Published Article / This study was conducted with 29 Grade 12 learners who were studying calculus. The purpose was to explore how the learners responded to questions based on the derivative and why they did so. Data was collected from the written responses of the learners to two assessments carried out over a six-month period as well as interviews with four of the learners. It was found that learners made extensive use of inappropriate formulae, drawn from other sections of the curriculum The study recommends that teachers should not focus solely on how to carry out procedures, but they should also pay attention to why certain procedures are appropriate or not.

In this thesis, we study the existence of random periodic solutions for both nonlinear dissipative stochastic functional differential equations (SFDEs) and semilinear nondissipative SFDEs in C([-r,0],R^d). Under some sufficient conditions for the existence of global semiflows for SFDEs, by using pullback-convergence technique to SFDE, we obtain a general theorem about the existence of random periodic solutions. By applying coupled forward-backward infinite horizon integral equations method, we perform the argument of the relative compactness of Wiener-Sobolev spaces in C([0,τ],C([-r,0]L²(Ω))) and the generalized Schauder's fixed point theorem to show the existence of random periodic solutions.

<p>Recently, wireless sensor network (WSN) has become a promising technologywith a wide range of applications such as supply chain monitoringand environment surveillance. It is typically composed of multiple tiny devicesequipped with limited sensing, computing and wireless communicationcapabilities. Design of such networks presents several technique challengeswhile dealing with various requirements and diverse constraints. Performanceanalysis techniques are required to provide insight on design parametersand system behaviors.</p><p>Based on network calculus, we present a deterministic analysis methodfor evaluating the worst-case delay and buffer cost of sensor networks. Tothis end, three general traffic flow operators are proposed and their delayand buffer bounds are derived. These operators can be used in combinationto model any complex traffic flowing scenarios. Furthermore, the methodintegrates a variable duty cycle to allow the sensor nodes to operate at lowrates thus saving power. In an attempt to balance traffic load and improveresource utilization and performance, traffic splitting mechanisms areintroduced for mesh sensor networks. Based on network calculus, the delayand buffer bounds are derived in non-splitting and splitting scenarios.In addition, analysis of traffic splitting mechanisms are extended to sensornetworks with general topologies. To provide reliable data delivery in sensornetworks, retransmission has been adopted as one of the most popularschemes. We propose an analytical method to evaluate the maximum datatransmission delay and energy consumption of two types of retransmissionschemes: hop-by-hop retransmission and end-to-end retransmission.</p><p>We perform a case study of using sensor networks for a fresh food trackingsystem. Several experiments are carried out in the Omnet++ simulationenvironment. In order to validate the tightness of the two bounds obtainedby the analysis method, the simulation results and analytical results arecompared in the chain and mesh scenarios with various input traffic loads.From the results, we show that the analytic bounds are correct and tight.Therefore, network calculus is useful and accurate for performance analysisof wireless sensor network.</p> / Ipack VINN Excellence Center

In this thesis we study a reasoning module for agents that have cognitive abilities, such as memory, perception, action, and are expected to function autonomously for long periods of time. The module provides the ability to reason about action and change using the language of the situation calculus and variants of the basic action theories. The main focus of this thesis is on the logical problem of progressing an action theory. First, we investigate the conjecture by Lin and Reiter that a practical first-order definition of progression is not appropriate for the general case. We show that Lin and Reiter were indeed correct in their intuitions by providing a proof for the conjecture, thus resolving the open question about the first-order definability of progression and justifying the need for a second-order definition. Then we proceed to identify three cases where it is possible to obtain a first-order progression with the desired properties: i) we extend earlier work by Lin and Reiter and present a case where we restrict our attention to a practical class of queries that may only quantify over situations in a limited way; ii) we revisit the local-effect assumption of Liu and Levesque that requires that the effects of an action are fixed by the arguments of the action and show that in this case a first-order progression is suitable; iii) we investigate a way that the local-effect assumption can be relaxed and show that when the initial knowledge base is a database of possible closures and the effects of the actions are range-restricted then a first-order progression is also suitable under a just-in-time assumption. Finally, we examine a special case of the action theories with range-restricted effects and present an algorithm for computing a finite progression. We prove the correctness and the complexity of the algorithm, and show its application in a simple example that is inspired by video games.

Sequent calculi for BDI logics is a research object of the thesis. BDI logics are widely used for agent system description and implementation. Agents are autonomous systems, those acts in some environment and aspire to achieve preassigned goals. Implementation of the decision making is the main and the most complicated part in agent systems implementation. Logic calculi may be used for the decision making implementation. In this thesis, there are researched sequent calculi for BDI logics. Sequent calculi for BDI logics, like sequent calculi for other modal logics, use loop-check technique to get decidability. Inefficient loop-check takes a major part of the resources used for the derivation. For some modal logics, there are known loop-check free sequent calculi or calculi with an efficient loop-check. In this thesis, there is presented loop-check free sequent calculus for KD45 logic, which is the main fragment of the BDI logics. Introduced calculus not only eliminates loop-check, but also simplifies sequent derivation. For the branching time logic (another BDI logic fragment) there is presented sequent calculus with an efficient loop-check. Obtained results are adapted for creation sequent calculi for monoagent and multiagent BDI logics. Introduced calculi use only restricted loop-check. Moreover, loop-check is totally eliminated for some types of the loops. These results enables to create more efficient agent systems, those are based on the BDI logics. / Darbe nagrinėjami sekvenciniai skaičiavimai BDI logikoms. BDI logikos yra plačiai naudojamos agentinių sistemų aprašymui ir realizavimui. Agentai yra autonomiškos sistemos, kurios veikia kažkurioje aplinkoje ir siekia įvykdyti iš anksto apibrėžtus tikslus. Sprendimų priėmimo realizavimas yra svarbiausia ir sudėtingiausia dalis realizuojant agentines sistemas. Sprendimo priėmimo realizavimui gali būti naudojami logikos skaičiavimai. Šiame darbe ir yra nagrinėjami sekvenciniai skaičiavimai BDI logikoms. BDI logikose, kaip ir kitose modalumo logikose, yra naudojama ciklų paieška išsprendžiamumui gauti. Neefektyvi ciklų paieška užima didesnę išvedimų paieškos resursų dalį. Kai kurioms modalumo logikoms yra žinomi becikliai skaičiavimai ar skaičiavimai naudojantys efektyvią ciklų paiešką. Šiame darbe yra pateikiamas beciklis sekvencinis skaičiavimas KD45 logikai, kuri yra esminis BDI logikų fragmentas. Pateiktas skaičiavimas ne tik eliminuoja ciklų paiešką, bet ir supaprastina patį sekvencijos išvedimą. Skaidaus laiko logikai (kitam BDI logikų fragmentui) yra pateikiamas sekvencinis skaičiavimas naudojantis efektyvią ciklų paiešką. Gauti rezultatai yra pritaikyti sukuriant sekvencinius skaičiavimus vianaagentinei ir daugiaagentinei BDI logikoms. Pristatyti skaičiavimai naudoja tik apribotą ciklų paiešką. Be to, kai kurių tipų ciklus eliminuoja visiškai. Šie rezultatai įgalina kurti efektyvesnes agentines sistemas, paremtas BDI logikomis.

Darbe nagrinėjami sekvenciniai skaičiavimai BDI logikoms. BDI logikos yra plačiai naudojamos agentinių sistemų aprašymui ir realizavimui. Agentai yra autonomiškos sistemos, kurios veikia kažkurioje aplinkoje ir siekia įvykdyti iš anksto apibrėžtus tikslus. Sprendimų priėmimo realizavimas yra svarbiausia ir sudėtingiausia dalis realizuojant agentines sistemas. Sprendimo priėmimo realizavimui gali būti naudojami logikos skaičiavimai. Šiame darbe ir yra nagrinėjami sekvenciniai skaičiavimai BDI logikoms. BDI logikose, kaip ir kitose modalumo logikose, yra naudojama ciklų paieška išsprendžiamumui gauti. Neefektyvi ciklų paieška užima didesnę išvedimų paieškos resursų dalį. Kai kurioms modalumo logikoms yra žinomi becikliai skaičiavimai ar skaičiavimai naudojantys efektyvią ciklų paiešką. Šiame darbe yra pateikiamas beciklis sekvencinis skaičiavimas KD45 logikai, kuri yra esminis BDI logikų fragmentas. Pateiktas skaičiavimas ne tik eliminuoja ciklų paiešką, bet ir supaprastina patį sekvencijos išvedimą. Skaidaus laiko logikai (kitam BDI logikų fragmentui) yra pateikiamas sekvencinis skaičiavimas naudojantis efektyvią ciklų paiešką. Gauti rezultatai yra pritaikyti sukuriant sekvencinius skaičiavimus vianaagentinei ir daugiaagentinei BDI logikoms. Pristatyti skaičiavimai naudoja tik apribotą ciklų paiešką. Be to, kai kurių tipų ciklus eliminuoja visiškai. Šie rezultatai įgalina kurti efektyvesnes agentines sistemas, paremtas BDI logikomis. / Sequent calculi for BDI logics is a research object of the thesis. BDI logics are widely used for agent system description and implementation. Agents are autonomous systems, those acts in some environment and aspire to achieve preassigned goals. Implementation of the decision making is the main and the most complicated part in agent systems implementation. Logic calculi may be used for the decision making implementation. In this thesis, there are researched sequent calculi for BDI logics. Sequent calculi for BDI logics, like sequent calculi for other modal logics, use loop-check technique to get decidability. Inefficient loop-check takes a major part of the resources used for the derivation. For some modal logics, there are known loop-check free sequent calculi or calculi with an efficient loop-check. In this thesis, there is presented loop-check free sequent calculus for KD45 logic, which is the main fragment of the BDI logics. Introduced calculus not only eliminates loop-check, but also simplifies sequent derivation. For the branching time logic (another BDI logic fragment) there is presented sequent calculus with an efficient loop-check. Obtained results are adapted for creation sequent calculi for monoagent and multiagent BDI logics. Introduced calculi use only restricted loop-check. Moreover, loop-check is totally eliminated for some types of the loops. These results enables to create more efficient agent systems, those are based on the BDI logics.

Fractional calculus has been used as a research tool in the fields of pharmacology, biology, chemistry, and other areas [3]. The main purpose of this thesis is to calculate Green's functions of fractional difference equations, and to model problems in pharmacokinetics. We claim that the discrete fractional calculus yields the best prediction performance compared to the continuous fractional calculus in the application of a one-compartmental model of drug concentration. In Chapter 1, the Gamma function and its properties are discussed to establish a theoretical basis. Additionally, the basics of discrete fractional calculus are discussed using particular examples for further calculations. In Chapter 2, we use these basic results in the analysis of a linear fractional difference equation. Existence of solutions to this difference equation is then established for both initial conditions (IVP) and two-point boundary conditions (BVP). In Chapter 3, Green's functions are introduced and discussed, along with examples. Instead of using Cauchy functions, the technique of finding Green's functions by a traditional method is demonstrated and used throughout this chapter. The solutions of the BVP play an important role in analysis and construction of the Green's functions. Then, Green's functions for the discrete calculus case are calculated using particular problems, such as boundary value problems, discrete boundary value problems (DBVP) and fractional boundary value problems (FBVP). Finally, we demonstrate how the Green's functions of the FBVP generalize the existence results of the Green's functions of DVBP. In Chapter 4, different compartmental pharmacokinetic models are discussed. This thesis limits discussion to the one-compartmental model. The Mathematica FindFit command and the statistical computational techniques of mean square error (MSE) and cross-validation are discussed. Each of the four models (continuous, continuous fractional, discrete and discrete fractional) is used to compute the MSE numerically with the given data of drug concentration. Then, the best fit and the best model are obtained by inspection of the resulting MSE. In the last Chapter, the results are summarized, conclusions are drawn, and directions for future work are stated.

No description available.

Seit W. Diffie und M. Hellman im Jahr 1976 ihren Ansatz für einen sicheren kryptographischen Schlüsselaustausch vorgestellten, ist der sogenannte Diskrete Logarithmus zu einem zentrales Thema der Kryptoanalyse geworden. Dieser stellt eine Erweiterung des bekannten Logarithmus auf beliebige endliche Gruppen dar. In der vorliegenden Dissertation werden zwei von C. Diem eingeführte Algorithmen untersucht, mit deren Hilfe der diskrete Logarithmus in der Picardgruppe glatter, nichthyperelliptischer Kurven vom Geschlecht g > 3 bzw. g > 4 über endlichen Körpern berechnet werden kann. Beide Ansätze basieren auf der sogenannten Indexkalkül-Methode und benutzen zur Erzeugung der dafür benötigten Relationen spezielle Linearsysteme, welche durch Schneiden von ebenen Modellen der Kurve mit Geraden erzeugt werden. Um Aussagen zur Laufzeit der Algorithmen tätigen zu können, werden verschiedene Sätze über die Geometrie von Kurven bewiesen. Als zentrale Aussage wird zum einem gezeigt, dass ebene Modelle niedrigen Grades effizient berechnet werden können. Zum anderen wird bewiesen, dass sich bei genügend großem Grundkörper die Anzahl der vollständig über dem Grundkörper zerfallenden Geraden wie heuristisch erwartet verhällt. Für beide Aussagen werden dabei Familien von Kurven betrachtet und diese gelten daher uniform für alle glatten, nichthyperelliptischen Kurven eines festen Geschlechts. Die genannten Resultate führen schlussendlich zu dem Beweis einer erwarteten Laufzeit von O(q^(2-2/(g-1))) für den ersten der beiden Algorithmen, wobei q die Anzahl der Elemente im Grundkörper darstellt. Der zweite Algoritmus verbessert dies auf eine heuristische Laufzeit in O(q^(2-2/(g-2))), imdem er Divisoren von höherem Spezialiätsgrad erzeugt. Es wird bewiesen, dass dieser Ansatz für einen uniform gegen 1 konvergierenden Anteil an glatten, nichthyperelliptischen Kurven eines festen Geschlechts über Grundkörpern großer Charakteristik eine große Anzahl an Relationen erzeugt. Wiederum werden zum Beweis der zugrundeliegenden geometrischen Aussagen Familien von Kurven betrachtet, um so die Uniformität zu gewährleisten. Beide Algorithmen wurden zudem implementiert. Zum Abschluss der Arbeit werden die Ergebnisse der entsprechenden Experimente vorgestellt und eingeordnet.

ropositional proof complexity is the study of the resources that are needed to prove formulas in propositional logic. In this thesis we are concerned with the size and space of proofs, and in particular with the latter. Different approaches to reasoning are captured by corresponding proof systems. The simplest and most well studied proof system is resolution, and we try to get our understanding of other proof systems closer to that of resolution. In resolution we can prove a space lower bound just by showing that any proof must have a large clause. We prove a similar relation between resolution width and polynomial calculus space that lets us derive space lower bounds, and we use it to separate degree and space. For cutting planes we show length-space trade-offs. This is, there are formulas that have a proof in small space and a proof in small length, but there is no proof that can optimize both measures at the same time. We introduce a new measure of space, cumulative space, that accounts for the space used throughout a proof rather than only its maximum. This is exploratory work, but we can also prove new results for the usual space measure. We define a new proof system that aims to capture the power of current SAT solvers, and we show a landscape of length-space trade-offs comparable to those in resolution. To prove these results we build and use tools from other areas of computational complexity. One area is pebble games, very simple computational models that are useful for modelling space. In addition to results with applications to proof complexity, we show that pebble game cost is PSPACE-hard to approximate. Another area is communication complexity, the study of the amount of communication that is needed to solve a problem when its description is shared by multiple parties. We prove a simulation theorem that relates the query complexity of a function with the communication complexity of a composed function. / <p>QC 20170509</p>