Approximate zero polynomials of polynomial matrices and linear systems

This paper introduces the notions of approximate and optimal approximate zero polynomial of a polynomial matrix by deploying recent results on the approximate GCD of a set of polynomials Karcaniaset al. (2006) 1 and the exterior algebra Karcanias and Giannakopoulos (1984) 4 representation of polynomial matrices. The results provide a new definition for the "approximate", or "almost" zeros of polynomial matrices and provide the means for computing the distance from non-coprimeness of a polynomial matrix. The computational framework is expressed as a distance problem in a projective space. The general framework defined for polynomial matrices provides a new characterization of approximate zeros and decoupling zeros Karcanias et al. (1983) 2 and Karcanias and Giannakopoulos (1984) 4 of linear systems and a process leading to computation of their optimal versions. The use of restriction pencils provides the means for defining the distance of state feedback (output injection) orbits from uncontrollable (unobservable) families of systems, as well as the invariant versions of the "approximate decoupling polynomials". The overall framework that is introduced provides the means for introducing measures for the distance of a system from different families of uncontrollable, or unobservable systems, which may be feedback dependent, or feedback invariant as well as the notion of "approximate decoupling polynomials"

Clinical Correlates and Cognitive Underpinnings of Uncontrollable Worry

Uncontrollable anxious thought characterizes a number of emotional disorders and has been linked to impaired emotional and physical health. The present research aimed to clarify the nature of uncontrollable worry as a clinical and cognitive construct. Chapter 1 evaluated uncontrollability of worry as a diagnostic criterion for generalized anxiety disorder (GAD). This study was conducted in response to the proposed removal of the uncontrollability of worry requirement from the diagnostic criteria for GAD in DSM-5. Uncontrollability of worry incrementally predicted an array of clinical severity indicators--including GAD severity, co-occurrence of other mood and anxiety disorders, and treatment utilization--over and above existing GAD criteria, underscoring the importance of uncontrollability of worry to the conceptualization of GAD. In Chapter 2, the Beliefs about Thought Control scale (BATC) was developed to investigate metacognitive beliefs about thought control as a cognitive correlate of--and potential risk factor for--GAD and other disorders characterized by uncontrollable thought. The BATC showed strong convergent and discriminant validity and incrementally predicted trait worry and GAD severity over and above an existing worry-specific measure. Chapter 3 aimed to identify the cognitive underpinnings of attempts to control worry using an experimental approach in which worry was induced in the laboratory. Two components of executive control--working memory and inhibition--were impaired during attempts to control worry, suggesting that these processes may play a role in the worry control process. Collectively, these studies highlight the importance of uncontrollability in defining clinically significant worry, point to cognitive and metacognitive processes that may contribute to the onset or maintenance of uncontrollable worry, and suggest several promising directions for future basic and applied research