A method for computing quadratic Brunovsky forms

In this paper, for continuous, linearly-controllable quadratic control systems with a single input, an explicit, constructive method is proposed for studying their Brunovsky forms, initially studied in [W. Kang and A. J. Krener, Extended quadratic controller normal form and dynamic state feedback linearization of nonlinear systems, SIAM Journal on Control and Optimization, 30:1319-1337, 1992]. In this approach, the computation of Brunovsky forms and transformation matrices and the proof of their existence and uniqueness are carried out simultaneously. In addition, it is shown that quadratic transformations in the aforementioned paper can be simplified to prevent multiplicity in Brunovsky forms. This method is extended for studying discrete quadratic systems. Finally, computation algorithms for both continuous and discrete systems are summarized, and examples demonstrated.Comment: Author's name was listed as Wenlong Ji

Generalized modularity matrices

Various modularity matrices appeared in the recent literature on network analysis and algebraic graph theory. Their purpose is to allow writing as quadratic forms certain combinatorial functions appearing in the framework of graph clustering problems. In this paper we put in evidence certain common traits of various modularity matrices and shed light on their spectral properties that are at the basis of various theoretical results and practical spectral-type algorithms for community detection

Exact arithmetic on the Stern–Brocot tree

AbstractIn this paper we present the Stern–Brocot tree as a basis for performing exact arithmetic on rational numbers. There exists an elegant binary representation for positive rational numbers based on this tree [Graham et al., Concrete Mathematics, 1994]. We will study this representation by investigating various algorithms to perform exact rational arithmetic using an adaptation of the homographic and the quadratic algorithms that were first proposed by Gosper for computing with continued fractions. We will show generalisations of homographic and quadratic algorithms to multilinear forms in n variables. Finally, we show an application of the algorithms for evaluating polynomials

A mesh algorithm for principal quadratic forms

In 1970 a negative solution to the tenth Hilbert problem, concerning the determination of integral solutions of diophantine equations, was published by Y. W. Matiyasevich. Despite this result, we can present algorithms to compute integral solutions (roots) to a wide class of quadratic diophantine equations of the form q(x) = d, where q : Z is a homogeneous quadratic form. We will focus on the roots of one (i.e., d = 1) of quadratic unit forms (q11 = ... = qnn = 1). In particular, we will describe the set of roots Rq of positive definite quadratic forms and the set of roots of quadratic forms that are principal. The algorithms and results presented here are successfully used in the representation theory of finite groups and algebras. If q is principal (q is positive semi-definite and Ker q={v ∈ Zn; q(v) = 0}= Z · h) then |Rq| = ∞. For a given unit quadratic form q (or its bigraph), which is positive semi-definite or is principal, we present an algorithm which aligns roots Rq in a Φ-mesh. If q is principal (|Rq| is less than ∞), then our algorithm produces consecutive roots in Rq from finite subset of Rq, determined in an initial step of the algorithm