| 1. |
- Pejlare, Johanna, 1976
(författare)
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On Axioms and Images in the History of Mathematics
- 2007
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This dissertation deals with aspects of axiomatization, intuition and visualization in the history of mathematics. Particular focus is put on the end of the 19th century, before David Hilbert's (1862–1943) work on the axiomatization of Euclidean geometry. The thesis consists of three papers. In the first paper the Swedish mathematician Torsten Brodén (1857–1931) and his work on the foundations of Euclidean geometry from 1890 and 1912, is studied. A thorough analysis of his foundational work is made as well as an investigation into his general view on science and mathematics. Furthermore, his thoughts on geometry and its nature and what consequences his view has for how he proceeds in developing the axiomatic system, is studied. In the second paper different aspects of visualizations in mathematics are investigated. In particular, it is argued that the meaning of a visualization is not revealed by the visualization and that a visualization can be problematic to a person if this person, due to a limited knowledge or limited experience, has a simplified view of what the picture represents. A historical study considers the discussion on the role of intuition in mathematics which followed in the wake of Karl Weierstrass' (1815–1897) construction of a nowhere differentiable function in 1872. In the third paper certain aspects of the thinking of the two scientists Felix Klein (1849–1925) and Heinrich Hertz (1857–1894) are studied. It is investigated how Klein and Hertz related to the idea of naïve images and visual thinking shortly before the development of modern axiomatics. Klein in several of his writings emphasized his belief that intuition plays an important part in mathematics. Hertz argued that we form images in our mind when we experience the world, but these images may contain elements that do not exist in nature.
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| 2. |
- Pejlare, Johanna, 1976, et al.
(författare)
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On the relations between geometry and algebra in Gestrinius' edition of Euclid's Elements
- 2016
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Ingår i: Radford, L., Furinghetti, F., & Hausberger, T. (Eds.) (2016). Proceedings of the 2016 ICME Satellite Meeting of the International Study Group on the Relations Between the History and Pedagogy of Mathematics. Montpellier, France: IREM de Montpellier.. - 2909916510 ; , s. 513-523
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Konferensbidrag (refereegranskat)abstract
- In 1637 the Swedish mathematician Martinus Erici Gestrinius contributed with a commented edition of Euclid’s Elements. In this article we analyse the relationship between geometry and algebra in Gestrinius’ Elements, as presented in Book II. Of particular interest are Propositions 4, 5, and 6, dealing with straight lines cut into equal and unequal parts, and the three kinds of quadratic equations Gestrinius associates with them. We argue that Gestrinius followed Clavius translation of the Elements, but was influenced by Ramus to include algebra.
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| 3. |
- Bandara, L., et al.
(författare)
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Eigenvalue asymptotics for weighted Laplace equations on rough Riemannian manifolds with boundary
- 2021
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Ingår i: Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze. - : Scuola Normale Superiore - Edizioni della Normale. - 0391-173X .- 2036-2145. ; 22:4, s. 1843-1878
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Tidskriftsartikel (refereegranskat)abstract
- Our topological setting is a smooth compact manifold of dimension two or higher with smooth boundary. Although this underlying topological structure is smooth, the Riemannian metric tensor is only assumed to be bounded and measurable. This is known as a rough Riemannian manifold. For a large class of boundary conditions we demonstrate a Weyl law for the asymptotics of the eigenvalues of the Laplacian associated to a rough metric. Moreover, we obtain eigenvalue asymptotics for weighted Laplace equations associated to a rough metric. Of particular novelty is that the weight function is not assumed to be of fixed sign, and thus the eigenvalues may be both positive and negative. Key ingredients in the proofs were demonstrated by Birman and Solomjak nearly fifty years ago in their seminal work on eigenvalue asymptotics. In addition to determining the eigenvalue asymptotics in the rough Riemannian manifold setting for weighted Laplace equations, we also wish to promote their achievements which may have further applications to modern problems.
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| 4. |
- Bauer, M., et al.
(författare)
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Semi-invariant Riemannian metrics in hydrodynamics
- 2020
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Ingår i: Calculus of Variations and Partial Differential Equations. - : Springer Science and Business Media LLC. - 0944-2669 .- 1432-0835. ; 59:2
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Tidskriftsartikel (refereegranskat)abstract
- Many models in mathematical physics are given as non-linear partial differential equation of hydrodynamic type; the incompressible Euler, KdV, and Camassa-Holm equations are well-studied examples. A beautiful approach to well-posedness is to go from the Eulerian to a Lagrangian description. Geometrically it corresponds to a geodesic initial value problem on the infinite-dimensional group of diffeomorphisms with a right invariant Riemannian metric. By establishing regularity properties of the Riemannian spray one can then obtain local, and sometimes global, existence and uniqueness results. There are, however, many hydrodynamic-type equations, notably shallow water models and compressible Euler equations, where the underlying infinite-dimensional Riemannian structure is not fully right invariant, but still semi-invariant with respect to the subgroup of volume preserving diffeomorphisms. Here we study such metrics. For semi-invariant metrics of Sobolev Hk-type we give local and some global well-posedness results for the geodesic initial value problem. We also give results in the presence of a potential functional (corresponding to the fluid's internal energy). Our study reveals many pitfalls in going from fully right invariant to semi-invariant Sobolev metrics; the regularity requirements, for example, are higher. Nevertheless the key results, such as no loss or gain in regularity along geodesics, can be adopted.
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| 5. |
- Lindahl, Karl-Olof, 1975-
(författare)
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Applied Algebraic Dynamics
- 2010
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Ingår i: P-Adic Numbers, Ultrametric Analysis, and Applications. - : Pleiades Publishing, Ltd.. - 2070-0466 .- 2070-0474. ; 2:4, s. 360-362
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Tidskriftsartikel (refereegranskat)
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| 6. |
- Yun, Yixiao, 1987, et al.
(författare)
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Maximum-Likelihood Object Tracking from Multi-View Video by Combining Homography and Epipolar Constraints
- 2012
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Ingår i: 6th ACM/IEEE Int'l Conf on Distributed Smart Cameras (ICDSC 12), Oct 30 - Nov.2, 2012, Hong Kong. - 9781450317726 ; , s. 6 pages-
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Konferensbidrag (refereegranskat)abstract
- This paper addresses problem of object tracking in occlusion scenarios, where multiple uncalibrated cameras with overlapping fields of view are used. We propose a novel method where tracking is first done independently for each view and then tracking results are mapped between each pair of views to improve the tracking in individual views, under the assumptions that objects are not occluded in all views and move uprightly on a planar ground which may induce a homography relation between each pair of views. The tracking results are mapped by jointly exploiting the geometric constraints of homography, epipolar and vertical vanishing point. Main contributions of this paper include: (a) formulate a reference model of multi-view object appearance using region covariance for each view; (b) define a likelihood measure based on geodesics on a Riemannian manifold that is consistent with the destination view by mapping both the estimated positions and appearances of tracked object from other views; (c) locate object in each individual view based on maximum likelihood criterion from multi-view estimations of object position. Experiments have been conducted on videos from multiple uncalibrated cameras, where targets experience long-term partial or full occlusions. Comparison with two existing methods and performance evaluations are also made. Test results have shown effectiveness of the proposed method in terms of robustness against tracking drifts caused by occlusions.
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| 7. |
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Discrete Geometry and Mathematical Morphology : First International Joint Conference, DGMM 2021, Uppsala, Sweden, May 24–27, 2021, Proceedings
- 2021
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Samlingsverk (redaktörskap) (refereegranskat)abstract
- This book constitutes the proceedings of the First IAPR International Conference on Discrete Geometry and Mathematical Morphology, DGMM 2021, which was held during May 24-27, 2021, in Uppsala, Sweden.The conference was created by joining the International Conference on Discrete Geometry for computer Imagery, DGCI, with the International Symposium on Mathematical Morphology, ISMM.The 36 papers included in this volume were carefully reviewed and selected from 59 submissions. They were organized in topical sections as follows: applications in image processing, computer vision, and pattern recognition; discrete and combinatorial topology; discrete geometry - models, transforms, visualization; discrete tomography and inverse problems; hierarchical and graph-based models, analysis and segmentation; learning-based approaches to mathematical morphology; multivariate and PDE-based mathematical morphology, morphological filtering.The book also contains 3 invited keynote papers.
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| 8. |
- Ross, J., et al.
(författare)
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ON THE MAXIMAL RANK PROBLEM FOR THE COMPLEX HOMOGENEOUS MONGE-AMPERE EQUATION
- 2019
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Ingår i: Analysis & Pde. - : Mathematical Sciences Publishers. - 1948-206X .- 2157-5045. ; 12:2, s. 493-504
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Tidskriftsartikel (refereegranskat)abstract
- We give examples of regular boundary data for the Dirichlet problem for the complex homogeneous Monge-Ampere equation over the unit disc, whose solution is completely degenerate on a nonempty open set and thus fails to have maximal rank.
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| 9. |
- Gerken, Jan, 1991, et al.
(författare)
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Equivariance versus augmentation for spherical images
- 2022
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Ingår i: Proceedings of Machine Learning Resaerch. ; , s. 7404-7421
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Konferensbidrag (refereegranskat)abstract
- We analyze the role of rotational equivariance in convolutional neural networks (CNNs) applied to spherical images. We compare the performance of the group equivariant networks known as S2CNNs and standard non-equivariant CNNs trained with an increasing amount of data augmentation. The chosen architectures can be considered baseline references for the respective design paradigms. Our models are trained and evaluated on single or multiple items from the MNIST- or FashionMNIST dataset projected onto the sphere. For the task of image classification, which is inherently rotationally invariant, we find that by considerably increasing the amount of data augmentation and the size of the networks, it is possible for the standard CNNs to reach at least the same performance as the equivariant network. In contrast, for the inherently equivariant task of semantic segmentation, the non-equivariant networks are consistently outperformed by the equivariant networks with significantly fewer parameters. We also analyze and compare the inference latency and training times of the different networks, enabling detailed tradeoff considerations between equivariant architectures and data augmentation for practical problems.
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| 10. |
- Gimperlein, Heiko, et al.
(författare)
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Nonclassical Spectral Asymptotics and Dixmier Traces: from Circles to Contact Manifolds
- 2017
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Ingår i: Forum of Mathematics, Sigma. - : Cambridge University Press (CUP). - 2050-5094. ; 5, s. Article no e3 -
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Tidskriftsartikel (refereegranskat)abstract
- We consider the spectral behavior and noncommutative geometry of commutators [P, f], where P is an operator of order 0 with geometric origin and f a multiplication operator by a function. When f is Holder continuous, the spectral asymptotics is governed by singularities. We study precise spectral asymptotics through the computation of Dixmier traces; such computations have only been considered in less singular settings. Even though a Weyl law fails for these operators, and no pseudodifferential calculus is available, variations of Connes' residue trace theorem and related integral formulas continue to hold. On the circle, a large class of nonmeasurable Hankel operators is obtained from Holder continuous functions f, displaying a wide range of nonclassical spectral asymptotics beyond the Weyl law. The results extend from Riemannian manifolds to contact manifolds and noncommutative tori.
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