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Ο κλασματικός λογισμός είναι ο κλάδος της μαθηματικής ανάλυσης που μελετά παραγώγους και ολοκληρώματα κλασματικής τάξης, και επομένως επιτρέπει την διατύπωση κλασματικών διαφορικών εξισώσεων (FDEs). Aν και ο κλασματικός λογισμός εισήχθη για πρώτη φορά από τον Leibniz περισσότερα από 300 χρόνια πριν, εν τούτοις η εφαρμογή του σε προβλήματα της μαθηματικής φυσικής ξεκίνησε τις τελευταίες δεκαετίες. Συγκεκριμένα, η κλασματική ανάλυση ξεκίνησε βρίσκοντας εφαρμογή σε πολλούς τομείς των φυσικών επιστημών και της επιστήμης της μηχανικής, και μόλις το 2009 εισήχθη για πρώτη φορά στον τομέα της φαρμακοκινητικής. Η φαρμακοκινητική είναι η επιστήμη η οποία μελετά την κινητική της απορρόφησης, της κατανομής και της απομάκρυνσης των φαρμάκων, δηλαδή περιγράφει τη χρονική εξέλιξη του φαρμάκου στον ανθρώπινο οργανισμό και χρησιμοποιεί κυρίως διαμερισματικά μοντέλα. Έχει αποδειχθεί ότι συγκεκριμένα είδη φαρμάκων, μετά τη χορήγησή τους στο ανθρώπινο σώμα, ακολουθούν κινητική η οποία περιγράφεται καλύτερα με τη χρήση κλασματικών διαφορικών εξισώσεων. Ο κλασματικός λογισμός και οι εφαρμογές του είναι ένας αναπτυσσόμενος τομέας ενεργούς έρευνας. Σε ό,τι αφορά τη φαρμακοκινητική, πρόκειται για ένα πολλά υποσχόμενο εργαλείο και η αντίστοιχη βιβλιογραφία αυξάνεται ολοένα και περισσότερο. Στην παρούσα εργασία μελετάται η εφαρμογή του κλασματικού λογισμού στη φαρμακοκινητική. Συγκεκριμένα, δίνουμε αναλυτική λύση σε γραμμικά συστήματα κλασματικών διαφορικών εξισώσεων, τα οποία αντιπροσωπεύουν φαρμακοκινητικά μοντέλα που έχουν προκύψει από την έως τώρα βιβλιογραφία. Όλα τα φαρμακοκινητικά μοντέλα που έχουν μελετηθεί δίνουν μόνο αριθμητικές λύσεις. Aυτό που επιχειρείται για πρώτη φορά στην παρούσα εργασία, είναι να δοθούν οι αναλυτικές λύσεις των μοντέλων αυτών, έστω και αν η μορφή τους είναι πολύπλοκη. Αναλυτικότερα, το πρώτο κεφάλαιο της εργασίας περιέχει μια ανασκόπηση των βασικότερων στοιχείων της θεωρίας της κλασματικής ανάλυσης που θα χρησιμοποιήσουμε, όπως: συναρτήσεις Mittag-Leffler, βασικές ιδιότητες αυτών και υπολογισμός μετασχηματισμού Laplace συγκεκριμένων μορφών αυτών των συναρτήσεων, καθώς επίσης και ορισμός του κλασματικού ολοκληρώματος και της κλασματικής παραγώγου συναρτήσεων. Στο δεύτερο κεφάλαιο αναλύεται η σύνδεση της διαμερισματικής ανάλυσης με την φαρμακοκινητική. Στο τρίτο κεφάλαιο περιγράφεται η σύνδεση του κλασματικού λογισμού με τη φαρμακοκινητική, καθώς και οι λόγοι για τους οποίους υπερτερεί η προσέγγιση αυτή έναντι των προσεγγίσεων που χρησιμοποιούνταν έως και το 2009. Tο τέταρτο κεφάλαιο αφορά εφαρμογές του κλασματικού λογισμού, ενώ δίνονται οι αναλυτικές λύσεις των γραμμικών συστημάτων κλασματικών διαφορικών εξισώσεων που προκύπτουν. Ακόμη, στο Παράρτημα Α αναφέρονται κάποια στοιχεία που αφορούν στο ισοζύγιο μάζας, στο Παράρτημα Β δίνονται τα αποτελέσματα και οι γραφικές παραστάσεις των εφαρμογών που μελετήθηκαν στο τέταρτο κεφάλαιο, και, τέλος, στο Παράρτημα Γ δίνονται οι εντολές του Mathematica που χρησιμοποιήθηκαν για την απεικόνιση των αναλυτικών λύσεων. / Fractional calculus is the sector of mathematical analysis that deals with derivatives and fractional order integrals, resulting the derivation of Fractional Differential Equations (FDEs). Fractional calculus was first introduced by Leibniz more than 300 years ago. Nevertheless, its application on mathematical physics problems has just started the last few decades. In particular, fractional analysis started being applied on sciences of physics and mechanics . Furthermore, fractional analysis was introduced in the field of pharmacokinetics only a few years ago (2009). Pharmacokinetics is the science that deals with the kinetics of the absorption, the distribution and the excretion of drugs. In other words, it describes the time course of the drug inside the human body. Pharmacokinetics mostly uses compartmental models . It has been demonstrated that several types of drugs, follow a kinetic operation after entering in the human body, which is better described by Fractional Differential Equations. Fractional calculus and its applications is a developing sector of active research. Pharmacokinetics, in particular, is a promising tool and the corresponding literature is increasingly growing. The present thesis deals with the application of fractional calculus in pharmacokinetics. In particular, we provide an analytical solution in fractional differential equations linear systems, which represent pharmacokinetic models that have emerged of the existing literature. All the pharmacokinetic models that have been studied provide only arithmetical solutions. The new aspect of the present thesis is an attempt to provide the analytical solutions of these models, even if their form is complicated. In more detail, the first chapter of the study contains a review of the most fundamental fractional-analysis-theory elements that we will use, such as: Mittag-Leffler functions, their basic properties, calculation of Laplace transformation for specific forms of these functions, definition of the fractional integral and the fractional derivative of functions. In the second chapter the binding of compartmental analysis with pharmacokinetics is analyzed. In the third chapter the binding of fractional calculus with pharmacokinetics is described, as well as the reasons why this approach is superior to the previous approaches that were used until 2009. The fourth chapter contains applications of fractional calculus. The analytical solutions for the fractional differential equations linear systems that arise are also given. Furthermore, Appendix A includes some elements related to the mass balance, while Appendix B contains the results and graphs of the applications that were studied in the fourth chapter. Finally, Appendix C provides the Mathematica code that were used for the illustration of the analytical solutions.

Change in adaptive outcomes of children who are treated for brain tumors is examined using longitudinal data. The children received different types of treatment from none to any combinations of three treatments, which are surgery, radiation and chemotherapy. In this thesis, we use mixed model to find the significant variables that predict change in outcomes of communication skill, daily living skills and socialization skill. Fractional polynomial transformation method and Gompertz method are applied to build non-linear longitudinal curves. We use PRESS as the criterion to compare these two methods. Comparison analysis shows the effect of each significant variable on adaptive behaviors over time. In most cases, model with Gompertz method is better than that with Transformation method. Significant predictors of change in adaptive outcomes include Time, Gender, Surgery, SES classes, interaction between Time and Radiation, interaction between Time and Gender, interaction between Age and Gender.

The individual growth model is a relatively new statistical technique. It is now widely used to examine the trajectories of individuals and groups in repeated measures data. This study examines the association of the receptive vocabulary over time and characteristics of children who were treated for brain tumors. The children undertook different types of treatment from one to any combinations of surgery, radiation and chemotherapy. The individual growth model is used to analyze the longitudinal data and to address the issues behind the data. Results of this study present several factors' influences to the rate of change of PPVT scores. The conclusions of this thesis indicate that the decline in the PPVT scores is associated with gender, age at diagnosis, socioeconomic status, type of treatment and Neurological Predictor Scale.

Bakalauro darbe nagrinėjamas trupmeninių tiesinių uždavinių sprendimas MS Office Excel Solver ir AMPL pagalba. Aprašomas taikomųjų projektų, skirtų trupmeniniams tiesiniams programavimo uždaviniams spręsti, kūrimas. Pateikti sprendimo rezultatai bei atliktas projektų palyginimas. Aptartos iškilusios problemos ir nurodyta, kaip jas išspręsti. / In this Bachelor work the derivation of fractional linear tasks are being researched and analyzed via MS Office Excel Solver and AMPL. Described in the application projects for fractional linear programming problem solution development. Provide the results and the comparison of the schemes was care out. Problems which appeared in this work were discussed as well as the solutions provided.

In recent decades, significant interest, based on physics and engineering applications, has developed on so-called anomalous diffusion processes that possess different spread functions with classical ones. The resulting differential equation whose fundamental solution matches this decay process is best modeled by an equation containing a fractional order derivative. This dissertation mainly focuses on some inverse problems for fractional diffusion equations. After some background introductions and preliminaries in Section 1 and 2, in the third section we consider our first inverse boundary problem. This is where an unknown boundary condition is to be determined from overposed data in a time- fractional diffusion equation. Based upon the fundamental solution in free space, we derive a representation for the unknown parameters as the solution of a nonlinear Volterra integral equation of second kind with a weakly singular kernel. We are able to make physically reasonable assumptions on our constraining functions (initial and given boundary values) to be able to prove a uniqueness and reconstruction result. This is achieved by an iterative process and is an immediate result of applying a certain fixed point theorem. Numerical examples are presented to illustrate the validity and effectiveness of the proposed method. In the fourth section a reaction-diffusion problem with an unknown nonlinear source function, which has to be determined from overposed data, is considered. A uniqueness result is proved and a numerical algorithm including convergence analysis under some physically reasonable assumptions is presented in the one-dimensional case. To show effectiveness of the proposed method, some results of numerical simulations are presented. In Section 5, we also attempted to reconstruct a nonlinear source in a heat equation from a number of known input sources. This represents a new research even for the case of classical diffusion and would be the first step in a solution method for the fractional diffusion case. While analytic work is still in progress on this problem, Newton and Quasi-Newton method are applied to show the feasibility of numerical reconstructions. In conclusion, the fractional diffusion equations have some different properties with the classical ones but there are some similarities between them. The classical tools like integral equations and fixed point theory still hold under slightly different assumptions. Inverse problems for fractional diffusion equations have applications in many engineering and physics areas such as material design, porous media. They are trickier than classical ones but there are also some advantages due to the mildly ill-conditioned singularity caused by the new kernel functions.

In industrial experimentation, time and material costs are often at a premium. In designing an experiment, one needs to balance the desire for sufficient experimental runs to provide adequate data analysis, with the need to conduct a cost-effective experiment. A common compromise is the use of fractional factorial (FF) designs, in which only a fraction of the total possible runs is utilized. After discussing the basic concepts of FF designs, we introduce the fractional factorial split-plot (FFSP) design. Such designs occur frequently, because certain factors are often harder to vary than others, thus imposing randomization restric- tions. This thesis examines two techniques aimed at reducing run size that have not been greatly explored in the FFSP setting — nonregular designs and semifoldover designs. We show that these designs offer competitive alternatives to the more standard regular and full foldover designs and we produce tables of optimal designs in both scenarios.

As CMOS technologies advance to 22-nm dimensions and below, constructing analog circuits are difficult to design within permitted specifications. One of the reasons for this is a limit of voltage resolution. In this situation, time-mode processing is a technique that is believed to be well suited for solving many of these challenges. A primary advantage of this technique is the ability to achieve analog functions using digital logic structures. Time difference amplifiers (TDA) can be a key component to realize fine time solutions. TDA are an innovative method to improve the time resolution as well as the evolution of ADC. This thesis introduces a TDA that amplifies the input time difference between two signals by a fractional gain. The closed loop gain control system used in this work consists of a pseudo differential current starved delay element (PDCSDE) and a monotonic digitally controlled delay element (DCDE). By using these elements to create a delay chain and a control loop, the result is a stable fractional time difference gain (TD gain). The system was designed and simulated in 65nm process at 1.2V power supply. The measured results show that this TDA achieves a fractional TD gain offset lower than 1.3%, with supply variation of ±15%, and input range as wide as ±250ps. The new design was also more resilient to process, voltage and temperature (PVT) variations

In industrial experimentation, time and material costs are often at a premium. In designing an experiment, one needs to balance the desire for sufficient experimental runs to provide adequate data analysis, with the need to conduct a cost-effective experiment. A common compromise is the use of fractional factorial (FF) designs, in which only a fraction of the total possible runs is utilized. After discussing the basic concepts of FF designs, we introduce the fractional factorial split-plot (FFSP) design. Such designs occur frequently, because certain factors are often harder to vary than others, thus imposing randomization restric- tions. This thesis examines two techniques aimed at reducing run size that have not been greatly explored in the FFSP setting — nonregular designs and semifoldover designs. We show that these designs offer competitive alternatives to the more standard regular and full foldover designs and we produce tables of optimal designs in both scenarios.

This dissertation comprises two essays that attempt to determine, empirically, the fiscal response of governments’ to international assistance. The first essay examines whether an increasingly popular recommendation in international aid policy to switch from tied foreign assistance to untied foreign assistance affects investment in critical development expenditure sectors by developing countries. In the past, most international aid has been in the form of tied assistance as donors believed that tying aid will improve its effectiveness. It has been argued, that if tied aid is well designed and effectively managed then its overall effectiveness can be improved. On the contrary, it is also believed that tied aid acts as an impediment to donor cooperation and the building of partnership with developing countries. In addition, it is also argued that it removes the ‘feeling’ of ownership and responsibility of projects from partner countries in aid supported development. Two other more popular arguments used to challenge the effectiveness of foreign aid is that it is compromised when tied to the goods and services of the donor countries because almost 30 percent of its value is eliminated and also because it does not allow recipient countries to act on their priorities for public spending. These problems bring into question whether tied aid is truly the most effective way to help poor countries. A recommendation by the international community is that a switch to untied aid would be necessary. With untied aid, the recipient country is not obligated to buy the goods of the donor country neither is it compelled to pursue the public expenditure priorities of donors. Instead with untied aid they will have greater flexibility over spending decisions and can more easily pursue the priorities of their countries as they see fit. Hence, one could expect that a one dollar increase in untied aid will increase spending in the critical priority sectors by more than a one dollar increase in tied assistance. The question therefore is whether national domestic priorities coincide or not with what the international community has traditionally deemed should be priority. Empirically, we test this prediction using country-by-country data for 57 countries for the period 1973 to 2006. The results suggest that on average untied aid has a greater impact on pro-poor spending than do tied aid. In addition, the results also suggest that fungibility is still an issue even after accounting for the effects of untied aid. However, one could argue that fungibility may not be as bad as it appears since the switch to untied aid improves spending in the sectors that are essential for growth and development. The second essay explores the hypothesis that the expectations of debt forgiveness can discourage developing countries from attaining fiscal independence through an improvement of their tax effort. On the one hand, the international financial community typically advises poor countries to improve revenue mobilization but, on the other hand, the same international community routinely continues to bail-out poor countries that fail to meet their loan repayment obligations. The act of bailing-out these countries creates an expectation on the part of developing country governments that they will receive debt forgiveness time and again in the future. Therefore, the expectation of future bail outs creates a moral hazard that leads to endemic lower tax efforts. The key prediction of our simple theoretical model is that in the presence of debt forgiveness, tax ratios will decline and this decline will be stronger the higher the frequency and intensity of the bailouts. Empirically, we test this prediction using country-level data for 66 countries for the period 1989 to 2006. The results strongly suggest that debt forgiveness plays a significant role in the low tax effort observed in developing countries. Our empirical model allows for the endogeneity of tax effort and debt forgiveness. Interestingly we find that more debt forgiveness is actually provided to countries with lower tax effort. The results are robust to various specifications.