| 11. |
- Helenius, Ola, 1970, et al.
(författare)
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Measuring temperature within the didactic space of preschool
- 2016
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Ingår i: Nordisk matematikkdidaktikk, NOMAD. - : Nationellt centrum för matematikutbildning (NCM). - 1104-2176. ; 21:4, s. 155-176
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Tidskriftsartikel (refereegranskat)abstract
- The informal character of preschool mathematics, engaged in during children’s play, places complex requirements on preschool teachers. It also leads to challenges in developing appropriate analytical tools for researching teacher work. In this paper a framework, the ”didaktic space”, is described and used to analyse interactions between preschool teachers and children in relationship to mathematical learning situations. An interaction between a preschool teacher and a group of children about how to compare their temperatures is analysed, using this framework. The analysis focuses on how the teacher’s contributions as well as those of the children changed as the role of the mathematics changed. The paper discusses how the didaktic space offers a nuanced understanding of preschool mathematical situations, both to researchers and to teachers.
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| 12. |
- Sorooshian, Shahryar, 1980, et al.
(författare)
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Expedited analytical hierarchical process for multicriteria decision making
- 2022
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Ingår i: ICIC Express Letters. - 1881-803X. ; 16:2, s. 145-151
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Tidskriftsartikel (refereegranskat)abstract
- In this study, a procedure to overcome analytical hierarchical process (AHP) limitations was developed and tested, and a tool created via pairwise comparison chart and weighted sum method hybridization was introduced. In addition, a multicriteria decision-making software was selected to validate the constructed hybrid decision-making tool. The software was formulated based on an established decision-making technique. Data for the comparison analysis were obtained from an ongoing research, and output from the comparison analysis provided evidence for the accuracy of the constructed tool. The proposed hybridized method aligned with AHP reasoning but demonstrated a faster process; hence, it was called ‘expedited AHP’.
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| 13. |
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| 14. |
- Brandes, Julia, et al.
(författare)
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On generating functions in additive number theory, II: lower-order terms and applications to PDEs
- 2021
-
Ingår i: Mathematische Annalen. - : Springer Science and Business Media LLC. - 0025-5831 .- 1432-1807. ; 379, s. 347-76
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Tidskriftsartikel (refereegranskat)abstract
- We obtain asymptotics for sums of the form Sigma(p)(n=1) e(alpha(k) n(k) + alpha(1)n), involving lower order main terms. As an application, we show that for almost all alpha(2) is an element of [0, 1) one has sup(alpha 1 is an element of[0,1)) | Sigma(1 <= n <= P) e(alpha(1)(n(3) + n) + alpha(2)n(3))| << P3/4+epsilon, and that in a suitable sense this is best possible. This allows us to improve bounds for the fractal dimension of solutions to the Schrodinger and Airy equations.
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| 15. |
- Berman, Robert, 1976
(författare)
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Convergence Rates for Discretized Monge-Ampere Equations and Quantitative Stability of Optimal Transport
- 2021
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Ingår i: Foundations of Computational Mathematics. - : Springer Science and Business Media LLC. - 1615-3375 .- 1615-3383. ; 21, s. 1099-1140
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Tidskriftsartikel (refereegranskat)abstract
- In recent works-both experimental and theoretical-it has been shown how to use computational geometry to efficiently construct approximations to the optimal transport map between two given probability measures on Euclidean space, by discretizing one of the measures. Here we provide a quantitative convergence analysis for the solutions of the corresponding discretized Monge-Ampere equations. This yields H-1-converge rates, in terms of the corresponding spatial resolution h, of the discrete approximations of the optimal transport map, when the source measure is discretized and the target measure has bounded convex support. Periodic variants of the results are also established. The proofs are based on new quantitative stability results for optimal transport maps, shown using complex geometry.
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| 16. |
- 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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| 17. |
- Bandara, Lashi, et al.
(författare)
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Riesz continuity of the Atiyah–Singer Dirac operator under perturbations of local boundary conditions
- 2019
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Ingår i: Communications in Partial Differential Equations. - : Informa UK Limited. - 0360-5302 .- 1532-4133. ; 44:12, s. 1253-1284
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Tidskriftsartikel (refereegranskat)abstract
- On a smooth complete Riemannian spin manifold with smooth compact boundary, we demonstrate that Atiyah–Singer Dirac operator /DB in L2 depends Riesz continuously on L∞ perturbations of local boundary conditions B. The Lipschitz bound for the map B→/DB(1+/D2B)−12 depends on Lipschitz smoothness and ellipticity of B and bounds on Ricci curvature and its first derivatives as well as a lower bound on injectivity radius away from a compact neighbourhood of the boundary. More generally, we prove perturbation estimates for functional calculi of elliptic operators on manifolds with local boundary conditions.
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| 18. |
- 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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| 19. |
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Pseudo-Differential Operators and Generalized Functions
- 2015. - 1
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Samlingsverk (redaktörskap) (refereegranskat)abstract
- At the ninth congress of the International Society for Analysis Applications and Computations (ISAAC), ISAAC Group in Pseudo-Differential Operators (IGPDO) and the ISAAC Group in Generalized Functions (IGGF) agreed to continue with the publication of a joint volume, as for the eight ISAAC Congress in Moscow 2013, of selecting papers from their two special sessions. Generalized functions as a general framework for almost all fields in analysis, and pseudo-differential operators as a basis of microlocal analysis in combination with harmonic and complex analysis with many applications, fit well and offer rich synergies for the further development of analysis in general.Moreover, the participants of both sessions agreed to dedicate this volume to Professor Michael Oberguggenberger at Insbruck university, Austria on his 60th birthday. Professor Oberguggenberger is one of the founder of the algebraic approach to generalized function theory with many contributions to the qualitative analysis of partial differential equations and a leader of the International Asso- ciation for Generalized Functions based in Vienna. Professor Oberguggenberger is highly appreciated as a scientist and a strong expert on generalized functions, especially Columbeau algebras. He is also very appreciated as a modest and encouraging person who supervised several PhD student to their examination. It is a pleasure for us to dedicate the volume to him.This joint volume is titledPseudo-Differential Operators and Generalized Functionsand consists of invited papers, mainly based on the scientific activities of the groups IGPDO and IGGF at the ninth ISAAC congress in Krakow, Poland, during August 2013. The volume is intended to be an independent sequel to the volumes “Advances in Pseudo-Differential Operators”, “Pseudo-Differential Operators and Related Topics”, “Modern Trends in Pseudo-Differential Operators”, “New Developments in Pseudo-Differential Operators”, “Pseudo-Differential Operators: Complex Analysis and Partial Differential Equations” and “Pseudo-Differential Operators, Generalized Functions and Asymptotics”. These volumes were based on, respectively, the 4th ISAAC congress in Toronto in 2003, conference in V ̈axjo ̈ 2004, 5th ISAAC congress in Catana in 2005, 6th ISAAC congress in Ankara in 2007, workshop in Toronto in 2008, 7th ISAAC congress in London in 2009, and 8th ISAAC congress in Moscow ISAAC in 2011.The volume consists of 19 peer-reviewed contributions representing modern trends in the theory of generalized functions and pseudo-differential operators. Topics include algebras of generalized functions, ultra-distributions, partial differential equations, micro-local analysis, harmonic analysis, global analysis, geometry, quantization, mathematical physics, and time-frequency analysis. Variety of applications especially in the framework of manifolds with singular metrics and general relativity, microlocal analysis and the analysis of equations with singularities will be interested for a wide audience including graduate students and researchers in partial differential equations, mathematical physics, various fields of analysis, stochastic analysis and geometry.The papers can be sorted roughly into two groups. The first group of papers is related to generalized functions and deals with various problems of equations with singular coefficients and data within algebras of generalized functions where the classical method of regularizations is well established. Moreover in this setting, local analysis is well adapted and analyzed towards Höder type spaces and in the direction of generalized manifolds and applications in general relativity.In the second group of papers, various kinds of Fourier analysis are more present, involving micro-local analysis, harmonic analysis, theory of ultra-distributions, time-frequency analysis, etc. For example, Wiener type Tauberian theorems related to generalized integral transforms stochastic equations are adapted to classical distribution theory. Ultradistribution spaces are analyzed in connection with global type operators and wave fronts. Gabor analysis via modulation spaces or Hermite expansions is developed for various Gelfand-Shilov classes. Time-frequency methods are applied on evolution operators and on random MIMO systems.
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| 20. |
- Önskog, Thomas, 1979-, et al.
(författare)
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Pricing and hedging of financial derivatives using a posteriori error estimates and adaptive methods for stochastic differential equations
- 2010
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Ingår i: Journal of Computational and Applied Mathematics. - : Elsevier. - 0377-0427 .- 1879-1778. ; 235, s. 563-592
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Tidskriftsartikel (refereegranskat)abstract
- The efficient and accurate calculation of sensitivities of the price of financial derivatives with respect to perturbations of the parameters in the underlying model, the so-called `Greeks', remains a great practical challenge in the derivative industry. This is true regardless of whether methods for partial differential equations or stochastic differential equations (Monte Carlo techniques) are being used. The computation of the `Greeks' is essential to risk management and to the hedging of financial derivatives and typically requires substantially more computing time as compared to simply pricing the derivatives. Any numerical algorithm (Monte Carlo algorithm) for stochastic differential equations produces a time-discretization error and a statistical error in the process of pricing financial derivatives and calculating the associated `Greeks'. In this article we show how a posteriori error estimates and adaptive methods for stochastic differential equations can be used to control both these errors in the context of pricing and hedging of financial derivatives. In particular, we derive expansions, with leading order terms which are computable in a posteriori form, of the time-discretization errors for the price and the associated `Greeks'. These expansions allow the user to simultaneously first control the time-discretization errors in an adaptive fashion, when calculating the price, sensitivities and hedging parameters with respect to a large number of parameters, and then subsequently to ensure that the total errors are, with prescribed probability, within tolerance.
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