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  • 1.
    Lockby, Andreas
    et al.
    School of Science and Technology, Örebro University, Örebro, Sweden..
    Sandin, Patrik
    Örebro University, School of Science and Technology.
    Ögren, Magnus
    Örebro University, School of Science and Technology.
    Gulliksson, Mårten
    Örebro University, School of Science and Technology.
    Finding Stationary Solutions of PDEs with Constraints using Damped Dynamical Systems2016In: Comsol Conference 2016, 2016Conference paper (Refereed)
    Abstract [en]

    The dynamical functional particle method(DFPM) is a method for solving equations, e.g. PDEs, using a second order damped dynamical system. We show how the method can be extended to include constraints both explicitly as global constraints and adding the constraints as additional damped dynamical equations. These methods are implemented in Comsol and we show numerical tests for finding the stationary solution of a nonlinear heat equation with and without constraints (global and dynamical). The results show that DFPM is a very general and robust way of solving PDEs and it should be of interest to implement the approach more generally in Comsol.

  • 2.
    Møller-Andersen, Jakob
    et al.
    Department of Applied Mathematics and Computer Science, Technical University of Denmark, Lyngby, Denmark.
    Ögren, Magnus
    Örebro University, School of Science and Technology. Department of Applied Mathematics and Computer Science, Technical University of Denmark, Lyngby, Denmark; Nano Science Center, Department of Chemistry, University of Copenhagen, København, Denmark.
    Perturbative semiclassical trace formulae for harmonic oscillators2015In: Reports on mathematical physics, ISSN 0034-4877, E-ISSN 1879-0674, Vol. 75, no 3, 359-382 p.Article in journal (Refereed)
    Abstract [en]

    In this article we extend previous semiclassical studies by including more general perturbative potentials of the harmonic oscillator in arbitrary spatial dimensions. Our starting point is a radial harmonic potential with an arbitrary even monomial perturbation, which we use to study the resulting U(D) to O(D) symmetry breaking. We derive the gross structure of the semiclassical spectrum from periodic orbit theory, in the form of a perturbative (ħ → 0) trace formula. We then show how to apply the results to even-order polynomial potentials, possibly including mean-field terms. We have drawn the conclusion that the gross structure of the quantum spectrum is determined from only classical circular and diameter orbits for this class of systems.

  • 3.
    Sandin, Patrik
    et al.
    Örebro University, School of Science and Technology.
    Ögren, Magnus
    Örebro University, School of Science and Technology.
    Gulliksson, Mårten
    Örebro University, School of Science and Technology. Univ Orebro, Sch Sci & Technol, S-70182 Orebro, Sweden..
    Numerical solution of the stationary multicomponent nonlinear Schrodinger equation with a constraint on the angular momentum2016In: Physical Review E, ISSN 2470-0045, Vol. 93, no 3, 033301Article in journal (Refereed)
    Abstract [en]

    We formulate a damped oscillating particle method to solve the stationary nonlinear Schrodinger equation (NLSE). The ground-state solutions are found by a converging damped oscillating evolution equation that can be discretized with symplectic numerical techniques. The method is demonstrated for three different cases: for the single-component NLSE with an attractive self-interaction, for the single-component NLSE with a repulsive self-interaction and a constraint on the angular momentum, and for the two-component NLSE with a constraint on the total angular momentum. We reproduce the so-called yrast curve for the single-component case, described in [A. D. Jackson et al., Europhys. Lett. 95, 30002 (2011)], and produce for the first time an analogous curve for the two-component NLSE. The numerical results are compared with analytic solutions and competing numerical methods. Our method is well suited to handle a large class of equations and can easily be adapted to further constraints and components.

  • 4.
    Sandin, Patrik
    et al.
    Örebro University, School of Science and Technology.
    Ögren, Magnus
    Örebro University, School of Science and Technology.
    Gulliksson, Mårten
    Örebro University, School of Science and Technology.
    Smyrnakis, J.
    Technological Education Institute of Crete, Heraklion, Greece.
    Magiropoulos, M.
    Technological Education Institute of Crete, Heraklion, Greece.
    Kavoulakis, G. M.
    Technological Education Institute of Crete, Heraklion, Greece.
    Dimensional reduction in Bose-Einstein condensed clouds of atoms confined in tight potentials of any geometry and any interaction strength2017In: Physical Review E, ISSN 2470-0045, Vol. 95, no 1, 012142Article in journal (Refereed)
    Abstract [en]

    Motivated by numerous experiments on Bose-Einstein condensed atoms which have been performed in tight trapping potentials of various geometries (elongated and/or toroidal/annular), we develop a general method which allows us to reduce the corresponding three-dimensional Gross-Pitaevskii equation for the order parameter into an effectively one-dimensional equation, taking into account the interactions (i.e., treating the width of the transverse profile variationally) and the curvature of the trapping potential. As an application of our model we consider atoms which rotate in a toroidal trapping potential. We evaluate the state of lowest energy for a fixed value of the angular momentum within various approximations of the effectively one-dimensional model and compare our results with the full solution of the three-dimensional problem, thus getting evidence for the accuracy of our model.

  • 5.
    Schindler, Maike
    et al.
    Örebro University, School of Science and Technology.
    Lilienthal, Achim
    Örebro University, School of Science and Technology.
    Chadalavada, Ravi
    Örebro University, School of Science and Technology.
    Ögren, Magnus
    Örebro University, School of Science and Technology.
    Creativity in the eye of the student: Refining investigations of mathematical creativity using eye-tracking goggles2016In: Proceedings of the 40th Conference of the International Group for the Psychology of Mathematics Education (PME) / [ed] C. Csíkos, A. Rausch, & J. Szitányi, 2016Conference paper (Refereed)
    Abstract [en]

    Mathematical creativity is increasingly important for improved innovation and problem-solving. In this paper, we address the question of how to best investigate mathematical creativity and critically discuss dichotomous creativity scoring schemes. In order to gain deeper insights into creative problem-solving processes, we suggest the use of mobile, unobtrusive eye-trackers for evaluating students’ creativity in the context of Multiple Solution Tasks (MSTs). We present first results with inexpensive eye-tracking goggles that reveal the added value of evaluating students’ eye movements when investigating mathematical creativity—compared to an analysis of written/drawn solutions as well as compared to an analysis of simple videos.

  • 6.
    Sørensen, Mads Peter
    et al.
    Department of Applied Mathematics and Computer Science, Technical University of Denmark, Kongens Lyngby, Denmark.
    Falsig Pedersen, Niels
    Department of Applied Mathematics and Computer Science, Technical University of Denmark, Kongens Lyngby, Denmark.
    Ögren, Magnus
    Örebro University, School of Science and Technology.
    The dynamics of magnetic vortices in type II superconductors with pinning sites studied by the time dependent Ginzburg–Landau model2017In: Physica. C, Superconductivity, ISSN 0921-4534, E-ISSN 1873-2143, Vol. 533, 40-43 p.Article in journal (Refereed)
    Abstract [en]

    We investigate the dynamics of magnetic vortices in type II superconductors with normal state pinning sites using the Ginzburg–Landau equations. Simulation results demonstrate hopping of vortices between pinning sites, influenced by external magnetic fields and external currents. The system is highly nonlinear and the vortices show complex nonlinear dynamical behaviour.

  • 7.
    Ögren, Magnus
    Örebro University, School of Science and Technology. Nano Science Center, Department of Chemistry, University of Copenhagen, Denmark.
    Local boundary conditions for NMR-relaxation in digitized porous media2014In: The European Physical Journal B, ISSN 1434-6028, Vol. 87, no 11, 255- p.Article in journal (Refereed)
    Abstract [en]

    We narrow the gap between simulations of nuclear magnetic resonance dynamics on digital domains (such as CT-images) and measurements in D-dimensional porous media. We point out with two basic domains, the ball and the cube in D dimensions, that due to a digital uncertainty in representing the real pore surfaces of dimension D − 1, there is a systematic error in simulated dynamics. We then reduce this error by introducing local Robin boundary conditions.

  • 8.
    Ögren, Magnus
    et al.
    Örebro University, School of Science and Technology.
    Abdullaev, Fatkhulla
    Physical-Technical Institute, Uzbekistan Academy of Sciences, Tashkent, Uzbekistan .
    Konotop, Vladimir
    Centro de Física Teórica e Computacional and Departamento de Física, Faculdade de Ciências, Universidade de Lisboa, Lisboa, Portugal .
    Solitons in a 𝒫𝒯-symmetric 𝜒(2) coupler2017In: Optics Letters, ISSN 0146-9592, E-ISSN 1539-4794, Vol. 42, no 20, 4079-4082 p.Article in journal (Refereed)
    Abstract [en]

    We consider the existence and stability of solitons in a 𝜒(2) coupler. Both the fundamental and second harmonics (SHs) undergo gain in one of the coupler cores and are absorbed in the other one. The gain and loss are balanced, creating a parity-time (𝒫𝒯) symmetric configuration. We present two types of families of 𝒫𝒯-symmetric solitons having equal and different profiles of the fundamental and SHs. It is shown that the gain and loss can stabilize solitons. The interaction of stable solitons is shown. In the cascading limit, the model is reduced to the 𝒫𝒯-symmetric coupler with effective Kerr-type nonlinearity and the balanced nonlinear gain and loss.

  • 9.
    Ögren, Magnus
    et al.
    Örebro University, School of Science and Technology. Department of Applied Mathematics and Computer Science, Technical University of Denmark, Kongens Lyngby, Denmark; Department of Chemistry, Nano-Science Center, University of Copenhagen, Copenhagen, Denmark.
    Nyström, Marcus
    Lund University Humanities Laboratory, Lund, Sweden.
    Jarodzka, Halszka
    Open University, Heerlen, The Netherlands.
    There’s more to the multimedia effect than meets the eye: is seeing pictures believing?2017In: Instructional science, ISSN 0020-4277, E-ISSN 1573-1952, Vol. 45, no 2, 263-287 p.Article in journal (Refereed)
    Abstract [en]

    Textbooks in applied mathematics often use graphs to explain the meaning of formulae, even though their benefit is still not fully explored. To test processes underlying this assumed multimedia effect we collected performance scores, eye movements, and think-aloud protocols from students solving problems in vector calculus with and without graphs. Results showed no overall multimedia effect, but instead an effect to confirm statements that were accompanied by graphs, irrespective of whether these statements were true or false. Eye movement and verbal data shed light on this surprising finding. Students looked proportionally less at the text and the problem statement when a graph was present. Moreover, they experienced more mental effort with the graph, as indicated by more silent pauses in thinking aloud. Hence, students actively processed the graphs. This, however, was not sufficient. Further analysis revealed that the more students looked at the statement, the better they performed. Thus, in the multimedia condition the graph drew students’ attention and cognitive capacities away from focusing on the statement. A good alternative strategy in the multimedia condition was to frequently look between graph and problem statement, and thus to integrate their information. In conclusion, graphs influence where students look and what they process, and may even mislead them into believing accompanying information. Thus, teachers and textbook designers should be very critical on when to use graphs and carefully consider how the graphs are integrated with other parts of the problem.

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