numbers ci, 1 ≤ i ≤ 6, there exist integers x1,…, x6such that

The inhomogeneous minima of indefinite quadratic forms

If f(x) is a real indefinite quadratic form in n variables with determinant d ≠ 0, we set for any real α.