| 1. |
- Aichholzer, Oswin, et al.
(författare)
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Bicolored Order Types
- 2024
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Ingår i: Computing in Geometry and Topology. - 2750-7823. ; 3:2
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Tidskriftsartikel (refereegranskat)abstract
- In their seminal work on Multidimensional Sorting, Goodman and Pollack introduced the so-called order type, which for each ordered triple of a point set in the plane gives its orientation, clockwise or counterclockwise. This information is sufficient to solve many problems from discrete geometry where properties of point sets do not depend on the exact coordinates of the points but only on their relative positions. Goodman and Pollack showed that an efficient way to store an order type in a matrix λ of quadratic size (w.r.t. the number of points) is to count for every oriented line spanned by two points of the set how many of the remaining points lie to the left of this line. We generalize the concept of order types to bicolored point sets (every point has one of two colors). The bicolored order type contains the orientation of each bicolored triple of points, while no information is stored for monochromatic triples. Similar to the uncolored case, we store the number of blue points that are to the left of an oriented line spanned by two red points or by one red and one blue point in λB. Analogously the number of red points is stored in λR. As a main result, we show that the equivalence of the information contained in the orientation of all bicolored point triples and the two matrices λB and λR also holds in the colored case. This is remarkable, as in general the bicolored order type does not even contain sufficient information to determine all extreme points (points on the boundary of the convex hull of the point set).We then show that the information of a bicolored order type is sufficient to determine whether the two color classes can be linearly separated and how one color class can be sorted around a point of the other color class. Moreover, knowing the bicolored order type of a point set suffices to find bicolored plane perfect matchings or to compute the number of crossings of the complete bipartite graph drawn on a bicolored point set in quadratic time.
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| 2. |
- Bagheri, Alireza, et al.
(författare)
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Minsum m watchmen’s routes in Stiegl polygons
- 2023
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Ingår i: XX Spanish Meeting on Computational Geometry. ; , s. 41-44
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Konferensbidrag (refereegranskat)abstract
- We present an O(n2 · min{m, n}) time and O(n · min{m, n}) storage algorithm to compute the minsum set of m watchmen routes given their starting points in a Stiegl polygon ― a staircase polygon where the floor solely consists of one horizontal and one vertical edge ― with n vertices.
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| 3. |
- Oswin, Aichholzer, et al.
(författare)
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Two Equivalent Representations of Bicolored Order Types
- 2023
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Ingår i: 39th European Workshop on Computational Geometry. ; , s. 12-17
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Konferensbidrag (övrigt vetenskapligt/konstnärligt)abstract
- In their seminal work on Multidimensional Sorting, Goodman and Pollack introduced the so-called order type, which for each ordered triple of a point set in the plane gives its orientation, clockwise or counterclockwise. This information is sufficient to solve many problems from discrete geometry where properties of point sets do not depend on the exact coordinates of the points but only on their relative positions. Goodman and Pollack showed that an efficient way to store an order type in a matrix λ of quadratic size (w.r.t. the number of points) is to count for every oriented line spanned by two points of the set how many of the remaining points lie to the left of this line.We generalize the concept of order types to bicolored point sets (every point has one of two colors). The bicolored order type contains the orientation of each bicolored triple of points, while no information is stored for monochromatic triples. Similar to the uncolored case, we store the number of blue points that are to the left of an oriented line spanned by two red points or by one red and one blue point in λB. Analogously the number of red points is stored in λR.We show that the equivalence of the information contained in the orientation of all bicolored point triples and the two matrices λB and λR also holds in the colored case. This is remarkable, as in general the bicolored order type does not even contain sufficient information to determine all extreme points (points on the boundary of the convex hull of the point set).
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| 4. |
- Lindström, Torsten, 1965-, et al.
(författare)
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Destabilization, Stabilization, and Multiple Attractors in Saturated Mixotrophic Environments
- 2020
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Ingår i: SIAM Journal on Applied Mathematics. - United States : Society for Industrial and Applied Mathematics. - 0036-1399 .- 1095-712X. ; 80:6, s. 2338-2364
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Tidskriftsartikel (refereegranskat)abstract
- The ability of mixotrophs to combine phototrophy and phagotrophy is now well recognized and found to have important implications for ecosystem dynamics. In this paper, we examine the dynamical consequences of the invasion of mixotrophs in a system that is a limiting case of the chemostat. The model is a hybrid of a competition model describing the competition between autotroph and mixotroph populations for a limiting resource, and a predator--prey-type model describing the interaction between autotroph and herbivore populations. Our results show that mixotrophs are able to invade in both autotrophic environments and environments described by interactions between autotrophs and herbivores. The interaction between autotrophs and herbivores might be in equilibrium or cycle. We find that invading mixotrophs have the ability to both stabilize and destabilize autotroph-herbivore dynamics depending on the competitive ability of mixotrophs. The invasion of mixotrophs can also result in multiple attractors.
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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. |
- Ebbelind, Andreas, et al.
(författare)
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Learning fractions : transformations between representations from a social semiotic perspective of multimodality
- 2012
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Ingår i: Proceedings of Norma 11. - Reykjavík : University of Iceland Press. - 9789979549659 ; , s. 217-226
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Konferensbidrag (refereegranskat)abstract
- This study presents a tentative framework for studying the learning of fractions in the context of transformations between different forms of representations. The framework is used in an empirical sample of how eight 10-year-old students express understanding of activities which were developed to challenge them to reflect on different ways of representing aspects of the concept of fractions. The framework is based on a social semiotic perspective of multimodality.The analysis discloses how the framework helps in structuring our understanding of the interplay between representations in the learning of fractions. Specifically, we saw how concrete physical material and gestures complemented the symbolic and spoken language in the students’ solution strategies of different tasks.
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| 7. |
- Filtser, Omrit, et al.
(författare)
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Guarding Polyominoes Under k-Hop Visibility
- 2024
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Ingår i: LATIN 2024. - : Springer Publishing Company. - 9783031555978 - 9783031555985 ; , s. 288-302
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Konferensbidrag (refereegranskat)abstract
- We study the ART GALLERY Problem under k-hop visibility in polyominoes. In this visibility model, two unit squares of a polyomino can see each other if and only if the shortest path between the respective vertices in the dual graph of the polyomino has length at most k. In this paper, we show that the VC dimension of this problem is 3 in simple polyominoes, and 4 in polyominoes with holes. Furthermore, we provide a reduction from PLANAR MONOTONE 3SAT, thereby showing that the problem is NP-complete even in thin polyominoes (i.e., polyominoes that do not a contain a 2 x 2 block of cells). Complementarily, we present a linear-time 4-approximation algorithm for simple 2-thin polyominoes (which do not contain a 3 x 3 block of cells) for all k is an element of N.
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| 8. |
- Lingefjärd, Thomas, 1952, et al.
(författare)
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Matematik med mobiltelefoner
- 2009
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Ingår i: Nämnaren. - : NCM och Nämnaren. - 0348-2723. ; 36:4, s. 1-10
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Tidskriftsartikel (övrigt vetenskapligt/konstnärligt)
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| 9. |
- Sayers, Judy, et al.
(författare)
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Opportunities to learn foundational number sense in three Swedish year one textbooks : implications for the importation of overseas-authored materials
- 2021
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Ingår i: International Journal of Mathematical Education in Science and Technology. - : Taylor & Francis. - 0020-739X .- 1464-5211. ; 52:4, s. 506-526
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we present statistical analyses of three textbooks used by Swedish teachers to support year one children's learning of mathematics. One, Eldorado, is authored by Swedish teachers, another, Favorit, is a Swedish adaptation of a popular Finnish series and the third, Singma, is a Swedish adaptation of a Singapore series. Data were coded against the eight categories of foundational number sense, which are the number-related competences literature has shown to be essential for the later mathematical success of year one learners. Two analyses were undertaken; the first was a frequency analysis of the tasks coded for a particular category, the second was a time-series analysis highlighting the temporal location of such opportunities. The frequency analyses identified statistically significant differences with respect to children's opportunities to acquire foundational number sense. Additionally, the time series showed substantial differences in the ways in which such tasks were located in the structure of the textbooks. Such differences, we argue, offer substantial didactical challenges to teachers trying to adapt their practices to the expectations of such imports.
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| 10. |
- Sollervall, Håkan, et al.
(författare)
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Designing a didactical situation with mobile and web technologies
- 2015
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Ingår i: Proceedings of the Ninth Conference of the European Society for Research in Mathematics Education (cerme9). - : Charles Univ, Fac Educ. ; , s. 2410-2417
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Konferensbidrag (refereegranskat)abstract
- We have designed, implemented and evaluated a didactical situation where 27 students in grade 4 created large triangular constructions in an outdoor environment by involving a customised GPS-based mobile application. The students' strategies for construction were reflected upon during a teacher-led discussion involving web technologies and aiming at the formulation of principles for construction. The effective communication of underpinning theories and possible learning objectives, in combination with user-friendly mobile and web technologies, served to scaffold the teacher's successful orchestration of a logos-oriented mathematical discourse.
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