New Results on the Equivalence of Rational Representations of Behaviors

This article deals with the equivalence of representations of behaviors of linear differential systems. In general, the behavior of a given linear differential system has many different representations. In this paper we restrict ourselves to kernel and image representations. Two kernel representations are called equivalent if they represent one and the same behavior. For kernel representations defined by polynomial matrices, necessary and sufficient conditions for equivalence are well-known. In this paper, we deal with the equivalence of rational representations, i. e. kernel and image representations that are defined in terms of rational matrices. As the main result of this paper, we will derive a new condition for equivalence of rational kernel representations of possibly noncontrollable behaviors. This paper also deals with the equivalence of polynomial as well as rational image representations.</p

The clinical utility of patient-reported outcome measures in total hip replacement and lumbar spine surgery

Background: Beginning in the late 1990s, the Swedish Hip Arthroplasty Register (SHAR) and the Swespine have successfully implemented programs to collect patient-reported outcomes measures (PROMs). The use of PROMs has enabled assessment of patients’ health-related quality of life (HRQoL), physical function and pain following total hip replacement (THR) and lumbar spine surgery (LSS). The nationwide collection of PROMs has made it possible to evaluate changes of care, compare providers, investigate factors influencing outcomes that matter for patients, and it has contributed to improvement in clinical practice. Objective: The overall objective of this thesis is to investigate different ways to utilize PROMs following total hip replacement and lumbar spine surgery. Specifically, this thesis aims to: Investigate PROMs in patients who have undergone LSS prior to THR and in patients who have undergone THR prior to LSS compared to matched patients with isolated THR or LSS. Investigate if the order of THR and LSS affects PROMs one year following the last procedure in patients with both procedures performed within a period of two years. Investigate if PROMs can predict the risk for reoperation following THR. Assess the measurement properties of EQ-5D-5L compared to EQ-5D-3L in a Swedish THR population and to estimate how different severity levels of the two versions of the questionnaire conforms. Patients and methods: For Paper I-III, data including PROMs on patients with THR and LSS performed in 2002-2012 were obtained from SHAR and Swespine and linked to identify those who occurred in both registers. In Paper IV, data from SHAR on patients with THR in 2002-2014 were used to establish the relationship between PROMs and reoperation. For Paper V, patients eligible for THR in western Sweden during 2015 were invited to answer EQ-5D-3L and EQ-5D-5L with a two-week separation before and after surgery. Logistic and linear regression analyses were used to investigate research questions. Results: Patients with both THR and LSS performed had worse one-year PROMs following the last procedure compared to patients with surgery in only one location. Patients eligible for both THR and LSS within a short period of time had better outcomes following the last procedure if surgery started with LSS. PROMs collected one year following THR predicted the risk for subsequent reoperation. Patients frequently utilized the additional response options of EQ-5D-5L and ceiling effects at the one-year follow-up were reduced compared to EQ-5D-3L. EQ VAS estimates for different severity levels conformed well between questionnaires. Conclusion: This thesis contributes to the understanding of patient-reported outcomes for patients who undergo both THR and LSS. Given their ability to predict reoperations following THR, PROMs can be utilized to identify patients at increased risk, which may be used to improve follow-up routines and care. Since EQ-5D-5L better describes health-related quality of life in THR patients, the introduction of the extended questionnaire as a standard tool in SHAR will enable a more accurate assessment of the procedure

Realization and elimination in rational representations of behaviors

<p>This article deals with the relationship between rational representations of linear differential systems and their state representations. In particular we study the relationship between rational representations on the one hand, and output nulling and driving variable representations on the other. In the input output framework it is well-known that every controllable and observable realization of the transfer matrix of the system yields a minimal input/state/output representation. If a proper rational matrix is used for a rational kernel or image representation of the system, then the question arises under what conditions realizations of this rational matrix give rise to state representations. We will establish conditions under which realizations of the proper rational matrix appearing in a rational image representation yield minimal driving variable representations of the system. Likewise, we find conditions under which realization leads from rational kernel representations to minimal output nulling representations. We also study the converse problem, namely state elimination from driving variable and output nulling representations. We will establish closed form expressions for kernel and image representations of the external behaviors associated with these particular state representations. (C) 2013 Elsevier B.V. All rights reserved.</p>

Equivalence of rational representations of behaviors

This article deals with the equivalence of representations of behaviors of linear differential systems. In general, the behavior of a given linear differential system has many different representations. In this paper we restrict ourselves to kernel and image representations. Two kernel representations are called equivalent if they represent one and the same behavior. For kernel representations defined by polynomial matrices, necessary and sufficient conditions for equivalence are well known. In this paper, we deal with the equivalence of rational representations, i. e. kernel and image representations that are defined in terms of rational matrices. As the first main result of this paper, we will derive a new condition for the equivalence of rational kernel representations of possibly noncontrollable behaviors. Secondly we will derive conditions for the equivalence of rational representations of a given behavior in terms of the polynomial modules generated by the rows of the rational matrices. We will also establish conditions for the equivalence of rational image representations. Finally, we will derive conditions under which a given rational kernel representation is equivalent to a given rational image representation.