| 1. |
- Gerlee, Philip, 1980, et al.
(författare)
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Scientific Models : Red Atoms, White Lies and Black Boxes in a Yellow Book
- 2016
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Bok (övrigt vetenskapligt/konstnärligt)abstract
- A zebrafish, the hull of a miniature ship, a mathematical equation and a food chain - what do these things have in common? They are examples of models used by scientists to isolate and study particular aspects of the world around us. This book begins by introducing the concept of a scientific model from an intuitive perspective, drawing parallels to mental models and artistic representations. It then recounts the history of modelling from the 16th century up until the present day. The iterative process of model building is described and discussed in the context of complex models with high predictive accuracy versus simpler models that provide more of a conceptual understanding. To illustrate the diversity of opinions within the scientific community, we also present the results of an interview study, in which ten scientists from different disciplines describe their views on modelling and how models feature in their work. Lastly, it includes a number of worked examples that span different modelling approaches and techniques. It provides a comprehensive introduction to scientific models and shows how models are constructed and used in modern science. It also addresses the approach to, and the culture surrounding modelling in different scientific disciplines. It serves as an inspiration for model building and also facilitates interdisciplinary collaborations by showing how models are used in different scientific fields. The book is aimed primarily at students in the sciences and engineering, as well as students at teacher training colleges but will also appeal to interested readers wanting to get an overview of scientific modelling in general and different modelling approaches in particular.
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| 2. |
- Sorooshian, Shahryar, 1980, et al.
(författare)
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Performance of the decision-making trial and evaluation laboratory
- 2023
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Ingår i: AIMS Mathematics. - : American Institute of Mathematical Sciences (AIMS). - 2473-6988. ; 8:3, s. 7490-7514
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Tidskriftsartikel (refereegranskat)abstract
- Multiple attribute decision-making (MADM) techniques constitute a practical approach for solving complex problems involving multiple and often conflicting criteria. Decision-making trial and evaluation laboratory (DEMATEL) is a popular MADM technique with both admirers and critics. This study presents a comprehensive review of DEMATEL through bibliometric analysis using the Scopus database. This article examined 3,521 papers published in journals, conferences or books between 1981 and 2023. We examined a few parameters for commenting on the performance of the technique. Among them are research outputs, the network of DEMATEL users, implementation subject areas, research zones, financing opportunities and publication hosts and their impact trends. We conclude from the findings of this study that the DEMATEL is capable of dealing with modern problem-solving in future environments. Although the growth of new MADMs is obvious, based on the gathered data, we forecast that more than 776 documents will be published in 2025 using DEMATEL for problem-solving. This expanding tendency will continue in the future. As distinct MADMs have diverse constraints, foundations, computing complexity and standpoints, which result in different performances, outmoded low-performance MADM techniques must be reported by researchers to continue this paper’s objective to minimize ambiguity among decision-makers and practitioners. To facilitate such a comparison in the future, a quantitative performance coefficient was also developed here.
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| 3. |
- Wang, Siyang, et al.
(författare)
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CONVERGENCE OF FINITE DIFFERENCE METHODS FOR THE WAVE EQUATION IN TWO SPACE DIMENSIONS
- 2018
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Ingår i: Mathematics of Computation. - : American Mathematical Society (AMS). - 0025-5718 .- 1088-6842. ; 87:314, s. 2737-2763
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Tidskriftsartikel (refereegranskat)abstract
- When using a finite difference method to solve an initial-boundary-value problem, the truncation error is often of lower order at a few grid points near boundaries than in the interior. Normal mode analysis is a powerful tool to analyze the effect of the large truncation error near boundaries on the overall convergence rate, and has been used in many research works for different equations. However, existing work only concerns problems in one space dimension. In this paper, we extend the analysis to problems in two space dimensions. The two dimensional analysis is based on a diagonalization procedure that decomposes a two dimensional problem to many one dimensional problems of the same type. We present a general framework of analyzing convergence for such one dimensional problems, and explain how to obtain the result for the corresponding two dimensional problem. In particular, we consider two kinds of truncation errors in two space dimensions: the truncation error along an entire boundary, and the truncation error localized at a few grid points close to a corner of the computational domain. The accuracy analysis is in a general framework, here applied to the second order wave equation. Numerical experiments corroborate our accuracy analysis.
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| 4. |
- Asadzadeh, Mohammad, 1952, et al.
(författare)
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ON HP-STREAMLINE DIFFUSION AND NITSCHE SCHEMES FOR THE RELATIVISTIC VLASOV-MAXWELL SYSTEM
- 2019
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Ingår i: Kinetic and Related Models. - : American Institute of Mathematical Sciences (AIMS). - 1937-5093 .- 1937-5077. ; 12:1, s. 105-131
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Tidskriftsartikel (refereegranskat)abstract
- We study stability and convergence of hp-streamline diffusion (SD) finite element, and Nitsche's schemes for the three dimensional, relativistic (3 spatial dimension and 3 velocities), time dependent Vlasov-Maxwell system and Maxwell's equations, respectively. For the hp scheme for the Vlasov-Maxwell system, assuming that the exact solution is in the Sobolev space HS+1(Omega), we derive global a priori error bound of order O(h/p)(s+1/2), where h(= max(K) h(K)) is the mesh parameter and p(= max(K) p(K)) is the spectral order. This estimate is based on the local version with h(K) = diam K being the diameter of the phase-space-time element K and pR-is the spectral order (the degree of approximating finite element polynomial) for K. As for the Nitsche's scheme, by a simple calculus of the field equations, first we convert the Maxwell's system to an elliptic type equation. Then, combining the Nitsche's method for the spatial discretization with a second order time scheme, we obtain optimal convergence of O(h(2) +k(2)), where h is the spatial mesh size and k is the time step. Here, as in the classical literature, the second order time scheme requires higher order regularity assumptions. Numerical justification of the results, in lower dimensions, is presented and is also the subject of a forthcoming computational work [22].
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| 5. |
- 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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| 6. |
- 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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| 7. |
- 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
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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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| 8. |
- 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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| 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. |
- 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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