| 1. |
- Wedman, Lotta, 1975-
(författare)
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The concept concept in mathematics education : A concept analysis
- 2020
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The notion concept is used in different ways within the field of mathematics education. The aim of this study is to carry out a concept analysis of the notion concept, within some frequently used frameworks describing conceptual understanding. Building on a philosophical literature review resulting in distinctions that can be used for interpreting views on concept, the study addresses the question: Which views on concept may be found in texts using the chosen frameworks, from the perspective of the distinctions mental versus non-mental, intersubjective versus subjective and molecular versus holistic? The design involves a literature review in mathematics education, resulting in a selection of texts. Views on concept, and to some extent on concept image, conception, and schema, are then interpreted with the help of indicators, and represented in 3D matrices. There are two categories of views on concept within the texts: a mental and intersubjective category, and a non-mental and intersubjective category. One difference between the views is whether conceptual structures have molecular or holistic features. Concerning the notions concept image, conception, and schema, there are generally three different views: an individual view and two culturally dependent views. The different views are sometimes combined. One result is findings regarding how language is used within the texts, where non-mental and mental arenas, and terms and meanings of terms, are not always distinguished. The main contribution of the study is to deepen the understanding of views on the notion concept and how terminology is used in mathematics education. This opens the way for a discussion of how the terminology mentioned above may be used coherently within the field of mathematics education.
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| 2. |
- Holmbom, Anders
(författare)
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Some modes of convergence and their application to homogenization and optimal composites design
- 1996
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- In this thesis, we develop, extend, apply, and discuss a number of methods for the study of limits of sequences of functions and operators. The connection between the notion of two-scale convergence and more general concepts of convergence is investigated and some alternative classes of admissible test functions are characterized. These techniques are extended into compactness results suitable to prove homogenization and corrector results for linear parabolic equations. A further refinement of these methods, together with a characterization of the limits of certain sequences of parameter-dependent functions which has been subject to extension from a quite general class of periodic domains, is introduced. This provides an efficient tool for the homogenization of e.g. nonlinear evolution heat conduction in heterogeneous materials which vibrate with high frequencies or are perforated by periodically arranged nonconducting holes. Moreover, we prove compactness and homogenization for sequences of solutions of linear elliptic and monotone parabolic equations defined in some classes of nonperiodic domains and derive a Darcy's law for a type of nonperiodic porous media. In the linear elliptic case the convergence is strengthened by means of correctors. Finally, we present some numerical results for homogenized stiffness of fibre composites and demonstrate how homogenization techniques for elasticity in composite materials and for liquid flow in porous media can be combined with recent optimization techniques to obtain optimal layout of composite materials.
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| 3. |
- Bråting, Kajsa, 1975-
(författare)
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Studies in the Conceptual Development of Mathematical Analysis
- 2009
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This dissertation deals with the development of mathematical concepts from a historical and didactical perspective. In particular, the development of concepts in mathematical analysis during the 19th century is considered. The thesis consists of a summary and three papers. In the first paper we investigate the Swedish mathematician E.G. Björling's contribution to uniform convergence in connection with Cauchy's sum theorem from 1821. In connection to Björling's convergence theory we discuss some modern interpretations of Cauchy's expression x=1/n. We also consider Björling's convergence conditions in view of Grattan-Guinness distinction between history and heritage. In the second paper we study visualizations in mathematics from historical and didactical perspectives. We consider some historical debates regarding the role of intuition and visual thinking in mathematics. We also consider the problem of what a visualization in mathematics can achieve in learning situations. In an empirical study we investigate what mathematical conclusions university students made on the basis of a visualization. In the third paper we consider Cauchy's theorem on power series expansions of complex valued functions on the basis of a paper written by E.G. Björling in 1852. We discuss Björling's, Lamarle's and Cauchy's different conditions for expanding a complex valued function in a power seris. In the third paper we also discuss the problem of the ambiguites of fundamental concpets that existed during the mid-19th century. We argue that Cauchy's and Lamarle's proofs of Cauchy's theorem on power series expansions of complex valued functions are correct on the basis of their own definitions of the fundamental concepts involved.
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| 4. |
- Kurujyibwami, Celestin
(författare)
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Admissible transformations and the group classification of Schrödinger equations
- 2017
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- We study admissible transformations and solve group classification problems for various classes of linear and nonlinear Schrödinger equations with an arbitrary number n of space variables.The aim of the thesis is twofold. The first is the construction of the new theory of uniform seminormalized classes of differential equations and its application to solving group classification problems for these classes. Point transformations connecting two equations (source and target) from the class under study may have special properties of semi-normalization. This makes the group classification of that class using the algebraic method more involved. To extend this method we introduce the new notion of uniformly semi-normalized classes. Various types of uniform semi-normalization are studied: with respect to the corresponding equivalence group, with respect to a proper subgroup of the equivalence group as well as the corresponding types of weak uniform semi-normalization. An important kind of uniform semi-normalization is given by classes of homogeneous linear differential equations, which we call uniform semi-normalization with respect to linear superposition of solutions.The class of linear Schrödinger equations with complex potentials is of this type and its group classification can be effectively carried out within the framework of the uniform semi-normalization. Computing the equivalence groupoid and the equivalence group of this class, we show that it is uniformly seminormalized with respect to linear superposition of solutions. This allow us to apply the version of the algebraic method for uniformly semi-normalized classes and to reduce the group classification of this class to the classification of appropriate subalgebras of its equivalence algebra. To single out the classification cases, integers that are invariant under equivalence transformations are introduced. The complete group classification of linear Schrödinger equations is carried out for the cases n = 1 and n = 2.The second aim is to study group classification problem for classes of generalized nonlinear Schrödinger equations which are not uniformly semi-normalized. We find their equivalence groupoids and their equivalence groups and then conclude whether these classes are normalized or not. The most appealing classes are the class of nonlinear Schrödinger equations with potentials and modular nonlinearities and the class of generalized Schrödinger equations with complex-valued and, in general, coefficients of Laplacian term. Both these classes are not normalized. The first is partitioned into an infinite number of disjoint normalized subclasses of three kinds: logarithmic nonlinearity, power nonlinearity and general modular nonlinearity. The properties of the Lie invariance algebras of equations from each subclass are studied for arbitrary space dimension n, and the complete group classification is carried out for each subclass in dimension (1+2). The second class is successively reduced into subclasses until we reach the subclass of (1+1)-dimensional linear Schrödinger equations with variable mass, which also turns out to be non-normalized. We prove that this class is mapped by a family of point transformations to the class of (1+1)-dimensional linear Schrödinger equations with unique constant mass.
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| 5. |
- Johansson, Maria
(författare)
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Carleman type inequalities and Hardy type inequalities for monotone functions
- 2007
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This Ph.D. thesis deals with various generalizations of the inequalities by Carleman, Hardy and Polya-Knopp. In Chapter 1 we give an introduction and overview of the area that serves as a frame for the rest of the thesis. In Chapter 2 we consider Carleman's inequality, which may be regarded as a discrete version of Polya-Knopp's inequality and also as a natural limiting inequality of the discrete Hardy inequality. We present several simple proofs of and remarks (e.g. historical) about this inequality. In Chapter 3 we give some sharpenings and generalizations of Carleman's inequality. We discuss and comment on these results and put them into the frame presented in the previous chapter. We also include some new proofs and results. In Chapter 4 we prove a multidimensional Sawyer duality formula for radially decreasing functions and with general weights. We also state the corresponding result for radially increasing functions. In particular, these results imply that we can describe mapping properties of operators defined on cones of such monotone functions. Moreover, we point out that these results can also be used to describe mapping properties of operators between some corresponding general weighted multidimensional Lebesgue spaces. In Chapter 5 we give a weight characterization of the weighted Hardy inequality for decreasing functions and we use this results to give a new weight characterization of the weighted Polya-Knopp inequality for decreasing functions and we also give a new scale of weightconditions for the Hardy inequality for decreasing functions. In Chapter 6 we make a unified approach to Hardy type inequalitits for non-increasing functions and prove a result which covers both the Sinnamon result with one condition and Sawyer's result with two independent conditions for the case when one weight is non-decreasing. In all cases we point out that this condition is not unique and can even be chosen among some (infinte) scales of conditions. In Chapter 7 we obtain the characterization of the general Hardy operator restricted to monotone functions. In Chapter 8 we present some new integral conditions characterizing the embedding between some Lorentz spaces. Only one condition is necessary for each case which means that our conditions are different and simpler than other corresponding conditions in the literature. We even prove our results in a more general frame. In our proof we use a technique of discretization and anti-discretization developed by A. Gogatishvili and L. Pick, where they considered the opposite embedding.
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| 6. |
- Bazarganzadeh, Mahmoudreza, 1977-
(författare)
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Free Boundary Problems of Obstacle Type, a Numerical and Theoretical Study
- 2012
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis consists of five papers and it mainly addresses the theory and schemes to approximate the quadrature domains, QDs. The first deals with the uniqueness and some qualitative properties of the two QDs. The concept of two phase QDs, is more complicated than its one counterpart and consequently introduces significant and interesting open.We present two numerical schemes to approach the one phase QDs in the paper. The first method is based on the properties of the free boundary the level set techniques. We use shape optimization analysis to construct second method. We illustrate the efficiency of the schemes on a variety of experiments.In the third paper we design two finite difference methods for the approximation of the multi phase QDs. We prove that the second method enjoys monotonicity, consistency and stability and consequently it is a convergent scheme by Barles-Souganidis theorem. We also present various numerical simulations in the case of Dirac measures.We introduce the QDs in a sub domain of and Rn study the existence and uniqueness along with a numerical scheme based on the level set method in the fourth paper.In the last paper we study the tangential touch for a semi-linear problem. We prove that there is just one phase free boundary points on the flat part of the fixed boundary and it is also shown that the free boundary is a uniform C1-graph up to that part.
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| 7. |
- Malik, Adam, 1991
(författare)
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Mathematical Modelling of Cell Migration and Polarization
- 2020
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Cell migration plays a fundamental role in both development and disease. It is a complex process during which cells interact with one another and with their local environment. Mathematical modelling offers tools to investigate such processes and can give insights into the underlying biological details, and can also guide new experiments. The first two papers of this thesis are concerned with modelling durotaxis, which is the phenomena where cells migrate preferentially up a stiffness gradient. Two distinct mechanisms which potentially drive durotaxis are investigated. One is based on the hypothesis that adhesion sites of migrating cells become reinforced and have a longer lifespan on stiffer substrates. The second mechanism is based on cells being able to generate traction forces, the magnitude of which depend on the stiffness of the substrate. We find that both mechanisms can indeed give rise to biased migration up a stiffness gradient. Our results encourages new experiments which could determine the importance of the two mechanisms in durotaxis. The third paper is devoted to a population-level model of cancer cells in the brain of mice. The model incorporates diffusion tensor imaging data, which is used to guide the migration of the cells. Model simulations are compared to experimental data, and highlights the model’s difficulty in producing irregular growth patterns observed in the experiments. As a consequence, the findings encourage further model development. The fourth paper is concerned with modelling cell polarization, in the absence of environmental cues, referred to as spontaneous symmetry breaking. Polarization is an important part of cell migration, but also plays a role during division and differentiation. The model takes the form of a reaction diffusion system in 3D and describes the spatio-temporal evolution of three forms of Cdc42 in the cell. The model is able to produce biologically relevant patterns, and numerical simulations show how model parameters influence key features such as pattern formation and time to polarization.
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| 8. |
- Grundén, Helena, 1968-
(författare)
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Mathematics teaching through the lens of planning : actors, structures, and power
- 2020
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This dissertation explores mathematics teaching by focusing on planning. The planning is seen as a social phenomenon related to surrounding practices and power relations in and between practices. Hence, planning in this dissertation is explored beyond what teachers do when planning.The research questions that guided the studies developed during the research process and address meaning of planning, influence of practices surrounding mathematics teaching, and common ideas about mathematics teaching in society. To answer the research questions, three studies were conducted, individual interviews, focus group interviews, and a study of mathematics education in news media.In addition to the aim of contributing to a deeper understanding of mathematics teaching, this dissertation aims to contribute methodologically by answering research questions addressing consequences different views of meaning have for thinking about interviews and assessment of research quality, and the usefulness of theoretical concepts from Critical Discourse Analysis on interview material about planning for mathematics teaching. In the dissertation, Critical Discourse Analysis is used as a theoretical frame, and theoretical constructs, such as actors, structures, and power, are used to explore planning as embedded in the social practice of mathematics teaching.The findings show that planning is an ongoing emotional process that is considered to be different things, including choosing examples to use or producing manipulatives. Findings also reveal that planning varies between teachers and schools, but also varies for individual teachers depending on, for example, time of the year or students. Another result is that although teachers are responsible for planning, their considerations, decisions, and reflections are influenced by other actors both in terms of how planning is done and what is planned for. These influences are explicitly through actors with formal power and implicitly through, for example, common ideas about mathematics teaching that are prevalent in society.Findings that relate to the methodological questions emphasize the importance of considering theoretical standpoints when assessing the quality of research. The findings also show that concepts such as power, actors, and structures are helpful to see and discuss in what ways mathematics teaching is a socially embedded phenomenon.
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| 9. |
- Roos, Helena, 1974-
(författare)
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The meaning(s) of inclusion in mathematics in student talk : Inclusion as a topic when students talk about learning and teaching in mathematics
- 2019
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis contributes to research and practice within the field of special education in mathematics with more knowledge about, and an understanding of, students´ meaning(s) of inclusion in mathematics education. Three research questions guide the study: What meaning(s) is/are ascribed, and how is inclusion used, in mathematics education research? What meaning(s) do the students ascribe to inclusion in mathematics learning and teaching? And what frames students´ meaning(s) of inclusion in mathematics learning and teaching?The first part of this study began with a systematic literature review on the notion of inclusion in mathematics education research, and the search resulted in 1,296 research studies. Of these, 76 studies were retained after the criteria for time span and peer-reviewed research were applied and 19 duplicates had been removed. The second part of the study involves a case study of three students and their meaning(s) of inclusion in mathematics education. The selected school was a lower secondary school in an urban area of Sweden. The school had set out to work inclusively, meaning their aims were to include all students in the ordinary classroom teaching in every subject and to incorporate special education into the ordinary teaching with no fixed special education groups. Three students were chosen for this part of the study: one in Grade 7 and two in Grade 8. Edward, one of the students in Grade 8, was chosen because he was thought to be a student in access to mathematics education. The other two students were chosen because they were thought to be struggling to gain access to mathematics education: Veronica in Grade 7 and Ronaldo in Grade 8 (the same class as Edward). In this study, the object of the study is the meaning(s) of inclusion in student talk. This study is an instrumental and collective case (Stake, 1995), as it involves several students’ meaning(s) aimed at developing a more general understanding of inclusion in mathematics education. The case is also an information-rich case (Patton, 2002), with contributions from students in mathematics education at an inclusive school. Applying Flyvbjerg’s (2006; 2011) notions, one can also call this kind of selection “information-oriented”, and the case is an extreme one – a choice made in order to get “a best case scenario”. An extreme case is a case used to “obtain information on unusual cases which can be especially problematic or especially good in a more closely defined sense” (Flyvbjerg, 2011, p. 307). The data in this study consists of both observations and interviews conducted during the spring semester 2016. The observations took place in a Grade 7 and Grade 8 classroom at the same school where the interviewed students were enrolled. At least one mathematics lesson each month for each class was observed, and student interviews followed each observation. The observations were used to provide a context for the interviews and to support the analysis. In this study, discourse analysis (DA) as described by Gee (2014a; 2014b) was chosen as both the theoretical frame and as an analytical tool because of its explanatory view on discourse, with description foregrounded. With the help of DA, this study describes both the meaning(s) and the use of the notion of inclusion in mathematics education research. It also describes students’ meaning(s) of inclusion in mathematics education as well as framing issues in student talk of inclusion in mathematics education. From Gee´s point of view, DA encompasses all forms of interaction, both spoken and written, and he provides a toolkit for analysing such interaction by posing questions to the text. Gee distinguishes two theoretical notions, big and small discourses, henceforth referred to as Discourse (D) and discourse (d). Discourse represents a wider context, both social and political, and is constructed upon ways of saying, doing, and being: “If you put language, action, interaction, values, beliefs, symbols, objects, tools, and places together in such a way that other recognize you as a particular type of who (identity) engaged in a particular type of what (activity), here and now, then you have pulled of a Discourse” (Gee, 2014 a, p. 52, Gee’s italics). When looking at discourse (with a small d), it focuses on language in use – the “stretches of language” we can see in the conversations we investigate (Gee, 2014a, 2014b), meaning the relations between words and sentences and how these relations visualize the themes within the conversations. These small discourses can inform on how the language is used, what typical words and themes are visible, and how the speakers or writers design the language. According to Gee (2015), big Discourse sets a larger context for the analysis of small discourse. The results of the first part of the study answer to the research question, What meaning(s) is ascribed, and how is inclusion used in mathematics education research? They show that research on inclusion in mathematics education use the term inclusion when both referring to an ideology and a way of teaching, although these two uses are usually treated separately and independently of each other. The results of the second part of the study answer to the following research questions: What meaning(s) do the students ascribe to inclusion in mathematics learning and teaching? And what frames students´ meaning(s) of inclusion in mathematics learning and teaching? These questions show how meaning(s) of inclusion in student talk can be described by three overarching Discourses: the Discourse of mathematics classroom setting, of assessment, and of accessibility in mathematics education. Within these Discourses, smaller discourses make issues of meanings of inclusion for the students visible in terms of: testing, grades, tasks, the importance of the teacher, (not) being valued, the dislike of mathematics, the classroom organization, and being in a small group. This study shows the complexities and challenges of teaching mathematics, all while simultaneously handling students’ diversity and promoting the mathematical development of each student. To enhance students’ participation and access demands that the teacher knows her or his students, is flexible, has a pedagogical stance and tactfulness, and is knowledgeable in mathematics and mathematics education. It also demands that the teacher is able to take a critical stance and resist the prevailing discourse of assessment that can sometimes overshadow the mathematics education, and in a sense, almost become mathematics for the students. Furthermore, this study also shows how complex and challenging it is to be a mathematics student: they are required to relate to, understand, and participate in many Discourses existing at the same time in a single mathematics classroom. These Discourses interrelate and are embedded in power relations between students and teachers and institutions. This demands that the students are alert and able to use various symbols and objects as well as recognize patterns, and then act accordingly. Hence, to be able to fully participate, you have to be able to talk the talk and walk the walk (Gee, 2014a). This means that not only do you have to use the language correctly, but also you have to act properly at the right time and place.
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| 10. |
- Araaya, Tsehaye, 1962-
(författare)
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The Symmetric Meixner-Pollaczek polynomials
- 2003
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The Symmetric Meixner-Pollaczek polynomials are considered. We denote these polynomials in this thesis by pn(λ)(x) instead of the standard notation pn(λ) (x/2, π/2), where λ > 0. The limiting case of these sequences of polynomials pn(0) (x) =limλ→0 pn(λ)(x), is obtained, and is shown to be an orthogonal sequence in the strip, S = {z ∈ ℂ : −1≤ℭ (z)≤1}.From the point of view of Umbral Calculus, this sequence has a special property that makes it unique in the Symmetric Meixner-Pollaczek class of polynomials: it is of convolution type. A convolution type sequence of polynomials has a unique associated operator called a delta operator. Such an operator is found for pn(0) (x), and its integral representation is developed. A convolution type sequence of polynomials may have associated Sheffer sequences of polynomials. The set of associated Sheffer sequences of the sequence pn(0)(x) is obtained, and is foundto be ℙ = {{pn(λ) (x)} =0 : λ ∈ R}. The major properties of these sequences of polynomials are studied.The polynomials {pn(λ) (x)}∞n=0, λ < 0, are not orthogonal polynomials on the real line with respect to any positive real measure for failing to satisfy Favard’s three term recurrence relation condition. For every λ ≤ 0, an associated nonstandard inner product is defined with respect to which pn(λ)(x) is orthogonal. Finally, the connection and linearization problems for the Symmetric Meixner-Pollaczek polynomials are solved. In solving the connection problem the convolution property of the polynomials is exploited, which in turn helps to solve the general linearization problem.
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