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Institute
Mathematiklehrkräfte sind bisher unzureichend auf die Digitalisierung des Mathematikunterrichts vorbereitet. Daher ist es notwendig bereits im Studium passende Lernangebote zur Entwicklung professioneller Kompetenzen für den Einsatz digitaler Mathematikwerkzeuge zu schaffen. Im Rahmen der vorliegenden Arbeit wird eine fachdidaktische Lehrveranstaltung für angehende Mathematiklehrkräfte der Sekundarstufen
konzipiert, um deren professionellen Kompetenzen mit Blick auf den Einsatz digitaler Mathematikwerkzeuge im Kontext der Leitidee Strukturen und funktionaler Zusammenhang zu fördern. Neben Werkzeugkompetenzen zur Nutzung der digitalen Mathematikwerkzeuge sollen die Kompetenzen zur Planung und Gestaltung von Mathematikunterricht entwickelt werden. Für das Themenfeld funktionale Zusammenhänge sind die beiden digitalen Mathematikwerkzeuge GeoGebra und Tabellenkalkulationsprogramm besonders relevant. Deren Potentiale für das funktionale Denken werden insbesondere in Bezug auf Darstellungsformen und Repräsentationswechsel sowie Lernschwierigkeiten thematisiert. Ergänzend zu den genannten Kompetenzen werden zudem Überzeugungen zum Einsatz digitaler Mathematikwerkzeuge in den Blick genommen. Diese haben einen erheblichen Einfluss darauf, ob die erworbenen Kompetenzen im eigenen Unterricht eingesetzt werden oder nicht.
Zur Ermittlung des Ist-Zustands bei Kompetenzen und Überzeugungen der angehenden
Mathematiklehrkräfte vor dem Besuch der Lehrveranstaltung wurden zwei Erhebungsinstrumente entwickelt und eingesetzt. Neben einem Fragebogen zur Selbsteinschätzung erfolgte die Datenerhebung mittels eines neu entwickelten Kompetenztests
im Sinne eines Leistungstests. Es zeigte sich, dass die professionellen Kompetenzen und Vorerfahrungen bei den Studierenden äußerst heterogen sind.
Um Rückschlüsse auf die Wirksamkeit der Lehrveranstaltung ziehen zu können, wurden die beiden Erhebungsinstrumente im Pre-Post-Design eingesetzt. Die Ergebnisse nach Besuch der Lehrveranstaltung lassen auf positive Effekte schließen. Gleichzeitig wird deutlich, dass eine einzelne Lehrveranstaltungen im Studium nicht ausreicht. Zudem besteht Bedarf an weiterer Forschung in diesem Bereich, insbesondere was die Untersuchung langfristiger Effekte und die Zusammenhänge zwischen den verschiedenen Kompetenzfacetten und Überzeugungen betrifft.
In view of the requirement that students (e.g. in Rhineland-Palatinate) from grade 7 onwards
have to work independently with dynamic geometry software (e.g. GeoGebra) and a
spreadsheet program, teachers have a special role in the integration of digital mathematics
toolsin mathematics lessons. For example this can be implemented, with regard to the guiding
idea L4 (Functional connection), since the use of digital mathematics tools is recommended to
promote functional thinking. Within the qualitive offensive for pre-service-teacher training,
which has been established by the Ministry of Education, the project MoSAiK provided a
Digital Research Workshop Multiple Representations in Mathematics Education using the
topic of elementary functions on secondary school level. On the one hand this workshop
examined tool skills of students in relation to the two softwares GeoGebra and spreadsheets
that apply for secondary schools, high schools and vocational schools. On the other hand
(technology-related) beliefs of the students were researched according to the function
concept.
For that reason a specific didactic workshop, which included on the one hand the fostering of
operating skills for the tools GeoGebra and spreadsheet and on the other hand the
subsequent planning and teaching of lessons had been conceived. Before and after attending
the workshop the operating and selection skills in both tools have been measured in a sample
of 55 students using a self-developed competency test. In addition, the sub-areas TK, TPK and
TCK of the TPACK-framework, self-efficacy beliefs and technology-related beliefs regarding
the advantages and disadvantages of using digital mathematics tools were collected using a
student questionnaire. The workshop intervention achieved a significant effect on the
development of the students' operating skills for both tools – GeoGebra and spreadsheet. The
results of a qualitative content analysis with quantitative elements show that in particular the
visualizing of functions and the checking of results were viewed as advantages of the digital
tools when teaching functional thinking. The most frequently cited disadvantage was a danger
to manual calculation methods. With regard to the time required, on the one hand, savings
were expected when using digital tools, but on the other hand, there were fears of high
expenditure if pupils do not have the necessary tool skills.
In the last years, the public interest in epidemiology and mathematical modeling of disease spread has increased - mainly caused by the COVID-19 pandemic, which has emphasized the urgent need for accurate and timely modelling of disease transmission. However, even prior to that, mathematical modelling has been used for describing the dynamics and spread of infectious diseases, which is vital for developing effective interventions and controls, e.g., for vaccination campaigns and social restrictions like lockdowns. The forecasts and evaluations provided by these models influence political actions and shape the measures implemented to contain the virus.
This research contributes to the understanding and control of disease spread, specifically for Dengue fever and COVID-19, making use of mathematical models and various data analysis techniques. The mathematical foundations of epidemiological modelling, as well as several concepts for spatio-temporal diffusion like ordinary differential equation (ODE) models, are presented, as well as an originally human-vector model for Dengue fever, and the standard (SEIR)-model (with the potential inclusion of an equation for deceased persons), which are suited for the description of COVID-19. Additionally, multi-compartment models, fractional diffusion models, partial differential equations (PDE) models, and integro-differential models are used to describe spatial propagation of the diseases.
We will make use of different optimization techniques to adapt the models to medical data and estimate the relevant parameters or finding optimal control techniques for containing diseases using both Metropolis and Lagrangian methods. Reasonable estimates for the unknown parameters are found, especially in initial stages of pandemics, when little to no information is available and the majority of the population has not got in contact with the disease. The longer a disease is present, the more complex the modelling gets and more things (vaccination, different types, etc.) appear and reduce the estimation and prediction quality of the mathematical models.
While it is possible to create highly complex models with numerous equations and parameters, such an approach presents several challenges, including difficulties in comparing and evaluating data, increased risk of overfitting, and reduced generalizability. Therefore, we will also consider criteria for model selection based on fit and complexity as well as the sensitivity of the model with respect to specific parameters. This also gives valuable information on which political interventions should be more emphasized for possible variations of parameter values.
Furthermore, the presented models, particularly the optimization using the Metropolis algorithm for parameter estimation, are compared with other established methods. The quality of model calculation, as well as computational effort and applicability, play a role in this comparison. Additionally, the spatial integro-differential model is compared with an established agent-based model. Since the macroscopic results align very well, the computationally faster integro-differential model can now be used as a proxy for the slower and non-traditionally optimizable agent-based model, e.g., in order to find an apt control strategy.
Die Geometrie unseres Anschauungsraumes – die euklidische Geometrie – ist für einen allgemeinbildenden Mathematikunterricht elementar. Seitens der Mathematiklehrkraft stellt grundsätzlich ihr Fachwissen das Fundament des Unterrichtens dar. Als Teil ihres Professionswissens sollten Mathematiklehrkräfte prinzipiell über ein Fachwissen verfügen, das in Bezug zur akademischen Mathematik den unterrichtlichen Anforderungen der schulischen Mathematik gerecht wird.
Die im Rahmen der Dissertation entwickelte Theorie des metrisch-normalen euklidischen Raumes charakterisiert sich in ihrer perspektivischen Dualität, der mathematischen Stringenz eines axiomatisch-deduktiven Vorgehens auf der einen und der Berücksichtigung der fachdidaktischen Anforderungen an Mathematiklehrkräfte auf der anderen Seite; sie hebt sich darin von bestehenden Theorien ab.
Counts of SARS-CoV-2-related deaths have been key numbers for justifying severe political, social and economical measures imposed by authorities world-wide. A particular focus thereby was the concomitant excess mortality (EM), i.e. fatalities above the expected all-cause mortality (AM). Recent studies, inter alia by the WHO, estimated the SARS-CoV-2-related EM in Germany between 2020 and 2021 as high as 200 000. In this study, we attempt to scrutinize these numbers by putting them into the context of German AM since the year 2000. We propose two straightforward, age-cohort-dependent models to estimate German AM for the ‘Corona pandemic’ years, as well as the corresponding flu seasons, out of historic data. For Germany, we find overall negative EM of about −18 500 persons for the year 2020, and a minor positive EM of about 7000 for 2021, unveiling that officially reported EM counts are an exaggeration. In 2022, the EM count is about 41 200. Further, based on NAA-test-positive related death counts, we are able to estimate how many Germans have died due to rather than with CoViD-19; an analysis not provided by the appropriate authority, the RKI. Through 2020 and 2021 combined, our due estimate is at no more than 59 500. Varying NAA test strategies heavily obscured SARS-CoV-2-related EM, particularly within the second year of the proclaimed pandemic. We compensated changes in test strategies by assuming that age-cohort-specific NAA-conditional mortality rates during the first pandemic year reflected SARS-CoV-2-characteristic constants.
We consider variational discretization of three different optimal control problems.
The first being a parabolic optimal control problem governed by space-time measure controls. This problem has a nice sparsity structure, which motivates our aim to achieve maximal sparsity on the discrete level. Due to the measures on the right hand side of the partial differential equation, we consider a very weak solution theory for the state equation and need an embedding into the continuous functions for the pairings to make sense. Furthermore, we employ Fenchel duality to formulate the predual problem and give results on solution theory of both the predual and the primal problem. Later on, the duality is also helpful for the derivation of algorithms, since the predual problem can be differentiated twice so that we can apply a semismooth Newton method. We then retrieve the optimal control by duality relations.
For the state discretization we use a Petrov-Galerkin method employing piecewise constant states and piecewise linear and continuous test functions in time. For the space discretization we choose piecewise linear and continuous functions. As a result the controls are composed of Dirac measures in space-time, centered at points on the discrete space-time grid. We prove that the optimal discrete states and controls converge strongly in L^q and weakly-* in Μ, respectively, to their smooth counterparts, where q ϵ (1,min{2,1+2/d}] is the spatial dimension. The variational discrete version of the state equation with the above choice of spaces yields a Crank-Nicolson time stepping scheme with half a Rannacher smoothing step.
Furthermore, we compare our approach to a full discretization of the corresponding control problem, precisely a discontinuous Galerkin method for the state discretization, where the discrete controls are piecewise constant in time and Dirac measures in space. Numerical experiments highlight the sparsity features of our discrete approach and verify the convergence results.
The second problem we analyze is a parabolic optimal control problem governed by bounded initial measure controls. Here, the cost functional consists of a tracking term corresponding to the observation of the state at final time. Instead of a regularization term for the control in the cost functional, we consider a bound on the measure norm of the initial control. As in the first problem we observe a sparsity structure, but here the control resides only in space at initial time, so we focus on the space discretization to achieve maximal sparsity of the control. Again, due to the initial measure in the partial differential equation, we rely on a very weak solution theory of the state equation.
We employ a dG(0) approximation of the state equation, i.e. we choose piecewise linear and continuous functions in space, which are piecewise constant in time for our ansatz and test space. Then, the variational discretization of the problem together with the optimality conditions induce maximal discrete sparsity of the initial control, i.e. Dirac measures in space. We present numerical experiments to illustrate our approach and investigate the sparsity structure
As third problem we choose an elliptic optimal control governed by functions of bounded variation (BV) in one space dimension. The cost functional consists of a tracking term for the state and a BV-seminorm in terms of the derivative of the control. We derive a sparsity structure for the derivative of the BV control. Additionally, we utilize the mixed formulation for the state equation.
A variational discretization approach with piecewise constant discretization of the state and piecewise linear and continuous discretization of the adjoint state yields that the derivative of the control is a sum of Dirac measures. Consequently the control is a piecewise constant function. Under a structural assumption we even get that the number of jumps of the control is finite. We prove error estimates for the variational discretization approach in combination with the mixed formulation of the state equation and confirm our findings in numerical experiments that display the convergence rate.
In summary we confirm the use of variational discretization for optimal control problems with measures that inherit a sparsity. We are able to preserve the sparsity on the discrete level without discretizing the control variable.
This thesis addresses the reduced basis methods for parametrized quasilinear elliptic and parabolic partial differential equations with strongly monotone differential operator. It presents all of the ingredients of the reduced basis method: basis generation for reduced basis approximation, certification of the approximation error by suitable a-posteriori error control and an Offine-Online decomposition. The methodology is further applied to the magnetostatic and magnetoquasistatic approximations of Maxwell’s equations and its validity is confirmed by numerical examples.
Scientific and public interest in epidemiology and mathematical modelling of disease spread has increased significantly due to the current COVID-19 pandemic. Political action is influenced by forecasts and evaluations of such models and the whole society is affected by the corresponding countermeasures for containment. But how are these models structured?
Which methods can be used to apply them to the respective regions, based on real data sets? These questions are certainly not new. Mathematical modelling in epidemiology using differential equations has been researched for quite some time now and can be carried out mainly by means of numerical computer simulations. These models are constantly being refinded and adapted to corresponding diseases. However, it should be noted that the more complex a model is, the more unknown parameters are included. A meaningful data adaptation thus becomes very diffcult. The goal of this thesis is to design applicable models using the examples of COVID-19 and dengue, to adapt them adequately to real data sets and thus to perform numerical simulations. For this purpose, first the mathematical foundations are presented and a theoretical outline of ordinary differential equations and optimization is provided. The parameter estimations shall be performed by means of adjoint functions. This procedure represents a combination of static and dynamical optimization. The objective function corresponds to a least squares method with L2 norm which depends on the searched parameters. This objective function is coupled to constraints in the form of ordinary differential equations and numerically minimized, using Pontryagin's maximum (minimum) principle and optimal control theory. In the case of dengue, due to the transmission path via mosquitoes, a model reduction of an SIRUV model to an SIR model with time-dependent transmission rate is performed by means of time-scale separation. The SIRUV model includes uninfected (U) and infected (V ) mosquito compartments in addition to the susceptible (S), infected (I) and recovered (R) human compartments, known from the SIR model. The unknwon parameters of the reduced SIR model are estimated using data sets from Colombo (Sri Lanka) and Jakarta (Indonesia). Based on this parameter estimation the predictive power of the model is checked and evaluated. In the case of Jakarta, the model is additionally provided with a mobility component between the individual city districts, based on commuter data. The transmission rates of the SIR models are also dependent on meteorological data as correlations between these and dengue outbreaks have been demonstrated in previous data analyses. For the modelling of COVID-19 we use several SEIRD models which in comparison to the SIR model also take into account the latency period and the number of deaths via exposed (E) and deaths (D) compartments. Based on these models a parameter estimation with adjoint functions is performed for the location Germany. This is possible because since the beginning of the pandemic, the cumulative number of infected persons and deaths
are published daily by Johns Hopkins University and the Robert-Koch-Institute. Here, a SEIRD model with a time delay regarding the deaths proves to be particularly suitable. In the next step, this model is used to compare the parameter estimation via adjoint functions with a Metropolis algorithm. Analytical effort, accuracy and calculation speed are taken into account. In all data fittings, one parameter each is determined to assess the estimated number of unreported cases.
Gegeben sei eine Basis b>=10 und eine Ziffer a0 aus der Menge {0,..., b − 1}. Wir untersuchen, ob es unendlich viele Primzahlen gibt, die in ihrer b-adischen Zifferndarstellung die Ziffer a0 nicht besitzen. Ferner schätzen wir die Anzahl dieser Primzahlen, die kleiner sind als X = b^k, nach oben und unten ab.
Damit gelingt uns eine Verallgemeinerung von Maynards Beweis für den Fall b = 10 und wir nutzen hierzu auch die in seiner Arbeit verwendeten Werkzeuge. Unter Anderem benötigen wir die Hardy-Littlewoodsche Kreismethode sowie diverse Siebmethoden, um die Minor Arcs zu kontrollieren.
Schließlich sehen wir, dass wir Maynard's Aussage vor allem dann auf beliebige Basen b>= 10 und ausgeschlossene Ziffern a0 aus {0, ..., b − 1} übertragen können, wenn zwei betragsmäßig größte Eigenwerte von Matrizen, die von b und a0 parametrisiert werden, bestimmte Abschätzungen erfüllen. Dass diese Abschätzungen im Fall b>=102 erfüllt sind, beweisen wir im letzten Kapitel. Für die Fälle b = 10 und b = 11 liegt ebenfalls ein Mathematica-Code vor, der die Abschätzungen bestätigt.
While reading this sentence, you probably gave (more or less deliberately) instructions to approximately 100 to 200 muscles of your body. A sceptical face or a smile, your fingers scrolling through the text or holding a printed version of this work, holding your head, sitting, and much more.
All these processes take place almost automatically, so they seem to be no real achievement. In the age of digitalization it is a defined goal to transfer human (psychological and physiological) behavior to machines (robots). However, it turns out that it is indeed laborious to obtain human facial expression or walking from robots. To optimize this transfer, a deeper understanding of a muscle's operating principle is needed (and of course an understanding of the human brain, which will, however, not be part of this thesis).
A human skeletal muscle can be shortened willingly, but not lengthened, thereto it takes an antagonist. The muscle's change in length is dependent on the incoming stimulus from the central nervous system, the current length of the muscle itself, and certain muscle--specific quantities (parameters) such as the maximum force. Hence, a muscle can be mathematically described by a differential equation (or more exactly a coupled differential--algebraic system, DAE), whose structure will be revealed in the following chapters. The theory of differential equations is well-elaborated. A multitude of applicable methods exist that may not be known by muscle modelers. The purpose of this work is to link the methods from applied mathematics to the actual application in biomechanics.
The first part of this thesis addresses stability theory. Let us remember the prominent example from middle school physics, in which the resting position of a ball was obviously less susceptible towards shoves when lying in a bowl rather than balancing at the tip of a hill. Similarly, a dynamical (musculo-skeletal) system can attain equilibrium states that react differently towards perturbations.
We are going to compute and classify these equilibria.
In the second part, we investigate the influence of individual parameters on model equations or more exactly their solutions. This method is known as sensitivity analysis.
Take for example the system "car" containing a value for the quantity "pressure on the break pedal while approaching a traffic light". A minor deviation of this quantity upward or downward may lead to an uncomfortable, abrupt stop or even to a collision, instead of a smooth stop with a sufficient gap.
The considered muscle model contains over 20 parameters that, if changed slightly, have varying effects on the model equation solutions at different instants of time. We will investigate the sensitivity of those parameters regarding different sub--models, as well as the whole model among different dynamical boundary conditions.
The third and final part addresses the \textit{optimal control} problem (OCP).
The muscle turns a nerve impulse (input or control) into a length change and therefore a force response (output). This forward process is computable by solving the respective DAE. The reverse direction is more difficult to manage. As an everyday example, the OCP is present regarding self-parking cars, where a given path is targeted and the controls are the position of the
steering wheel as well as the gas pedal.
We present two methods of solving OCPs in muscle modeling: the first is a conjunction of variational calculus and optimization in function spaces, the second is a surrogate-based optimization.