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• 1.
Linnaeus University, Faculty of Technology, Department of Mathematics.
University of Oslo, Norway. University of Oslo, Norway. University of Oslo, Norway.
Optimal stopping, randomized stopping and singular control with partial information flowIn: Article in journal (Refereed)

The purpose of this paper is two-fold: We extend the well-known relation between optimal stopping and randomized stopping of a given stochastic process to a situation where the available information flow is a sub-filtration of the filtration of the process. We call these problems optimal stopping and randomized stopping with partial information. Following an idea of Krylov [K] we introduce a special singular stochastic control problem with partial information and show that this is also equivalent to the partial information optimal stopping and randomized stopping problems. Then we show that the solution of this singular control problem can be expressed in terms of (partial information) variational inequalities, which in turn can be rewritten as a reflected backward stochastic differential equation (RBSDE) with partial information.

• 2.
Linnaeus University, Faculty of Technology, Department of Mathematics.
Linnaeus University, Faculty of Technology, Department of Mathematics. University of Oslo, Norway.
Singular control of SPDEs with space-mean dynamics2019In: Mathematical Control & Related Fields, ISSN 2156-8472Article in journal (Refereed)

We consider the problem of optimal singular control of a stochastic partial differential equation (SPDE) with space-mean dependence. Such systems are proposed as models for population growth in a random environment. We obtain sufficient and necessary maximum principles for such control problems. The corresponding adjoint equation is a reflected backward stochastic partial differential equation (BSPDE) with space-mean dependence. We prove existence and uniqueness results for such equations. As an application we study optimal harvesting from a population modelled as an SPDE with space-mean dependence.

• 3.
Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
Nelson-type Limits for α-Stable Lévy Processes2010Licentiate thesis, comprehensive summary (Other academic)

Brownian motion has met growing interest in mathematics, physics and particularly in finance since it was introduced in the beginning of the twentieth century. Stochastic processes generalizing Brownian motion have influenced many research fields theoretically and practically. Moreover, along with more refined techniques in measure theory and functional analysis more stochastic processes were constructed and studied. Lévy processes, with Brownian motionas a special case, have been of major interest in the recent decades. In addition, Lévy processes include a number of other important processes as special cases like Poisson processes and subordinators. They are also related to stable processes.

In this thesis we generalize a result by S. Chandrasekhar [2] and Edward Nelson who gave a detailed proof of this result in his book in 1967 [12]. In Nelson’s first result standard Ornstein-Uhlenbeck processes are studied. Physically this describes free particles performing a random and irregular movement in water caused by collisions with the water molecules. In a further step he introduces a nonlinear drift in the position variable, i.e. he studies the case when these particles are exposed to an external field of force in physical terms.

In this report, we aim to generalize the result of Edward Nelson to the case of α-stable Lévy processes. In other words we replace the driving noise of a standard Ornstein-Uhlenbeck process by an α-stable Lévy noise and introduce a scaling parameter uniformly in front of all vector fields in the cotangent space, even in front of the noise. This corresponds to time being sent to infinity. With Chandrasekhar’s and Nelson’s choice of the diffusion constant the stationary state of the velocity process (which is approached as time tends to infinity) is the Boltzmann distribution of statistical mechanics.The scaling limits we obtain in the absence and presence of a nonlinear drift term by using the scaling property of the characteristic functions and time change, can be extended to other types of processes rather than α-stable Lévy processes.

In future, we will consider to generalize this one dimensional result to Euclidean space of arbitrary finite dimension. A challenging task is to consider the geodesic flow on the cotangent bundle of a Riemannian manifold with scaled drift and scaled Lévy noise. Geometrically the Ornstein-Uhlenbeck process is defined on the tangent bundle of the real line and the driving Lévy noise is defined on the cotangent space.

• 4.
Vienna University of Technology, Austria.
Linnaeus University, Faculty of Technology, Department of Building and Energy Technology. Vienna University of Technology, Austria. Vienna University of Technology, Austria. Vienna University of Technology, Austria.
A Matrix-Vector Operation-Based Numerical Solution Method for Linear m-th Order Ordinary Differential Equations: Application to Engineering Problems2013In: Advances in Applied Mathematics and Mechanics, ISSN 2070-0733, E-ISSN 2075-1354, Vol. 5, no 3, p. 269-308Article in journal (Refereed)

Many problems in engineering sciences can be described by linear, inhomogeneous, m-th order ordinary differential equations (ODEs) with variable coefficients. For this wide class of problems, we here present a new, simple, flexible, and robust solution method, based on piecewise exact integration of local approximation polynomials as well as on averaging local integrals. The method is designed for modern mathematical software providing efficient environments for numerical matrix-vector operation-based calculus. Based on cubic approximation polynomials, the presented method can be expected to perform (i) similar to the Runge-Kutta method, when applied to stiff initial value problems, and (ii) significantly better than the finite difference method, when applied to boundary value problems. Therefore, we use the presented method for the analysis of engineering problems including the oscillation of a modulated torsional spring pendulum, steady-state heat transfer through a cooling web, and the structural analysis of a slender tower based on second-order beam theory. Related convergence studies provide insight into the satisfying characteristics of the proposed solution scheme.

• 5.
Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering. Matematik.
A modiﬁed Schwarz–Christoffel mapping for regions with piecewise smooth boundaries2008In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, Vol. 213, no 1, p. 56-70Article in journal (Refereed)

A method where polygon corners in Schwarz-Christoffel mappings are rounded, is used to construct mappings from the upper half-plane to regions bounded by arbitrary piecewise smooth curves. From a given curve, a polygon is constructed by taking tangents to the curve in a number of carefully chosen so called tangent points. The Schwarz-Christoffel mapping for that polygon is then constructed and modified to round the corners.

Since such a modification causes effects on the polygon outside the rounded corners, the parameters in the mapping have to be re-determined. This is done by comparing side-lengths in tangent polygons to the given curve and the curve produced by the modified Schwarz-Christoffel mapping. The set of equations that this comparison gives, can normally be solved using a quasi--Newton method.

The resulting function maps the upper half--plane on a region bounded by a curve that apart from possible vertices is smooth, i.e., one time continuously differentiable, that passes through the tangent points on the given curve, has the same direction as the given curve in these points and changes direction monotonically between them. Furthermore, where the original curve has a vertex, the constructed curve has a vertex with the same inner angle.

The method is especially useful for unbounded regions with smooth boundary curves that pass infinity as straight lines, such as channels with parallel walls at the ends. These properties are kept in the region produced by the constructed mapping.

• 6.
Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
Modified Schwarz–Christoffel mappings using approximate curve factors2009In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 233, no 4, p. 1117-1127Article in journal (Refereed)

The Schwarz–Christoffel mapping from the upper half-plane to a polygonal region in the complex plane is an integral of a product with several factors, where each factor corresponds to a certain vertex in the polygon. Different modifications of the Schwarz–Christoffel mapping in which factors are replaced with the so-called curve factors to achieve polygons with rounded corners are known since long times. Among other requisites, the arguments of a curve factor and its correspondent scl factor must be equal outside some closed interval on the real axis.

In this paper, the term approximate curve factor is defined such that many of the already known curve factors are included as special cases. Additionally, by alleviating the requisite on the argument from exact to asymptotic equality, new types of curve factors are introduced. While traditional curve factors have a C1 regularity, C regular approximate curve factors can be constructed, resulting in smooth boundary curves when used in conformal mappings.

Applications include modelling of wave scattering in waveguides. When using approximate curve factors in modified Schwarz–Christoffel mappings, numerical conformal mappings can be constructed that preserve two important properties in the waveguides. First, the direction of the boundary curve can be well controlled, especially towards infinity, where the application requires two straight parallel walls. Second, a smooth (C) boundary curve can be achieved.

• 7.
Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
Numerical Conformal mappings for regions Bounded by Smooth Curves2006Licentiate thesis, monograph (Other academic)

In many applications, conformal mappings are used to transform two-dimensional regions into simpler ones. One such region for which conformal mappings are needed is a channel bounded by continuously differentiable curves. In the applications that have motivated this work, it is important that the region an approximate conformal mapping produces, has this property, but also that the direction of the curve can be controlled, especially in the ends of the channel.

This thesis treats three different methods for numerically constructing conformal mappings between the upper half-plane or unit circle and a region bounded by a continuously differentiable curve, where the direction of the curve in a number of control points is controlled, exact or approximately.

The first method is built on an idea by Peter Henrici, where a modified Schwarz-Christoffel mapping maps the upper half-plane conformally on a polygon with rounded corners. His idea is used in an algorithm by which mappings for arbitrary regions, bounded by smooth curves are constructed.

The second method uses the fact that a Schwarz-Christoffel mapping from the upper half-plane or unit circle to a polygon maps a region Q inside the half-plane or circle, for example a circle with radius less than 1 or a sector in the half--plane, on a region Omega inside the polygon bounded by a smooth curve. Given such a region Omega, we develop methods to find a suitable outer polygon and corresponding Schwarz-Christoffel mapping that gives a mapping from Q to Omega.

Both these methods use the concept of tangent polygons to numerically determine the coefficients in the mappings.

Finally, we use one of Don Marshall's zipper algorithms to construct conformal mappings from the upper half--plane to channels bounded by arbitrary smooth curves, with the additional property that they are parallel straight lines when approaching infinity.

• 8.
Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
Numerical Conformal Mappings for Waveguides2009Doctoral thesis, comprehensive summary (Other academic)

Acoustic or electro-magnetic scattering in a waveguide with varying direction and cross-section can be re-formulated as a two-dimensional scattering problem, provided that the variations take place in only one dimension at a time. By using the so-called Building Block Method, it is possible to construct the scattering properties of a combination of scatterers when the properties of each scatterer are known. Hence, variations in the waveguide geometry or in the boundary conditions can be treated one at a time. Using the Building Block Method, the problem takes the form of the Helmholtz equation for stationary waves in a waveguide of infinite length and with smoothly varying geometry and boundary conditions. A conformal mapping is used to transform the problem into a corresponding problem in a straight horizontal waveguide, and by expanding the field in Fourier trigonometric series, the problem can be reformulated as an infinite-dimensional ordinary differential equation. From this, numerically solvable differential equations for the reflection and transmission operators are derived. To be applicable in the Building Block Method, the numerical conformal mapping must be constructed such that the direction of the boundary curve can be controlled. At the channel ends ,it is an indispensable requirement, that the two boundary curves are (at least) asymptotically parallel and straight. Furthermore, to achieve bounded operators in the differential equations, the boundary curves must satisfy different regularity conditions, depending on the boundary conditions. In this work, several methods to accomplish such conformal mappings are presented. The Schwarz–Christoffel mapping, which is a natural starting point and for which also efficient numerical software exists, can be modified in different ways in order to achieve polygons with rounded corners. We present algorithms by which the parameters in the mappings can be determined after such modifications. We show also how the unmodified Schwarz–Christoffel mapping can be used for regions with a smooth boundary. This is done by constructing an appropriate outer polygon to the considered region.Finally, we introduce one method that is not Schwarz–Christoffel-related, by showing how one of the so-called zipper algorithms can be used for waveguides. Keywords: waveguides, building block method, numerical conformalmappings, Schwarz–Christoffel mapping, rounded corners method, approximate curve factors, outer polygon method, boundary curvature, zipper method, geodesic algorithm, acoustic wave scattering, electro-magnetic wave scattering

• 9.
Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
Numerical Conformal Mappings for Waveguides2010In: Computational Mathematics: Theory, Methods and Applications / [ed] Peter G. Chareton, Hauppauge, NY, USA: Nova Science Publishers, Inc., 2010Chapter in book (Other academic)

A number of newly developed numerical conformal mapping techniques are described. Their purpose is to achieve conformal mappings with good accuracy for regions bounded by smooth or piecewise smooth curves in which the boudary curve direction can be controllod, especially towards infinity in unbounded regions as for example waveguides.

Most of the mappings are variants of the Schwarz-Christoffel mappings.

• 10.
Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
On the curvature of an inner curve in a Schwarz-Christoffel mapping2007Report (Other academic)

In the so called outer polygon method, an approximative conformal mapping for a given simply connected region \Omega is constructed using a Schwarz-­Christoffel mapping for an outer polygon, a polygonal region of which \Omega is a subset. The resulting region is then bounded by a C^\infty -curve, which among other things means that its curvature is bounded.

In this work, we study the curvature of an inner curve in a polygon, i.e., the image under the Schwarz-­Christoffel mapping from R, the unit disk or upper half­plane, to a polygonal region P of a curve inside R. From the Schwarz-­Christoffel formula, explicit expressions for the curvature are derived, and for boundary curves, appearing in the outer polygon method, estimations of boundaries for the curvature are given.

• 11.
Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
Schwarz–Christoffel Mappings for Nonpolygonal Regions2008In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 31, no 1, p. 94-111Article in journal (Refereed)

An approximate conformal mapping for an arbitrary region $\varOmega$ bounded by a smooth curve $\varGamma$ is constructed using the Schwarz–Christoffel mapping for a polygonal region in which $\varOmega$ is embedded. An algorithm for finding this so-called outer polygon is presented. The resulting function is a conformal mapping from the upper half-plane or the unit disk to a region $R$, approximately equal to $\varOmega$. $R$ is bounded by a $C^\infty$ curve, and since the mapping function originates from the Schwarz–Christoffel mapping and tangent polygons are used to determine it, important properties of $\Gamma$ such as direction, linear asymptotes, and inflexion points are preserved in the boundary of $R$. The method makes extensive use of existing Schwarz–Christoffel software in both the determination of outer polygons and the calculation of function values. By the use suggested here, the capabilities of such well-written software are extended.

• 12.
Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering. Matematik.
Using a Zipper Algorithm to find a Conformal Map for a Channel with Smooth Boundary2006In: AIP Conference Proceedings: Second Conference on Mathematical Modeling of Wave Phenomena, American Institute of Physics, New York , 2006, p. 378-Conference paper (Refereed)

The so called geodesic algorithm, which is one of the zipper algorithms for conformal mappings, is combined with a Schwarz–Christoffel mapping, in its original or in a modiﬁed form, to produce a conformal mapping function between the upper half-plane and an arbitrary channel with smooth boundary and parallel walls at the end.

• 13.
Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
Acoustic Transmission in Ducts of Various Shapes with an Impedance Condition2008In: International Conference on Numerical Analysis and Applied Mathematics 2008, AIP, Melville, USA , 2008, p. 33-36Conference paper (Refereed)

Propagation of acoustic waves in a two-dimensional duct with an impedance condition at the boundary, is studied. The duct is assumed to have two ends at infinity being asymptotically straight, but otherwise to be arbitrarily shaped.The so called Building Block Method allows us to synthesize propagation properties for ducts with complicated geometries from results for simpler ducts. Conformal mappings can be used to transform these simple ducts to straight ducts with constant cross-sections.By using recently developed techniques for numerical conformal mappings, it is possible to construct a transformation between an infinite strip and an arbitrarily shaped duct with smooth or piecewise smooth boundary, keeping both smoothness and the well controlled boundary direction towards infinity that the above mentioned method requires.To accomplish a stable formulation of the problem, we express it in terms of scattering operators. The resulting differential equation is solved using wave splitting and invariant embedding techniques. We expand the involved functions in Fourier series, and hence, it is possible to give the operators a matrix representation. Numerical results are produced using truncated matrices.

• 14.
Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
Electro-Magnetic Scattering in Variously Shaped Waveguides with an Impedance Condition2009In: Mathematical modelling of wave phenomena: 3rd Conference on Mathematical Modelling of Wave Phenomena, Växjö, Sweden, 9 – 13 June 2008, Melville, New York: American Institute of Physics , 2009, p. 36-45Conference paper (Refereed)

Electro-magnetic scattering is studied in a waveguide with varying shape and crosssection. Furthermore, an impedance or admittance condition is applied to two of the waveguide walls. Under the condition that variations in geometry or impedance take place in only one plane at the time, the problem can be solved as a two-dimensional wave-scattering problems. By using newly developed numerical conformal mapping techniques, the problem is transformed into a wave-scattering problem in a straight two-dimensional channel. A numerically stable formulation is reached in terms of transmission and reflection operators. Numerical results are given for a slowly varying waveguide with a bend and for one more complex geometry.

• 15.
Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
Gymnasiematematik på distans: Varför så många avbryter sina distansstudier i matematik2006Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis

Syftet med studien är att undersöka varför det är så många distansstudenter som avbryter sina distansstudier i matematik. Distansutbildning (DU) är en studieform i stark frammarsch. Nyckelorden för en lyckad DU är bland annat flexibilitet och individanpassning. Medan flexibiliteten framförallt ökar tillgängligheten står individanpassningen som garant för en god lärsituation. Faktorer som i hög grad påverkar DU och genomströmningen är studiemotiv, artefakter och hur det sociala sammanhanget upplevs. Jag har i min undersökning samlat in data med hjälp av kvalitativa intervjuer och dessutom granskat distansupplägget som eleverna på den aktuella skolan haft. Urvalsgruppen består av fem stycken elever i olika åldersgrupper som redan klarat av halva kursen. Även analysen har skett med en kvalitativ ansats. I den aktuella studien kan man se att många av de riskfaktorer som tros ligga bakom många avhopp även föreligger här med det aktuella studieupplägget.

• 16. Araujo, Juan
Linnaeus University, Faculty of Technology, Department of Mathematics.
Computation of scattering resonances in absorptive and dispersive media with applications to metal-dielectric nano-structures2020In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716Article in journal (Refereed)

In this paper we consider scattering resonance computations in optics when the resonators consist of frequency dependent and lossy materials, such as metals at optical frequencies. The proposed computational approach combines a novel hp-FEM strategy, based on dispersion analysis for complex frequencies, with a fast implementation of the nonlinear eigenvalue solver NLEIGS. Numerical computations illustrate that the pre-asymptotic phase is significantly reduced compared to standard uniform h and p strategies. Moreover, the efficiency grows with the refractive index contrast, which makes the new strategy highly attractive for metal-dielectric structures. The hp-refinement strategy together with the efficient parallel code result in highly accurate approximations and short runtimes on multi processor platforms.

• 17.
Umeå university, Sweden.
Umeå university, Sweden. KTH Royal Instute of Technology, Sweden.
Efficient resonance computations for Helmholtz problems based on a Dirichlet-to-Neumann map2018In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 330, p. 177-192Article in journal (Refereed)

We present an efficient procedure for computing resonances and resonant modes of Helmholtz problems posed in exterior domains. The problem is formulated as a nonlinear eigenvalue problem (NEP), where the nonlinearity arises from the use of a Dirichlet-to-Neumann map, which accounts for modeling unbounded domains. We consider a variational formulation and show that the spectrum consists of isolated eigenvalues of finite multiplicity that only can accumulate at infinity. The proposed method is based on a high order finite element discretization combined with a specialization of the Tensor Infinite Arnoldi method (TIAR). Using Toeplitz matrices, we show how to specialize this method to our specific structure. In particular we introduce a pole cancellation technique in order to increase the radius of convergence for computation of eigenvalues that lie close to the poles of the matrix-valued function. The solution scheme can be applied to multiple resonators with a varying refractive index that is not necessarily piecewise constant. We present two test cases to show stability, performance and numerical accuracy of the method. In particular the use of a high order finite element discretization together with TIAR results in an efficient and reliable method to compute resonances.

• 18.
Russian Academy of Sciences, Russia.
University of Niš, Serbia. Linnaeus University, Faculty of Technology, Department of Mathematics. Russian Academy of Sciences, Russia. University of Belgrade, Serbia. Russian Academy of Sciences, Russia.
p-Adic mathematical physics and B. Dragovich research2017In: P-Adic Numbers, Ultrametric Analysis, and Applications, ISSN 2070-0466, E-ISSN 2070-0474, Vol. 9, no 1, p. 82-85Article in journal (Refereed)

We present a brief review of some parts of p-adic mathematical physics related to the scientific work of Branko Dragovich on the occasion of his 70th birthday.

• 19.
Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering. Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
Diffractive Index Determination by Tikhonov Regularization on Forced String Vibration Data2009In: Mathematical modelling of wave phenomena: 3rd Conference on Mathematical Modelling of Wave Phenomena, Växjö, Sweden, 9-13 June 2008, Melville, New York: American Institute of Physics , 2009, p. 224-232Conference paper (Refereed)

Wave analysis is efficient for investigating the interior of objects. Examples are ultra sound examination of humans and radar using elastic and electromagnetic waves. A common procedure is inverse scattering where both transmitters and receivers are located outside the object or on its boundary. A variant is when both transmitters and receivers are located on the scattering object. The canonical model is a finite inhomogeneous string driven by a harmonic point force. The inverse problem for the determination of the diffractive index of the string is studied. This study is a first step to the problem for the determination of the mechanical strength of wooden logs. An inverse scattering theory is formulated incorporating two regularizing strategies. The results of simulations using this theory show that the suggested method works quite well and that the regularization methods based on the couple of spaces (L2; H1 ) could be very useful in such problems.

• 20.
Linnaeus University, Faculty of Technology, Department of Mathematics. National Research University of Information Technologies, Russia.
At the Crossroads of Three Seemingly Divergent Approaches to Quantum Mechanics2018In: Quantum Foundations, Probability and Information / [ed] A. Khrennikov, B. Toni, Springer, 2018, p. 13-21Chapter in book (Refereed)

Several concepts stemming from three apparently divergent approaches to quantum mechanics—Bohmian Mechanics, QBism, and Time-Symmetric Quantum Mechanics—are interwoven in an information-theoretic Darwinian scheme applied to fundamental physical systems that might contribute to shed light on some long-standing quantum mechanical conundrums.

• 21.
Linnaeus University, Faculty of Technology, Department of Mathematics.
Linnaeus University, Faculty of Technology, Department of Mathematics.
Decision-making and cognition modeling from the theory of mental instruments2017In: The Palgrave Handbook of Quantum Models in Social Science: Applications and Grand Challenges / [ed] Emmanuel Haven, Andrei Khrennikov, Palgrave Macmillan, 2017, p. 75-93Chapter in book (Other academic)

The authors present the theory of quantum measurements in a humanities friendly way. The most general process of decision-making is represented with the aid of the formalism of quantum apparatuses and instruments. This measurement formalism generalizes the standard one based on the von Neumann–Lüders projection postulate. Generalized quantum observables are mathematically represented as positive operator valued measures (POVMs) and state transformers resulting from the feedback of measurements to the states of systems that are given by quantum instruments. The quantum scheme of indirect measurements (a special realization of quantum instruments) is applied to model decision-making as resulting from the interaction between the belief and decision states. The authors analyze the specific features of quantum instruments which are important for cognitive and social applications. In particular, the state transformers given by quantum instruments are in general less invasive than the state projections. Thus quantum-like decision-making need not be viewed as a kind of state collapse.

• 22.
Linnaeus University, Faculty of Health, Social Work and Behavioural Sciences, School of Education, Psychology and Sport Science.
Linnaeus University, Faculty of Health, Social Work and Behavioural Sciences, School of Education, Psychology and Sport Science.
Vardagsmatematik: Pedagogers gruppsamtal om vardagsmatematiken i de tidigare skolåren2011Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis

Syftet med vår studie var att synliggöra hur pedagoger som undervisar i grundskolans tidiga år samtalar om undervisningen av matematikens användning i vardagen utifrån den socio­kulturella teorin. Gruppsamtalen utgick från rubrikerna konkretisering, individ­uali­se­ring, mate­matik och språk samt matematiksvårigheter kopplade till vardagsmatematiken. Vi använ­de oss av fokusgrupper som undersökningsinstrument. I studien ingick 13 peda­goger som alla undervisade i matematik på skolans lågstadium. Deltagarna i studien var verk­samma på fyra olika skolor i två olika kommuner.

Undersökningens resultat bearbetades utifrån en hermeneutisk innehållsanalys. I resultatet synliggörs en omfattande och medveten undervisning kring vardagsmatematiken hos de med­­verkande pedagogerna. Det som upplevs som hinder eller skapande av möjligheter är resurstilldelningen både materiellt och personellt. Gruppstorlek och gruppsammansättning upp­levs också som avgörande för en undervisning som lever upp till pedagogernas ambi­tioner. Konkretiseringen ges en stor betydelse för att fånga och förklara den abstrakta mate­matiken. Individualiseringen ses av pedagogerna som nödvändig för att få med sig alla elev­erna i matematikundervisningens gemenskap och för att alla elever ska nå målen. Den sam­talande matematiken beskrivs som ovärderlig för elevernas tänkande och därmed för­ståelse och lärandet. Det framgår att för elever med matematiska svårigheter är det konkreta materi­alet, tid tillsammans med pedagog och ett varierat undervisningssätt faktorer som skapar för­utsättningar för lärande.

• 23.
Linnaeus University, Faculty of Technology, Department of Mathematics.
University of Agder, Norway. Linnaeus University, Faculty of Technology, Department of Mathematics.
Hilbert space embeddings for Gelfand–Shilov and Pilipović spaces2017In: Generalized Functions and Fourier Analysis: Dedicated to Stevan Pilipović on the Occasion of his 65th Birthday / [ed] Michael Oberguggenberger, Joachim Toft, Jasson Vindas, Patrik Wahlberg, Springer, 2017, p. 31-44Chapter in book (Refereed)

We consider quasi-Banach spaces that lie between a Gelfand–Shilov space, or more generally, Pilipovi´c space, H, and its dual, H′. We prove that for such quasi-Banach space B, there are convenient Hilbert spaces, Hk, k=1,2ss, with normalized Hermite functions as orthonormal bases and such that B lies between H1 and H1, and the latter spaces lie between H and H′.

• 24.
Max-Planck-Institut für Informatik, Saarbrücken,.
Technical University of Denmark. Linnaeus University, Faculty of Technology, Department of Mathematics. Technical University of Denmark.
Evolutionary Algorithms for the Detection of Structural Breaks in Time Series: extended abstract2013In: Proceedings of the 15th annual conference on Genetic and evolutionary computation, ACM Press, 2013, p. 119-120Conference paper (Refereed)

Detecting structural breaks is an essential task for the statistical analysis of time series, for example,  for fitting parametric models to it. In short, structural breaks  are points in time at which the behavior of the time series changes. Typically, no solid background knowledge of the time series under consideration is available. Therefore, a black-box optimization approach is our method of choice for detecting structural breaks. We describe a \ea framework which easily adapts to a large number of statistical settings. The experiments on artificial and real-world time series show that the algorithm detects break points with high precision and is computationally very efficient.

A reference implementation is availble at the following address:

http://www2.imm.dtu.dk/\~\/pafi/SBX/launch.html

• 25.
Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
Individualisering i matematik: En studie om matematikundervisningens individualisering, f-6, i jämförelse med Lpo 94.2006Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis

Syftet med vår studie var att se om lärarna i år f - 6 i sin matematikundervisning följde de intentioner som Lpo 94 ger angående individualisering. Vi ville även se vilken innebörd ordet individualisering hade för olika lärare och om de ansåg att det fanns några svårigheter med att individualisera sin matematikundervisning. För att få svar på våra frågeställningar gjorde vi fyra intervjuer och en enkätundersökning med lärare på åtta olika skolor i södra Sverige. Enkätundersökningen byggde vi utifrån svaren vi fått från våra intervjuer för att få bekräftelse på om åsikterna var gällande för flertalet lärare. Vi redovisar vad tidigare forskning säger om kunskap, inlärning och individualisering. Vi definierar ordet individualisering och talar om vilka riktlinjer som Lpo 94 ger angående detta. Resultatet visar att de flesta lärare försöker följa intentionerna i Lpo 94 men att det finns svårigheter som gör att de inte lyckas anpassa undervisningen för den enskilde individen. Många lärare använder fortfarande samma läromedel till samtliga elever och hastighetsindividualiserar sin undervisning.

• 26.
University of Belgrade, Serbia;Serbian Academy of Sciences and Arts, Serbia.
Linnaeus University, Faculty of Technology, Department of Mathematics. National Research University of Information Technologies, Russia. Steklov Mathematical Institute of the Russian Academy of Sciences, Russia. Steklov Mathematical Institute of the Russian Academy of Sciences, Russia. Steklov Mathematical Institute of the Russian Academy of Sciences, Russia.
p-Adic mathematical physics: the first 30 years2017In: P-Adic Numbers, Ultrametric Analysis, and Applications, ISSN 2070-0466, E-ISSN 2070-0474, Vol. 9, no 2, p. 87-121Article in journal (Refereed)

p-Adic mathematical physics is a branch of modern mathematical physics based on the application of p-adic mathematical methods in modeling physical and related phenomena. It emerged in 1987 as a result of efforts to find a non-Archimedean approach to the spacetime and string dynamics at the Planck scale, but then was extended to many other areas including biology. This paper contains a brief review of main achievements in some selected topics of p-adic mathematical physics and its applications, especially in the last decade. Attention is mainly paid to developments with promising future prospects.

• 27.
ETH Zurich, Switzerland.
ETH Zurich, Switzerland. ETH Zurich, Switzerland.
Linearization techniques for band structure calculations in absorbing photonic crystals2012In: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 89, no 2, p. 180-191Article in journal (Refereed)

Band structure calculations for photonic crystals require the numerical solution of eigenvalue problems. In this paper, we consider crystals composed of lossy materials with frequency-dependent permittivities. Often, these frequency dependencies are modeled by rational functions, such as the Lorentz model, in which case the eigenvalue problems are rational in the eigenvalue parameter. After spatial discretization using an interior penalty discontinuous Galerkin method, we employ a recently developed linearization technique to deal with the resulting rational matrix eigenvalue problems. In particular, the efficient implementation of Krylov subspace methods for solving the linearized eigenvalue problems is investigated in detail. Numerical experiments demonstrate that our new approach is considerably cheaper in terms of memory and computing time requirements compared with the naive approach of turning the rational eigenvalue problem into a polynomial eigenvalue problem and applying standard linearization techniques. Copyright © 2011 John Wiley & Sons, Ltd.

• 28.
FOI, Swedish Defence Research Agency.
Uppsala university. Uppsala university ; FOI, Swedish Defence Research Agency.
The influence of weak and strong solid wall boundary conditions on the convergence to steady-state of the Navier-Stokes equations2009In: 19th AIAA Computational Fluid Dynamics Conference 2009, American Institute of Aeronautics and Astronautics, 2009, article id 3551Conference paper (Refereed)

In the present paper we study the influence of weak and strong no-slip solid wall boundary conditions on the convergence to steady-state. Our Navier-Stokes solver is edge based and operates on unstructured grids. The two types of boundary conditions are applied to no-slip adiabatic walls. The two approaches are analyzed for a simplified model problem and the reason for the different convergence rates are discussed in terms of the theoretical findings for the model problem. Numerical results for a 2D viscous steady state low Reynolds number problem show that the weak boundary conditions often provide faster convergence. It is shown that strong boundary conditions can prevent the steady state convergence. It is also demonstrated that the two approaches converge to the same solution. Similar results are obtained for high Reynolds number flow in two and three dimensions.

• 29.
ETH Zurich, Switzerland.
On a high-order discontinuous Galerkin method applied to canonical scattering problems2010In: International Symposium on Electromagnetic Theory: EMTS, 2010 URSI, IEEE, 2010, p. 752-755Conference paper (Refereed)

A high-order interior penalty method for scattering problems in two-dimensions is presented. Results for perfectly conducting objects illustrate the high accuracy of the method at low computational cost.

• 30.
Umeå university, Sweden.
Durham University, UK. University of Zagreb, Croatia.
Efficient and reliable hp-FEM estimates for quadratic eigenvalue problems and photonic crystal applications2016In: Computers and Mathematics with Applications, ISSN 0898-1221, E-ISSN 1873-7668, Vol. 72, no 4, p. 952-973Article in journal (Refereed)

We present a-posteriori analysis of higher order finite element approximations (hp-FEM) for quadratic Fredholm-valued operator functions. Residual estimates for approximations of the algebraic eigenspaces are derived and we reduce the analysis of the estimator to the analysis of an associated boundary value problem. For the reasons of robustness we also consider approximations of the associated invariant pairs. We show that our estimator inherits the efficiency and reliability properties of the underlying boundary value estimator. As a model problem we consider spectral problems arising in analysis of photonic crystals. In particular, we present an example where a targeted family of eigenvalues cannot be guaranteed to be semisimple. Numerical experiments with hp-FEM show the predicted convergence rates. The measured effectivities of the estimator compare favorably with the performance of the same estimator on the associated boundary value problem. We also present a benchmark estimator, based on the dual weighted residual (DWR) approach, which is more expensive to compute but whose measured effectivities are close to one.

• 31.
Umeå University, Sweden.
University of Zagreb, Croatia.
A subspace iteration algorithm for Fredholm valued functions2015In: Mathematical problems in engineering (Print), ISSN 1024-123X, E-ISSN 1563-5147, Vol. 2015, p. 1-14, article id 459895Article in journal (Refereed)

We present an algorithm for approximating an eigensubspace of a spectral component of an analytic Fredholm valued function. Our approach is based on numerical contour integration and the analytic Fredholm theorem. The presented method can be seen as a variant of the FEAST algorithm for infinite dimensional nonlinear eigenvalue problems. Numerical experiments illustrate the performance of the algorithm for polynomial and rational eigenvalue problems.

• 32.
Lund University, Sweden.
Lund University, Sweden.
A Comparison of two numerical methods for homogenization of Maxwell's equations2004In: Computational Electromagnetics- Methods and Applications (EMB04), Gothenburg, Sweden, October 18-19, 2004, 2004, p. 50-57Conference paper (Refereed)

When the wavelength is much larger than the typical scale of the microstructure in a material, it is possible to define effective or homogenized material coefficients. The classical way of determination of the homogenized coefficients consists of solving an elliptic problem in a unit cell. This method and the Floquet-Bloch method, where an eigenvalue problem is solved, are numerically compared with respect to accuracy and contrast sensitivity. The Floquet-Bloch method is shown to be a good alternative to the classical homogenization method, when the contrast is modest.

• 33.
Lund University, Sweden.
Lund University, Sweden.
On two numerical methods for homogenization of Maxwell's equations2007In: Journal Electromagnetic Waves and Applications, ISSN 0920-5071, E-ISSN 1569-3937, Vol. 21, no 13, p. 1845-1856Article in journal (Refereed)

When the wavelength is much larger than the typical scale of the microstructure in a material, it is possible to define effective or homogenized material coefficients. The classical way of determination of the homogenized coefficients consists of solving an elliptic problem in a unit cell. This method and the Floquet-Bloch method, where an eigenvalue problem is solved, are numerically compared with respect to accuracy and contrast sensitivity. Moreover, we provide numerical bounds on the effective permittivity. The Floquet-Bloch method is shown to be a good alternative to the classical homogenization method, when the contrast is modest.

• 34.
ETH Zurich, Switzerland.
ETH Zurich, Switzerland.
Complex dispersion relation calculations with the applications to absorptive photonic crystals2010In: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 84, p. 849-863Article in journal (Refereed)

A high-order discontinuous Galerkin method for calculations of complex dispersion relations of two-dimensional photonic crystals is presented. The medium is characterized by a complex-valued permittivityand we relate for this absorptive system the spectral parameter to the time frequency. We transform thenon-linear eigenvalue problem for a Lorentz material in air into a non-Hermitian linear eigenvalue problemand uses a Krylov space method to compute approximate eigenvalues. Moreover, we study the impact ofthe penalty term numerically and illustrate the high convergence rate of the method.

• 35.
Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
Problemlösning i form av räknesagor2009Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis

Abstrakt

Syftet med arbetet är att undersöka hur barn funderar ut lösningar på matematiska problem som de stöter på när de arbetar med räknesagor. Fokus har legat på vilka strategier eleverna använder sig av när de löser räknesagor och vilken betydelse kommunikationen har för hur eleverna utvecklar sina strategier. Den första kontakten med matematik kan vara avgörande för det fortsatta intresset. Läraren har därför en stor uppgift att fylla genom att introducera matematiken på ett intressant och lustfyllt sätt. Det är därför angeläget att det finns en balans mellan teori och praktik. Barn behöver uppleva matematiken genom flera olika tillvägagångssätt för att förstå den, där av vårt val av ämne.

Den metod vi valde i denna fallstudie var observationer enskilt och i grupp när barn gjorde räknesagor, för att finna svar på vårt syfte. Även intervjufrågor ställdes i samband med de enskilda räknesagorna.

Resultatet visade att barnen använder sig av olika lösningsstrategier. De vanligaste strategierna i de enskilda arbetena var att de ritade bilder medan de föredrog att använda siffror i grupparbetena. Flera av barnen hade god användning av sina bilder när de löste sina räknesagor. Detta kunde vi se genom att de hela tiden behövde gå tillbaka till föregående bild, när de skulle rita nästa. Resultatet i vår undersökning av räknesagor visade även att barnen inte var vana att kommunicera med varandra. Detta visade sig genom att det oftast var ett barn som tog initiativet och utförde räknesagan själv utan att fråga de andra.

• 36.
Technische Universität Darmstadt, Germany.
A Dual Consistent Finite Difference Method with Narrow Stencil Second Derivative Operators2018In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 75, no 2, p. 906-940Article in journal (Refereed)

We study the numerical solutions of time-dependent systems of partial differential equations, focusing on the implementation of boundary conditions. The numerical method considered is a finite difference scheme constructed by high order summation by parts operators, combined with a boundary procedure using penalties (SBP-SAT). Recently it was shown that SBP-SAT finite difference methods can yield superconvergent functional output if the boundary conditions are imposed such that the discretization is dual consistent. We generalize these results so that they include a broader range of boundary conditions and penalty parameters. The results are also generalized to hold for narrow-stencil second derivative operators. The derivations are supported by numerical experiments.

• 37.
Uppsala University.
Stable Numerical Methods with Boundary and Interface Treatment for Applications in Aerodynamics2012Doctoral thesis, comprehensive summary (Other academic)

In numerical simulations, problems stemming from aerodynamics pose many challenges for the method used. Some of these are addressed in this thesis, such as the fluid interacting with objects, the presence of shocks, and various types of boundary conditions.

Scenarios of the kind mentioned above are described mathematically by initial boundary value problems (IBVPs). We discretize the IBVPs using high order accurate finite difference schemes on summation by parts form (SBP), combined with weakly imposed boundary conditions, a technique called simultaneous approximation term (SAT). By using the energy method, stability can be shown.

The weak implementation is compared to the more commonly used strong implementation, and it is shown that the weak technique enhances the rate of convergence to steady state for problems with solid wall boundary conditions. The analysis is carried out for a linear problem and supported numerically by simulations of the fully non-linear Navier–Stokes equations.

Another aspect of the boundary treatment is observed for fluid structure interaction problems. When exposed to eigenfrequencies, the coupled system starts oscillating, a phenomenon called flutter. We show that the strong implementation sometimes cause instabilities that can be mistaken for flutter.

Most numerical schemes dealing with flows including shocks are first order accurate to avoid spurious oscillations in the solution. By modifying the SBP-SAT technique, a conservative and energy stable scheme is derived where the order of accuracy can be lowered locally. The new scheme is coupled to a shock-capturing scheme and it retains the high accuracy in smooth regions.

For problems with complicated geometry, one strategy is to couple the finite difference method to the finite volume method. We analyze the accuracy of the latter on unstructured grids. For grids of bad quality the truncation error can be of zeroth order, indicating that the method is inconsistent, but we show that some of the accuracy is recovered.

We also consider artificial boundary closures on unbounded domains. Non-reflecting boundary conditions for an incompletely parabolic problem are derived, and it is shown that they yield well-posedness. The SBP-SAT methodology is employed, and we prove that the discretized problem is stable.

• 38.
Uppsala University.
Uppsala University. Linköping University.
A stable and conservative method for locally adapting the design order of finite difference schemes2011In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 230, no 11, p. 4216-4231Article in journal (Refereed)

A procedure to locally change the order of accuracy of finite difference schemes is developed. The development is based on existing Summation-By-Parts operators and a weak interface treatment. The resulting scheme is proven to be accurate and stable.

Numerical experiments verify the theoretical accuracy for smooth solutions. In addition, shock calculations are performed, using a scheme where the developed switching procedure is combined with the MUSCL technique.

• 39.
Uppsala University.
Uppsala University. Uppsala University ; University of the Witvatersrand, South Africa ; FOI, The Swedish Defence Research Agency .
A stable and conservative method of locally adapting the design order of finite difference schemes2010In: Proceedings of the 7th South African Conference on Computational and Applied Mechanics / [ed] S. Kok, D. Wilke & H. Inglis, South African Association for Theoretical and Applied Mechanics , 2010, p. 128-136Conference paper (Other academic)

A procedure to switch the order of accuracy of finite difference schemes is developed. The development is based on existing Summation-By-Parts operators and a weak interface treatment. The resulting scheme is proven to be stable and accurate.

Numerical experiments verify the theoretical accuracy for smooth solutions. In addition shock calculations is performed, using a scheme where the developed switching procedure is combined with the MUSCL technique for shock capturing.

• 40.
Uppsala University.
Uppsala University. Uppsala University.
Shock Calculations using a Very High Order Accurate Euler and Navier-Stokes Solver2009In: Proceedings of the Sixth South African Conference on Computational and Applied Mechanics, South African Association for Theoretical and Applied Mechanics , 2009, p. 63-73Conference paper (Other academic)
• 41.
Uppsala University.
FOI, Swedish Defence Research Agency.
Analysis of mesh and boundary effects on the accuracy of node-centered finite volume schemes2009In: 19th AIAA Computational Fluid Dynamics Conference, American Institute of Aeronautics and Astronautics, 2009, article id 3651Conference paper (Refereed)

The accuracy of the node-centered finite volume method in one-dimension is analyzed. Numerical simulations and analysis are performed for both a hyperbolic and a elliptic case, for various types of grids. The results from the simulations agree with the analysis. The boundary conditions are implemented weakly using penaly technique. For the hyperbolic case we see that the type of grid has large impact on the order of accuracy, whereas the choice of penaly parameter only affect the error constant. For the elliptic case the grid has less impact on the order of accuracy. For both the hyperbolic and elliptic problem we show that the error contribution from the primal and dual grid can be treated separately.

• 42.
Uppsala University .
Uppsala University ; The Swedish Defence Research Agency.
Analysis of the order of accuracy for node-centered finite volume schemes2009In: Applied Numerical Mathematics, ISSN 0168-9274, E-ISSN 1873-5460, Vol. 59, no 10, p. 2659-2676Article in journal (Refereed)

The order of accuracy of the node-centered finite volume methods is analyzed, and the analysis is based on an exact derivation of the numerical errors in one dimension. The accuracy for various types of grids are considered. Numerical simulations and analysis are performed for both a hyperbolic and a elliptic case, and the results agree. The impact of weakly imposed boundary conditions is analyzed and verified numerically. We show that the error contribution from the primal and dual grid can be treated separately.

• 43.
Uppsala University.
Uppsala University.
Analysis of the order of accuracy for node-centered finite volume schemes2009Report (Other academic)

The order of accuracy of the node-centered finite volume methods is analyzed, and the analysis is based on an exact derivation of the numerical errors in one dimension. The accuracy for various types of grids are considered. Numerical simulations and analysis are performed for both a hyperbolic and a eliptic case, and the results agree. The impact of weakly imposed boundary conditions is analyzed and verified numerically. We show that the error contribution from the primal and dual grid can be treated separately.

• 44.
Technische Universität Darmstadt, Germany.
Exact Non-reflecting Boundary Conditions Revisited: Well-Posedness and Stability2017In: Foundations of Computational Mathematics, ISSN 1615-3375, E-ISSN 1615-3383, Vol. 17, no 4, p. 957-986Article in journal (Refereed)

Exact non-reflecting boundary conditions for a linear incompletely parabolic system in one dimension have been studied. The system is a model for the linearized compressible Navier-Stokes equations, but is less complicated which allows for a detailed analysis without approximations. It is shown that well-posedness is a fundamental property of the exact non-reflecting boundary conditions. By using summation by parts operators for the numerical approximation and a weak boundary implementation, it is also shown that energy stability follows automatically.

• 45.
Uppsala University, Sweden.
Exact non-reflecting boundary conditions revisited: well-posedness and stability2012Report (Other academic)

Exact non-reflecting boundary conditions for an incompletely parabolic system have been studied. It is shown that well-posedness is a fundamental property of the non-reflecting boundary conditions. By using summation by parts operators for the numerical approximation and a weak boundary implementation, energy stability follows automatically. The stability in combination with the high order accuracy results in a reliable, efficient and accurate method. The theory is supported by numerical simulations.

• 46.
Finite difference schemes with transferable interfaces for parabolic problems2018Report (Other academic)

We derive a method to locally change the order of accuracy of finite difference schemes that approximate the second derivative. The derivation is based on summation-by-parts operators, which are connected at interfaces using penalty terms. At such interfaces, the numerical solution has a double representation, with one representation in each domain. We merge this double representation into a single one, yielding a new scheme with unique solution values in all grid points. The resulting scheme is proven to be stable, accurate and dual consistent.

• 47.
Uppsala University.
Well-posedness and stability of exact non-reflecting boundary conditions2013In: 21st AIAA Computational Fluid Dynamics Conference, Fluid Dynamics and Co-located Conferences, American Institute of Aeronautics and Astronautics, 2013, article id 2960Conference paper (Refereed)
• 48.
Uppsala University.
Uppsala University.
Simulations of Ground Effects on Wake Vortices at Runways2009In: Proceedings of the Sixth South African Conference on Computational and Applied Mechanics, South African Association for Theoretical and Applied Mechanics , 2009, p. 101-108Conference paper (Refereed)
• 49.
Uppsala University.
University of Oslo. Uppsala University.
Simulations of Ground Effects on Wake Vortices at Runways2007Report (Other academic)

n this paper the interaction between two counter-rotating vortices is examined, and the performance of a newly developed finite difference code is discussed. The Reynolds numbers considered are low to medium, and the flow is compressible. Most of the computations are performed in a two dimensional domain, with different grid sizes, Reynolds number and order of accuracy of the scheme. Finally, a three dimensional computation is made in order to examine the relevance of the two dimensional model.

• 50.
Linnaeus University, Faculty of Technology, Department of Mathematics.
Entreprenöriellt lärande som verktyg i en föränderlig värld: En studie om värdeskapande lärande, entreprenöriella förmågor och dess inverkningar på problemlösning i matematik.2018Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis

Studien är en pedagogisk designforskning som genom en intervention belyser hur de entreprenöriella förmågorna komplexitet, ansvar och samverkan framträder genom en kombination av metoderna värdeskapande lärande och matematisk modellering. Studien avser också att belysa om metoderna kan bidra till ett kontextuellt och strategiskt matematiskt kunnande. Resultatet består av en beskrivning och analys av de förmågor som framträtt samt en analys av de designprinciper som användes för att tillverka den uppgift som eleverna använde under projektets gång. Resultatet visar att kombinationen av metoderna värdeskapande lärande och matematisk modellering, dels genom sin verklighetsanknytning och uppmuntrande till elevers engagemang, skapar goda möjligheter för ovanstående förmågors framträdande. Den andra delen av resultatet visar att eleverna behöver ges en god förförståelse för att bibehålla ett matematiskt fokus igenom hela uppgiftsprocessen.

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