Browsing by Subject "mathematics"

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  • Widlund, Anna; Tuominen, Heta; Korhonen, Johan (2018)
    It has been suggested that both performance and academic well-being play a role in adolescent students' educational attainment and school dropout. In this study, we therefore examined, first, what kinds of academic well-being (i.e., school burnout, schoolwork engagement, and mathematics self-concept) and mathematics performance profiles can be identified among lower secondary school students (N-grade (7) = 583, N-grade 9 = 497); second, how stable these profiles are across one school year during the seventh and ninth grades; and, third, how students with different academic well-being and mathematics performance profiles differ with respect to their educational aspirations. By means of latent profile analyses, three groups of students in seventh grade: thriving (34%), average (51%), and negative academic well-being (15%) and four groups of students in ninth grade: thriving (25%), average (50%), negative academic well-being (18%), and low-performing (7%) with distinct well-being and mathematics performance profiles were identified. Configural frequency analyses revealed that the profiles were relatively stable across one school year; 60% of the students displayed identical profiles over time. The thriving students reported the highest educational aspirations compared to the other groups. In addition, the low-performing students in the ninth grade had the lowest educational aspirations just before the transition to upper secondary school. Practical implications as well as directions for future research are discussed.
  • Xu, Yongjun; Liu, Xin; Cao, Xin; Huang, Changping; Liu, Enke; Qian, Sen; Liu, Xingchen; Wu, Yanjun; Dong, Fengliang; Qiu, Cheng-Wei; Qiu, Junjun; Hua, Keqin; Su, Wentao; Wu, Jian; Xu, Huiyu; Han, Yong; Fu, Chenguang; Yin, Zhigang; Liu, Miao; Roepman, Ronald; Dietmann, Sabine; Virta, Marko; Kengara, Fredrick; Zhang, Ze; Zhang, Lifu; Zhao, Taolan; Dai, Ji; Yang, Jialiang; Lan, Liang; Luo, Ming; Liu, Zhaofeng; An, Tao; Zhang, Bin; He, Xiao; Cong, Shan; Liu, Xiaohong; Zhang, Wei; Lewis, James P.; Tiedje, James M.; Wang, Qi; An, Zhulin; Wang, Fei; Zhang, Libo; Huang, Tao; Lu, Chuan; Cai, Zhipeng; Wang, Fang; Zhang, Jiabao (2021)
    Y Artificial intelligence (AI) coupled with promising machine learning (ML) techniques well known from computer science is broadly affecting many aspects of various fields including science and technology, industry, and even our day-to-day life. The ML techniques have been developed to analyze high-throughput data with a view to obtaining useful insights, categorizing, predicting, and making evidence-based decisions in novel ways, which will promote the growth of novel applications and fuel the sustainable booming of AI. This paper undertakes a comprehensive survey on the development and application of AI in different aspects of fundamental sciences, including information science, mathematics, medical science, materials science, geoscience, life science, physics, and chemistry. The challenges that each discipline of science meets, and the potentials of AI techniques to handle these challenges, are discussed in detail. Moreover, we shed light on new research trends entailing the integration of AI into each scientific discipline. The aim of this paper is to provide a broad research guideline on fundamental sciences with potential infusion of AI, to help motivate researchers to deeply understand the state-of-the-art applications of AI-based fundamental sciences, and thereby to help promote the continuous development of these fundamental sciences.
  • Polkowska, Aleksandra; Räsänen, Sirpa; Nuorti, Pekka; Maunula, Leena; Jalava, Katri (2021)
    Seven major food- and waterborne norovirus outbreaks in Western Finland during 2014-2018 were re-analysed. The aim was to assess the effectiveness of outbreak investigation tools and evaluate the Kaplan criteria. We summarised epidemiological and microbiological findings from seven outbreaks. To evaluate the Kaplan criteria, a one-stage meta-analysis of data from seven cohort studies was performed. The case was defined as a person attending an implicated function with diarrhoea, vomiting or two other symptoms. Altogether, 22% (386/1794) of persons met the case definition. Overall adjusted, 73% of norovirus patients were vomiting, the mean incubation period was 44 h (4 h to 4 days) and the median duration of illness was 46 h. As vomiting was a more common symptom in children (96%, 143/149) and diarrhoea among the elderly (92%, 24/26), symptom and age presentation should drive hypothesis formulation. The Kaplan criteria were useful in initial outbreak assessments prior to faecal results. Rapid food control inspections enabled evidence-based, public-health-driven risk assessments. This led to probability-based vehicle identification and aided in resolving the outbreak event mechanism rather than implementing potentially ineffective, large-scale public health actions such as the withdrawal of extensive food lots. Asymptomatic food handlers should be ideally withdrawn from high-risk work for five days instead of the current two days. Food and environmental samples often remain negative with norovirus, highlighting the importance of research collaborations. Electronic questionnaire and open-source novel statistical programmes provided time and resource savings. The public health approach proved useful within the environmental health area with shoe leather field epidemiology, combined with statistical analysis and mathematical reasoning.
  • Weigang, Helene Camilla (Helsingin yliopisto, 2017)
    This thesis theoretically investigates dispersal evolution in a wider ecological context. It factors in ecological relevant dependencies e.g. trade-offs or spatial heterogeneity, and allows coevolutionary interactions between immigration and other traits. It extends well-known models to include more biological realism, reveals novel evolutionary mechanisms and helps to understand the complex dispersal patterns more accurately. In particular, this work studies the evolution of dispersal, i.e., natal emigration when it is under a trade-off with fecundity. Furthermore, dispersal is divided into its three phases and hence studied as emigration, transfer and immigration. Emigration and immigration are made dependent on the local conditions experienced by the individuals: the patch types. The coevolution of patch-type dependent immigration is investigated alone, but also the coevolution of patch-type dependent immigration and patch-type dependent emigration or local adaptation is studied. The evolutionary framework was chosen to be adaptive dynamics, a way of describing the long-term evolutionary outcomes of single populations that can lead to evolutionary diversification of strategies.
  • Niemi, Hannele; Niu, Shuanghong Jenny (2021)
    The aim of this study was to uncover how digital storytelling advances students’ self-efficacy in mathematics learning and what kinds of learning experiences contribute to self-efficacy. Four Chinese classes with 10- to 11-year-old students (N = 121) participated in the project. The mathematics learning theme was geometry. Quantitative data was collected with questionnaires. The qualitative data was based on teachers’ and students’ interviews and observations. Both data sets showed that the students’ self-efficacy increased significantly during the project. The most important mediator was students’ perception of the meaningfulness of mathematics learning; digital storytelling enhanced the students’ ability to see mathematics learning as useful. They became more confident that they could learn mathematics and understand what they had learned. They also felt more confident in talking with their classmates about mathematical concepts. The role of self-efficacy was twofold: it supported students’ learning during the project and it increased due to meaningful mathematics learning experiences.
  • Venieri, Laura (Helsingin yliopisto, 2017)
    In this dissertation we define a generalization of Kakeya sets in certain metric spaces. Kakeya sets in Euclidean spaces are sets of zero Lebesgue measure containing a segment of length one in every direction. A famous conjecture, known as Kakeya conjecture, states that the Hausdorff dimension of any Kakeya set should equal the dimension of the space. It was proved only in the plane, whereas in higher dimensions both geometric and arithmetic combinatorial methods were used to obtain partial results. In the first part of the thesis we define generalized Kakeya sets in metric spaces satisfying certain axioms. These allow us to prove some lower bounds for the Hausdorff dimension of generalized Kakeya sets using two methods introduced in the Euclidean context by Bourgain and Wolff. With this abstract setup we can deal with many special cases in a unified way, recovering some known results and proving new ones. In the second part we present various applications. We recover some of the known estimates for the classical Kakeya and Nikodym sets and for curved Kakeya sets. Moreover, we prove lower bounds for the dimension of sets containing a segment in a line through every point of a hyperplane and of an (n-1)-rectifiable set. We then show dimension estimates for Furstenberg type sets (already known in the plane) and for the classical Kakeya sets with respect to a metric that is homogeneous under non-isotropic dilations and in which balls are rectangular boxes with sides parallel to the coordinate axis. Finally, we prove lower bounds for the classical bounded Kakeya sets and a natural modification of them in Carnot groups of step two whose second layer has dimension one, such as the Heisenberg group. On the other hand, if the dimension is bigger than one we show that we cannot use this approach.
  • Fang, Chun (Helsingin yliopisto, 2013)
    One of the central problems in dynamical systems and differential equations is the analysis of the structures of invariant sets. The structures of the invariant sets of a dynamical system or differential equation reflect the complexity of the system or the equation. For example, any omega-limit set of a finite dimensional differential equation is a singleton implies that each bounded solution of the equation eventually stabilizes at some equilibrium state. In general, a dynamical system or differential equation can have very complicated invariant sets or so called chaotic sets. It is of great importance to classify those systems whose minimal invariant sets have certain simple structures and to characterize the complexity of chaotic type sets in general dynamical systems. In this thesis, we focus on the following two important problems: estimates for the dimension of chaotic sets and stable sets in a finite positive entropy system, and characterizations of minimal sets of nonautonomous tridiagonal competitive-cooperative systems.
  • Kairema, Anna (Helsingin yliopisto, 2013)
    This dissertation brings contribution to two interrelated topics. The first contribution concerns the so-called systems of dyadic cubes in the context of metric spaces. Second contribution is applications to one and two weight norm inequalities for linear and sublinear positive integral operators. Both of the topics are important in harmonic analysis and an ongoing area of study. The main novelties of the presented works consist of improving and extending existing results into more general frameworks. The work consists of four research articles and an introductory part. The first two articles, written in collaboration with T. Hytönen, study systems of dyadic cubes in metric spaces. In the Euclidean space, dyadic cubes are well-known and define a convenient structure with useful covering and intersection properties. Such dyadic structures are central especially in the modern trend of harmonic analysis. In the first article extensions of these structures are constructed in general geometrically doubling metric spaces. These consist of a refinement of existing constructions and a completely new construction of finitely many adjacent dyadic systems which behave like "translates" of a fixed system but without requiring a group structure. In this context, "cubes" are not properly cubes but rather more complicated sets that collectively have properties reminiscent of those in the Euclidean case. However, it is natural to ask what type of sets could or should be regarded cubes. In the second paper, we give a complete answer to this question in the general framework of a geometrically doubling metric space making use of the "plumpness" notion already appeared in the geometric measure theory. From another side; the two latter articles study weighted norm inequalities. Via the new construction of adjacent dyadic systems, weighted estimates for positive integral operators are obtained in a general framework. In the third paper, the two-weight problem is investigated for potential-type operators. Both strong and weak type estimates are characterized by "testing type" conditions: to show the full norm inequality it suffices to test the desired estimate on a specific class of simple test functions only. The results improve some previous results in the sense that the considered ambient space is more general (with more general measures and no additional geometric assumptions) and the testing is over a countable collection of test function only (instead of a significantly larger collection appearing in the previous works on the topic). The main technical novelty of the proof is a decomposition of the operator, along dyadic systems giving rise to certain finitely many "dyadic" versions of the original operator. In the fourth article, the focus is on sharp constant estimates for generalized fractional integral operators. A positive answer and its sharpness are given in the context of a space of homogeneous type. The result is reduced to weak-type inequalities using the results in the third paper. The sharpness requires a construction of functions that locally behave similarly to the basic power functions on the Euclidean space. The result extends a recent Euclidean result. Keywords: dyadic cube, adjacent dyadic systems, metric space, space of homogeneous type, potential type operator, testing condition, weighted norm inequality, sharp bound
  • Huuskonen, Milla (Helsingin yliopisto, 2020)
    Objectives. Mindset has been found to have a significant relationship with academic performance, which has been hypothesised to be based on the relationship between mindset and differential reactions to errors. The literature exploring the relationship between the mindset, which is conceptualised as a continuum between the extremes of growth-oriented mindset and conceptions of fixed ability, and the processing of errors has, however, been scarce. The aim of the present study was to examine the relationship between mathematical mindset and error-related positivity (Pe), which is among the most investigated event-related potentials linked to errors in electroencephalography (EEG) studies. The growth mindset was hypothesised to be related to enhanced Pe as well as to more accurate task performance. Furthermore, Pe was not expected to be related to behavioural adaptations related to errors, such as post-error slowing (PES). Methods. The sample consisted of 97 children in the third grade. The children participated in an EEG experiment, during which they performed a task involving challenging mathematical calculations. From the experiment, both data on Pe amplitudes and the task performance of the participants was used. The children also completed a questionnaire to assess their mathematical growth mindset. After examining the reliability and validity of the mindset instrument, the relationship between Pe, mathematical mindset and task performance was explored using linear mixed models. Results. Mathematical growth mindset was significantly related to enhanced early Pe amplitudes, and this relationship was stronger with increased task accuracy. Late Pe did not have a significant correlation with mindset, but enhanced late Pe amplitudes were observed with increased task accuracy. In behavioural data, a significant PES effect was observed indicating slower responding following errors. This PES effect had a trend-level association with higher endorsement of mathematical growth mindset and early Pe. Conclusions. The enhanced early Pe amplitudes associated with growth mindset may suggest that growth mindset is associated with more efficient information processing resulting in higher response conflict after a participant’s representation of the correct response is shifted after their erroneous response. This may also explain the trend-level association between growth mindset, early Pe and PES. Strong conclusions cannot be made regarding late Pe, as the peak of the component might not have been captured in the analysis time frame.
  • Franco, Eugenia (Helsingin yliopisto, 2022)
    In this thesis we analyse the solutions of two different types of integral and integro-differential equations modelling structured populations. More precisely, we study a linear renewal equation that models the dynamics of physiologically structured populations, as well as an extension of the Smoluchowski coagulation equation: the coagulation equation with a constant flux of particles from the origin. The focus is on measure-valued solutions, i.e., solutions that, evaluated in time, are measures. Two different assumptions on the kernel characterizing the linear renewal equation are proven to guarantee asynchronous exponential growth/decline or convergence to a stable distribution for the unique solution of the renewal equation. If the kernel is factorizable, then the proof is based on Feller’s classical Renewal Theorem. If, instead the kernel is regularizing the proof is based on the theory of positive operators and on Laplace transform methods. The third main result of this thesis is related with the asymptotic behaviour of the coagulation equation with source. Indeed, the existence of self-similar solutions for the coagulation equation with a constant flux of particles from the origin is proven under the assumption that the kernel satisfies a polynomial bound that also prevents gelation. These self-similar solutions are expected to describe the long term behaviour of the solutions of the coagulation equation with source.
  • Marcozzi, Matteo (Helsingin yliopisto, 2016)
    By time dependent stochastic systems we indicate efffective models for physical phenomena where the stochasticity takes into account some features whose analytic control is unattainable and/or unnecessary. In particular, we consider two classes of models which are characterized by the different role of randomness: (1) deterministic evolution with random initial data; (2) truly stochastic evolution, namely driven by some sort of random force, with either deterministic or random initial data. As an example of the setting (1) in this thesis we will deal with the discrete nonlinear Schrödinger equation (DNLS) with random initial data and we will mainly focus on its applications concerning the study of transport coefficients in lattice systems. Since the seminal work by Green and Kubo in the mid 50 s, when they discovered that transport coefficients for simple fluids can be obtained through a time integral over the respective total current correlation function, the mathematical physics community has been trying to rigorously validate these predictions and extend them also to solids. In particular, the main technical difficulty is to obtain at least a reliable asymptotic form of the time behaviour of the Green-Kubo correlation. To do this, one of the possible approaches is kinetic theory, a branch of the modern mathematical physics stemmed from the challenge of deriving the classical laws of thermodynamics from microscopic systems. Nowadays kinetic theory deals with models whose dynamics is transport dominated in the sense that typically the solutions to the kinetic equations, whose prototype is the Boltzmann equation, correspond to ballistic motion intercepted by collisions whose frequency is order one on the kinetic space-time scale. Referring to the articles in the thesis by Roman numerals [I]-[V], in [I] and [II] we build some technical tools, namely Wick polynomials and their connection with cumulants, to pave the way towards the rigorous derivation of a kinetic equation called Boltzmann-Peierls equation from the DNLS model. The paper [III] can be contextualized in the same framework of kinetic predictions for transport coefficients. In particular, we consider the velocity flip model which belongs to the family (2) of our previous classification, since it consists of a particle chain with harmonic interaction and a stochastic term which flips the velocity of the particles. In [III] we perform a detailed study of the position-momentum correlation matrix via two diffeerent methods and we get an explicit formula for the thermal conductivity. Moreover, in [IV] we consider the Lorentz model perturbed by an external magnetic field which can be categorized in the class (1): it is a gas of non interacting particles colliding with obstacles located at random positions in the plane. Here we show that under a suitable scaling limit the system is described by a kinetic equation where the magnetic field affects only the transport term, but not the collisions. Finally, in [IV] we studied a generalization of the famous Kardar-Parisi-Zhang (KPZ) equation which falls into the category (2) being a nonlinear stochastic partial differential equation driven by a space-time white noise. Spohn has recently introduced a generalized vector valued KPZ equation in the framework of nonlinear fluctuating hydrodynamics for anharmonic particle chains, a research field which is again strictly connected to the investigation of transport coefficients. The problem with the KPZ equation is that it is ill-posed. However, in 2013 Hairer succeded to give a rigorous mathematical meaning to the solution of the KPZ via an approximation scheme involving the renormalization of the nonlinear term by a formally infinite constant. In [V] we tackle a vector valued generalization of the KPZ and we prove local in time wellposedness by using a technique inspired by the so-called Wilsonian Renormalization Group.
  • Yang, Fan (Helsingin yliopisto, 2014)
    Dependence logic is a new logic which incorporates the notion of dependence , as well as independence between variables into first-order logic. In this thesis, we study extensions and variants of dependence logic on the first-order, propositional and modal level. In particular, the role of intuitionistic connectives in this setting is emphasized. We obtain, among others, the following results: 1. First-order intuitionistic dependence logic is proved to have the same expressive power as the full second-order logic. 2. Complete axiomatizations for propositional dependence logic and its variants are obtained. 3. The complexity of model checking problem for modal intuitionistic dependence logic is analyzed.
  • Lahtinen, Aatos (Suomen metsätieteellinen seura, 1988)
    A standard tree tapers off monotonically upwards. An algorithm is presented for constructing a monotony preserving taper curve using a quadratic spline. It is suggested that the resultant taper curve is better than the usual cubic spline.
  • Sumu, Virpi; Tuominen, Jenna (University of Helsinki Science Education Centre, 2019)
  • Mei, Peng (2013)
    In this thesis, the main goal is to study the kinetic dynamics based on Hubbard model in dimensions d>2. We analyse rigorously some basic properties of the dynamics in the following scaling limit. As the coupling constant converges to zero, we rescale the time variable to capture the slow dynamics: we let coupling constant go to zero, time to infinity in such a way that the product of square of the coupling constant and time has a finite limit. The main ingredient of the kinetic limit is that, combining some basic ideas of scattering theory (long time scale) and perturbation theory (small coupling parameter), it automatically selects from the microscopic dynamics some dominating terms. This gives rise to a new description of time evolution (given by quantum Boltzmann equation), whose form can already be postulated from the second order perturbative expansion. The limit equation is expected to capture nontrivial information about the original system. Besides the weak coupling dynamics, we also discuss two complementary results in related Fermion systems.
  • Soultanis, Elefterios (Helsingin yliopisto, 2016)
    This dissertation studies classical questions in the field of geometric analysis in the context of metric spaces. The dissertation is comprised of three research articles. The first is on the connection of quasiconformal maps and the quasihyperbolic metric. The remaining two concern notions of homotopy classes of Sobolev type maps between metric spaces, comparison with the manifold case, and the existence of minimizers of a p-energy in these homotopy classes. The unifying theme of all three articles is analysis on metric spaces. That is, all three papers deal with questions concerning maps between metric spaces. The particular type of metric spaces involved is generally referred to as PI-spaces. PI-spaces satisfy conditions allowing one to extend a large part of classical first order calculus, such as the theory of Sobolev maps and, á posteriori, differentiability of Lipschitz functions.
  • Tuominen, Heta; Juntunen, Henriikka Mira Maria; Niemivirta, Markku (2020)
    Most studies utilizing a person-oriented approach to investigating students’ achievement goal orientation profiles have been domain-general or focused on a single domain (usually mathematics), thus excluding the possibility of identifying distinct subject-specific motivational profiles. In this study, we looked into this by examining upper secondary school students’ subject-specific achievement goal orientation profiles simultaneously in mathematics and English. As distinct profiles might contribute to how students invest time and effort in studying, we also examined differences in perceived subject-specific cost (i.e., effort required, emotional cost, opportunity cost) among students with different profiles, and how this was linked with students’ more general academic well-being (i.e., school engagement, burnout). The 434 Finnish general upper secondary school students participating in the study were classified based on their achievement goal orientations in the two subjects using latent profile analysis, and the predictions of the latent profile on distal outcomes (i.e., measures of cost and academic well-being) were examined within the mixture model. Five divergent achievement goal orientation profiles were identified: indifferent (29%), success-oriented (26%), mastery-oriented (25%), English-oriented, math-avoidant (14%), and avoidance-oriented (6%). The English-oriented, math-avoidant students showed the most distinct domain-specificity in their profile but, in general, profiles indicated more cross-domain generality than specificity. Overall, mastery-oriented students showed the most adaptive academic well-being, while avoidance-oriented students were the least engaged. Success-oriented students were characterised by high multiple goals in both subjects, elevated costs, and high scores on both positive (engagement) and negative (burnout) well-being indicators. The English-oriented, math-avoidant students perceived studying math as costly. The findings suggest that addressing students’ achievement motivation in different subjects may be useful for recognising factors endangering or fostering student learning and well-being.
  • Vähävihu, Elina (Helsingfors universitet, 2008)
    In this study the researcher wanted to show the observed connection of mathematics and textile work. To carry this out the researcher designed a textbook by herself for the upper secondary school in Tietoteollisuuden Naiset - TiNA project at Helsinki University of Technology (URL: The assignments were designed as additional teaching material to enhance and reinforce female students confidence in mathematics and in the management of their textile work. The research strategy applied action research, out of which two cycles two have been carried out. The first cycle consists of establishing the textbook and in the second cycle its usability is investigated. The third cycle is not included in this report. In the second cycle of the action research the data was collected from 15 teachers, five textile teachers, four mathematics teachers and six teachers of both subjects. They all got familiar with the textbook assignments and answered a questionnaire on the basis of their own teaching experience. The questionnaire was established by applying the theories of usability and teaching material assessment study. The data consisted of qualitative and quantitative information, which was analysed by content analysis with computer assisted table program to either qualitative or statistical description. According to the research results, the textbook assignments seamed to be applied better to mathematics lessons than textile work. The assignments pointed out, however, the clear interconnectedness of textile work and mathematics. Most of the assignments could be applied as such or as applications in the upper secondary school textile work and mathematics lessons. The textbook assignments were also applicable in different stages of the teaching process, e.g. as introduction, repetition or to support individual work or as group projects. In principle the textbook assignments were in well placed and designed in the correct level of difficulty. Negative findings concerned some too difficult assignments, lack of pupil motivation and unfamiliar form of task for the teacher. More clarity for some assignments was wished for and there was especially expressed a need for easy tasks and assignments in geometry. Assignments leading to the independent thinking of the pupil were additionally asked for. Two important improvements concerning the textbook attainability would be to get the assignments in html format over the Internet and to add a handicraft reference book.
  • Donvil, Brecht (Helsingin yliopisto, 2020)
    Recent developments in experimental methods allow for the study of thermodynamic properties of quantum systems. In quantum integrated circuits, quantum systems are elements in an electric circuit that can straightforwardly be coupled to other elements. This manipulability allows one to construct quantum heat engines, Maxwell demons etc. Quantum integrated circuits also are one of the main potential settings to realise a working quantum computer. Calorimetric measurements in integrated circuits serve as a promising technique to probe thermodynamic laws of the quantum regime and to study the inner workings of quantum devices. Due to this experimental accessibility, the theoretical study of open quantum systems in the context of quantum integrated circuits is highly relevant. Open quantum systems are typically small systems, e.g. qubits or oscillators, in contact with one or more reservoirs. The research on which this thesis is based can roughly be divided into two parts. The first part is concerned with the thermodynamics of a driven qubit in contact with a thermal bath. This system is the archetype of a quantum out-ofequilibrium system. In one case the qubit is strongly driven by a semiclassical driving field. Building on earlier works, the thermodynamic relations of the system are found by proving the equivalence with an easier to study qubit-oscillator system. In the other, case the qubit is driven by being in contact with two baths with a temperature gradient. The full generating function is derived in a proper approximative scheme and a fluctuation-dissipation relation is found. The second part focusses on a specific experimental scheme to perform calorimetric measurements. The scheme relies on coupling a quantum system to a finite reservoir and performing fast temperature measurements on the reservoir. Doing so allows one to infer energy changes in the reservoir and therefore to obtain the heat exchanged with the system. The dynamics of this system are modelled for weak system-reservoir coupling and concrete experimental predictions are made. In new work, the dynamics for a toy model of a system interacting with a finite reservoir are derived from first principles for a specific model. The first principle derivation matches with the earlier modelled dynamics in the weak coupling and allows to consider strong system-reservoir coupling as well.
  • Koponen, Mika; Asikainen, Mervi A.; Viholainen, Antti; Hirvonen, Pekka E. (2019)
    In this article we present a new approach to investigating teacher knowledge. The essay data related to Finnish future teachers' (N = 18) perceptions of the "knowledge required for teaching mathematics" were transformed into a network. We classified the knowledge topics using the Mathematical Knowledge for Teaching (MKT) framework and examined the relationships between the issues raised with the aid of network analysis. According to the results, the future teachers see the six MKT domains in a hierarchical sequence. As it is not subject specific, this approach is also applicable in the investigation of teacher knowledge of other subjects. (C) 2018 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (