The aim of this thesis is to develop discrete fractional models of tumor growth for a given data and to estimate parameters of these models in order to have better data tting. We use discrete nabla fractional calculus because we believe the discrete counterpart of this mathematical theory will give us a better and more accurate outcome. This thesis consists of ve chapters. In the rst chapter, we give the history of the fractional calculus, and we present some basic de nitions and properties that are used in this theory. We de ne nabla fractional exponential and then nabla fractional trigonometric functions. In the second chapter, we concentrate on completely monotonic functions on R, and we introduce completely monotonic functions on discrete domain. The third chapter presents discrete Laplace N-transform table which is a great tool to nd solutions of -th order nabla fractional di erence equations. Furthermore, we nd the solution of nonhomogeneous up to rst order nabla fractional di erence equation using N-transform. In the fourth chapter, rst we give the de nition of Casoration for the set of solutions up to n-th order nabla fractional equation. Then, we state and prove some basic theorems about linear independence of the set of solutions. We focus on the solutions of up to second order nabla fractional di erence equation. We examine these solutions case by case namely, for the real and distinct characteristic roots, real and same, and complex ones. The fth chapter emphasizes the aim of this thesis. First, we give a vi brief introduction to parameter estimation with Gomperts and Logistic curves. In addition, we recall a statistical method called cross-validation for prediction. We state continuous, discrete, continuous fractional and discrete fractional forms of Gompertz and Logistic curves. We use the tumor growth data for twenty-eight mice for the comparison. These control mice were inoculated with tumors but did not receive any succeeding treatment. We claim that the discrete fractional type of sigmoidal curves have the best data tting results when they are compared to the other types of models.

This thesis consists of six chapters. In the first chapter, we review some basic definitions and concepts of fractional calculus. Then we introduce fractional difference equations involving the Riemann-Liouville operator of real number order between zero and one. In the second chapter, we apply the Brouwer fixed point and Contraction Mapping Theorems to prove that there exists a solution for up to the first order nabla fractional difference equation with an initial condition. In chapter three, we define a lower and an upper solution for up to the first order nabla fractional difference equation with an initial condition. Under certain assumptions we prove that a lower solution stays less than an upper solution. Some examples are given to illustrate our findings in this chapter. Then we give constructive proofs of existence of a solution by defining monotone sequences. In the fourth chapter, we derive a continuous form of the Mittag-Leffler function. Then we use successive approximations method to calculate a discrete form of the Mittag-Leffler function. In the fifth chapter, we focus on finding the model which fits best for the data of tumor growth for twenty-eight mice. The models contain either three parameters (Gompertz, Logistic) or four parameters (Weibull, Richards). For each model, we consider continuous, discrete, continuous fractional and discrete fractional forms. Nihan Acar who is a former graduate student in mathematics department has already worked on Gompertz and Logistic models [1]. Here we continue and work on Richards curve. The difference between Acar’s work and ours is the number of parameters in each model. Gompertz and Logistic models contain three parameters and an alpha parameter. The Richards model has four parameters and an alpha parameter. In addition, we use statistical computation techniques such as residual sum of squares and cross-validation to compare fitting and predictive performance of these models. In conclusion, we put three models together to conclude which model is fitting best for the data of tumor growth for twenty-eight mice. In the last chapter, we conclude this thesis and state our future work.

Previous decades of cartilage research have predominantly focused on decoupling the solid and fluid interactions of the mechanical response. The resulting biphasic and triphasic models are widely used in the biomechanics community. However, a simple viscoelastic model is able to account for the stress-relaxation behavior of cartilage, without the added complexity of solid and fluid interactions. Using a viscoelastic model, cartilage is considered a single material with elastic and dissipative properties. A mechanical characterization is made with fewer material parameters than are required by the conventional biphasic and triphasic models. This approach has tremendous utility when comparing cartilage of different species and varying healths. Additionally, the viscoelastic models can be easily extended in dynamic analysis and FEA programs. Cartilage primarily experiences compressive motion during exercise. Therefore, to mimic biological function, a mechanical test should also compress the cartilage. One such test is a viscoelastic stress-relaxation experiment. The Prony and fractional calculus viscoelastic models have shown promise in modeling stress-relaxation of equine articular cartilage. The elastic-viscoelastic correspondence principle is used to extend linear viscoelasticity to the frequency domain. This provides a comparison of articular cartilage based on stored and dissipated moduli. The storage and loss moduli metrics are hypothesized to serve as benchmarks for evaluating osteoarthritic cartilage, and provide guidelines for newly engineered bio-materials. The main goal of the current study is to test the applicability of modeling articular cartilage with viscoelastic models. A secondary goal is to establish a rigorous set of harvesting techniques that allows access to fresh explants with minimal environmental exposure. With a complex substance like cartilage, it is crucial to avoid unnecessary emph{in vitro} environmental exposure. Additional areas of study include: determining the strain-dependency of the mechanical response, exploring the response of cartilage in different fluid mediums such as saline, synovial fluid, and synthetic substitutes, and studying the time-dependent properties of cartilage during stress-relaxation experiments. Equine stifle joints, which are mechanically analogous to human knees, are harvested and used for analysis in this study. It is believed that the proposed viscoelastic models can model other articulating joints as well. If viscoelastic theory can be used to characterize cartilage, then comparisons can be drawn between real and artificial cartilage, leading to better joint replacement technology.

This thesis addresses the H∞ analysis and control of continuous commensurate time-delay systems by frequential methods. First, the asymptotic behavior of the chains of poles are studied, and the conditions of stability for neutral systems with poles approaching the imaginary axis are given. The same analysis is done for fractional systems. In the sequel, a numerical method able to locate all the stability windows as well as the unstable root-locus for classical and fractional system is given. We conclude the analysis part by providing the stability crossing curves of a class of distributed delay system. Starting the synthesis part, we design PID controllers for unstable fractional systems using a small-gain theorem approach. Finally, using the Rekasius substitution, we construct a linear time invariant comparison system that allows us to get information about stability and H∞-norm for classical time-delay systems. Using this approach it is possible to design state and output feedback controllers, as well as linear filters for this class of systems.

This work made publicly available electronically on July 7, 2011.This thesis developed a new practical tuning method for fractional order proportional and integral controllers (FO-PI / PI®) for varying time-delay systems like networked con- trol systems (NCS), sensor networks, etc. Based on previously proposed FO-PI controller tuning rules using fractional Ms constrained integral gain optimization (F-MIGO), simulta- neous maximization of the jitter margin and integrated time weighted absolute error (ITAE) performance for a set of hundred gain delay time-constant (KLT) systems having di®erent time-constants and time-delay values are achieved. A multi-objective optimization algo- rithm is used to simultaneously maximize the ITAE factor and jitter margin of the plants at initial F-MIGO gain parameters. The new values of controller gain parameters are gen- eralized to give a new set of optimal fractional order proportional integral (OFOPI) tuning rules such that the jitter margin and system performance of closed-loop KLT systems are maximized and yet the closed-loop feedback system is stable. This is further tested and veri¯ed by simulation techniques. Comparisons are made with other existing proportional integral derivative (PID) and fractional order proportional integral (PI) tuning rules to prove the e±ciency of the new technique. It is further shown that OFOPI tuning rules per- form better than traditional tuning methods for lag-dominated FOPDT systems, because it can take the varying time-delay better into account. The tuning method is modi¯ed to work with discrete-time controllers in the context of NCSs. Furthermore, experimental results in a NCS platform, Stand-alone Smart Wheel (omnidirectional networked control robot wheel), are reported using the tuning rules developed in this thesis. The optimization tuning method performed almost equally well in practice as in simulations. The thesis also shows that the tuning rule development procedure for OFOPI is not only valid for FOPDT systems but is also applicable for other general classes of plants which could be reduced to ¯rst order plant systems. Temperature control in heat °ow apparatus and water-level control in a coupled tank system using FO-PI tuning rules are other major contributions of this thesis work.

Iterative rounding has been an increasingly popular approach to solving network design optimization problems ever since Jain introduced the concept in his revolutionary 2-approximation for the Survivable Network Design Problem (SNDP). This paper looks at several important iterative rounding approximation algorithms and makes improvements to some of their proofs. We generalize a matrix restatement of Nagarajan et al.'s token argument, which we can use to simplify the proofs of Jain's 2-approximation for SNDP and Fleischer et al.'s 2-approximation for the Element Connectivity (ELC) problem. Lau et al. show how one can construct a (2,2B + 3)-approximation for the degree bounded ELC problem, and this thesis provides the proof. We provide some structural results for basic feasible solutions of the Prize-Collecting Steiner Tree problem, and introduce a new problem that arises, which we call the Prize-Collecting Generalized Steiner Tree problem.

Multimedia services over wireless networks are getting popular. With multicast many mobile stations can join the same video multicast group and share the same radio resource to increase frequency utilization efficiently. However users may locate at different positions so as to suffer different path loss, interference and receive different signal to interference and noise ratio (SINR). Users at the cell-edge receiving lower SINR may degrade the multicast efficiency. In this thesis we propose four schemes considering fractional frequency reuse (FFR) over relay networks to reuse frequency in multi-cells. With fractional frequency reuse, users close to the base station (BS) have more resources to improve the total frequency utilization. A resource allocation scheme is also proposed to efficiently allocate wireless resources. Compared to the conventional relay scheme, the proposed schemes can provide more than 10% video layers for all users and give better video quality for users near BS.

With the popularity of wireless networks, it needs to support user¡¦s mobility cross different base stations, hence, the handover mechanism becomes an important issue. When the user frequently moves between two cells, it will occur the Ping-Pong effect that increases the delay time and reduces the efficiency of system. In this thesis, we proposed a new handover mechanism by considering the fractional frequency reuse (FFR) over relay networks to reuse frequency in multi-cells. The proposed method can reduce the unnecessary handover caused by the interference in the system of FFR. It uses the value of signal to interference and noise ratio (SINR) and the parameter of distance to make handover decision. The simulation results indicate the proposed handover mechanism can reduce more than 8% of the handover number in average in comparison to the competing method in the best case.

Many empirical studies show that both equity volatility and its trading volume have long range dependence and can be modeled as fractional integrated processes. The objective of this study is to investigate relationship between volatility and volume.We adopt four estimators of volatility, which includes the squared log returns, historical volatility, iterative t estimators and $GARCH$ estimators. The results show that among the four estimators squared log returns usually have the largest integration orders and produce hightest ratios of fractional cointegration. The fractional integrated orders are estimated separately and jointly, and the cointegration parameters are estimated by ordinary least squares, a narrow band frequency domain least squares method and a semiparametric estimator of Whittle likelihood. Models are also established when volatility and volume are not fractional cointegrated.

This dissertation presents a 2.4 GHz wideband GFSK-modulated frequency synthesizer using two-point delta-sigma modulation (TPDSM). The two bottlenecks in this design have been rigorously investigated. One bottleneck is the nonlinear performance of the phase-locked loop (PLL). The other one is the inherent gain and delay mismatch between the two modulation points. Both nonlinear and mismatch factors dominate the modulation accuracy in the closed PLL. The proposed formulation can successfully predict the dependencies of the modulation accuracy on both factors. The comparison of the averaged frequency deviation and frequency-shift -keying (FSK) error between theory and measurement shows excellent agreement. The modulated frequency synthesizer implemented in this study can achieve a 2.5 Mbps data rate as well as a 15 £gs PLL stable time with only 2.2 % FSK error under good design and operating conditions.