Department of Mathematics and Statistics Graduate Projectshttp://hdl.handle.net/1974/61582017-09-20T03:53:47Z2017-09-20T03:53:47ZEnsemble Controllability of Linear Control SystemsAdu, Danielhttp://hdl.handle.net/1974/226512017-09-16T04:46:03Z2017-01-01T00:00:00ZEnsemble Controllability of Linear Control Systems
Adu, Daniel
This thesis is concerned about ensemble controllability. We give an overview of the notions of ensemble controllability, in particular, $L_{2}$-ensemble controllability, and uniform ensemble controllability. We review the results presented in~\cite{UH-MS:14} on uniform ensemble controllability of one-parameter time-invariant linear systems and in~\cite{JSL:11} on $L_{2}$-ensemble controllability. In contrast to the notions in~\cite{UH-MS:14} and~\cite{JSL:11}, we investigate on the possibility to steer an ensemble using constrained control signals in the unit interval, which we call uniform~\emph{null}~ensemble controllability for one-parameter time-invariant linear systems using constrained control signals in the unit interval. We give a necessary as well as a sufficient condition for uniform~\emph{null}~ensemble controllability of one-parameter time-invariant linear systems using constrained control signals in the unit interval. Using tools from complex approximation theory, we show that in the discrete-time scenario, the problem of uniform~\emph{null}~ensemble controllability of one-parameter time-invariant linear systems using control signals in the unit interval is equivalent to polynomial approximation problem.
2017-01-01T00:00:00ZDegeneracy of velocity constraints in rigid body systemsBrggs, Jonnyhttp://hdl.handle.net/1974/149752017-03-30T14:25:37Z2016-09-27T00:00:00ZDegeneracy of velocity constraints in rigid body systems
Brggs, Jonny
The equations governing the dynamics of rigid body systems with velocity constraints are singular at degenerate configurations in the constraint distribution. In this report, we describe the causes of singularities in the constraint distribution of interconnected rigid body systems with smooth configuration manifolds. A convention of defining primary velocity constraints in terms of orthogonal complements of one-dimensional subspaces is introduced. Using this convention, linear maps are defined and used to describe the space of allowable velocities of a rigid body. Through the definition of these maps, we present a condition for non-degeneracy of velocity constraints in terms of the one dimensional subspaces defining the primary velocity constraints. A method for defining the constraint subspace and distribution in terms of linear maps is presented. Using these maps, the constraint distribution is shown to be singular at configuration where there is an increase in its dimension.
2016-09-27T00:00:00ZRepresentation of numbers by quaternary quadratic formsKar, Arpitahttp://hdl.handle.net/1974/149742017-03-30T14:25:41Z2016-09-27T00:00:00ZRepresentation of numbers by quaternary quadratic forms
Kar, Arpita
2016-09-27T00:00:00ZAn R-package for the Estimation and Testing of Multiple Covariates and Biomarker Interactions for Survival Data Based on Local Partial LikelihoodZhang, Siweihttp://hdl.handle.net/1974/149732017-03-30T14:25:40Z2016-09-27T00:00:00ZAn R-package for the Estimation and Testing of Multiple Covariates and Biomarker Interactions for Survival Data Based on Local Partial Likelihood
Zhang, Siwei
When we study the variables that a ffect survival time, we usually estimate their eff ects by the Cox regression model. In biomedical research, e ffects of the covariates are often modi ed by a biomarker variable. This leads to covariates-biomarker interactions.
Here biomarker is an objective measurement of the patient characteristics at baseline.
Liu et al. (2015) has built up a local partial likelihood bootstrap model to estimate
and test this interaction e ffect of covariates and biomarker, but the R code developed
by Liu et al. (2015) can only handle one variable and one interaction term and can
not t the model with adjustment to nuisance variables. In this project, we expand
the model to allow adjustment to nuisance variables, expand the R code to take
any chosen interaction terms, and we set up many parameters for users to customize
their research. We also build up an R package called "lplb" to integrate the complex
computations into a simple interface.
We conduct numerical simulation to show that the new method has excellent fi nite
sample properties under both the null and alternative hypothesis. We also applied the
method to analyze data from a prostate cancer clinical trial with acid phosphatase
(AP) biomarker.
2016-09-27T00:00:00Z