| 1. |
- Johansson, Anders, et al.
(author)
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Countable state shifts and uniqueness of g-measures
- 2007
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In: American Journal of Mathematics. - : Project Muse. - 0002-9327 .- 1080-6377. ; 129:6, s. 1501-1511
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Journal article (peer-reviewed)abstract
- In this paper we present a new approach to studying g-measures which is based upon local absolute continuity. We extend an earlier result that square summability of variations of g ensures uniqueness of g-measures. The first extension is to the case of countably many symbols. The second extension is to some cases where g >= 0, relaxing the earlier requirement that inf g > 0.
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| 2. |
- Johansson, Anders, 1960-, et al.
(author)
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Multifractal analysis of non-uniformly hyperbolic systems
- 2010
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In: Israel Journal of Mathematics. - : Springer. - 0021-2172 .- 1565-8511. ; 177:1, s. 125-144
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Journal article (peer-reviewed)abstract
- We prove a multifractal formalismfor Birkhoff averages of continuous functions in the case of some non-uniformly hyperbolic maps, which includes interval examples such as the Manneville-Pomeau map.
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| 3. |
- Johansson, Anders, 1960-, et al.
(author)
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Phase transitions in long-range Ising models and an optimal condition for factors of g-measures
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Other publication (other academic/artistic)abstract
- We weaken the assumption of summable variations in a paper by Verbitskiy \cite{verb} to a weaker condition, Berbee's condition, in order for a 1-block factor (a single site renormalisation) of the full shift space on finitely many symbols to have a $g$-measure with a continuous $g$-function. But we also prove by means of a counterexample, that this condition is (within constants) optimal. The counterexample is based on the second of our main results, where we prove that there is an inverse critical temperature in a one-sided long-range Ising model which is at most 8 times the critical inverse temperature for the (two-sided) Ising model with long-range interactions.
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| 4. |
- Johansson, Anders, 1960-, et al.
(author)
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Phase transitions in long-range Ising models and an optimal condition for factors of g-measures
- 2019
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In: Ergodic Theory and Dynamical Systems. - : Cambridge University Press. - 0143-3857 .- 1469-4417. ; 39:5, s. 1317-1330
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Journal article (peer-reviewed)abstract
- We weaken the assumption of summable variations in a paper by Verbitskiy [On factors of g-measures. Indag. Math. (N.S.) 22 (2011), 315-329] to a weaker condition, Berbee's condition, in order for a one-block factor (a single-site renormalization) of the full shift space on finitely many symbols to have a g-measure with a continuous g-function. But we also prove by means of a counterexample that this condition is (within constants) optimal. The counterexample is based on the second of our main results, where we prove that there is a critical inverse temperature in a one-sided long-range Ising model which is at most eight times the critical inverse temperature for the (two-sided) Ising model with long-range interactions.
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| 5. |
- Johansson, Anders, et al.
(author)
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Square Summability of Variations and Convergence of the Transfer Operator
- 2008
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In: Ergodic Theory and Dynamical Systems. - 0143-3857 .- 1469-4417. ; 28:4, s. 1145-1151
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Journal article (peer-reviewed)abstract
- In this paper we study the one-sided shift operator on a state space defined by a finite alphabet. Using a scheme developed by Walters [P. Walters. Trans. Amer Math. Soc. 353(l) (2001), 327-347], we prove that the sequence of iterates of the transfer operator converges under square summability of variations of the g-function, a condition which gave uniqueness of a g-measure in our earlier work [A. Johansson and A. Oberg. Math. Res. Lett. 10(5-6) (2003), 587-601]. We also prove uniqueness of the so-called G-measures, introduced by Brown and Dooley [G. Brown and A. H. Dooley. Ergod. Th. & Dynam. Sys. 11 (1991), 279-307], under square summability of variations.
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| 6. |
- Johansson, Anders, 1960-, et al.
(author)
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Unique Bernoulli g-measures
- 2012
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In: Journal of the European Mathematical Society (Print). - : European Mathematical Society Publishing House. - 1435-9855 .- 1435-9863. ; 14:5, s. 1599-1615
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Journal article (peer-reviewed)abstract
- We improve and subsume the conditions of Johansson and O¨ berg [18] and Berbee [2]for uniqueness of a g-measure, i.e., a stationary distribution for chains with complete connections.In addition, we prove that these unique g-measures have Bernoulli natural extensions. In particular,we obtain a unique g-measure that has the Bernoulli property for the full shift on finitely manystates under any one of the following additional assumptions.(1)P1n=1(varn log g)2 < 1,(2) For any fixed ✏ > 0,P1n=1 e−(1/2+✏)(var1 log g+···+varn log g) = 1,(3) varn log g = o(1/pn) as n!1.That the measure is Bernoulli in the case of (1) is new. In (2) we have an improved version ofBerbee’s [2] condition (concerning uniqueness and Bernoullicity), allowing the variations of log gto be essentially twice as large. Finally, (3) is an example that our main result is new both foruniqueness and for the Bernoulli property.We also conclude that we have convergence in the Wasserstein metric of the iterates of theadjoint transfer operator to the g-measure.
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| 7. |
- Fransson, Göran, 1968-, et al.
(author)
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On the Swedish National Grade School for Digital Technologies in Education – GRADE : Expectations and experiences of doctorial students and supervisors
- 2018
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In: ICERI2018 Proceedings. - Sevilla : IATED. - 9788409059485 ; , s. 769-774
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Conference paper (peer-reviewed)abstract
- The Swedish National Graduate School for Digital Technologies in Education (GRADE) is a cooperative venture between six Swedish universities established during 2018. Within the field of educational sciences and in the area of digital technologies in education, GRADE aims to strengthen the expertise in the area and to increase national and international cooperation in research training activities.Over a number of years, and from multitude of sources (cf. Brown & Davis, 2004; Fisher, Higgins & Loveless, 2006; Kafai & Resnick 1996), research has stressed that increased digitalization in schools leads to a complexity that needs to be taken into account on different levels, from different perspectives and with different designs, methodologies and theoretical perspectives (cf. Olofsson, Lindberg, Fransson & Hauge, 2015; Price, Jewitt & Brown, 2013; Tondeur, Valcke & van Braak, 2008). At a micro-level, the learning situation of students, teachers and school-leaders changes and it becomes important to deepen the knowledge about the impact digital technologies has on the fundamental conditions for teaching and learning of different school subjects (cf. Chun, Kern & Smith 2016; Leung & Baccaglini-Frank, 2017). On a macro-level, conditions for education as such changes and digital technologies becomes an important object of study as agents of change (Wong & Li, 2008). The digitalization of K-12 schools has long been highlighted in policy as a necessity (cf. Kirkman et al, 2002; OECD, 2010). However, research and evaluations (cf. Fransson et al, 2012; OECD, 2015; Wastiau et al, 2013) show that many substantial challenges remain. One of the fundamental pillars of GRADE is the interdisciplinary approach. Several disciplines are present (Applied IT, Curriculum studies, Education, Informatics, Technology and Learning, Educational work, Work-interated Learning) in researching digital technologies in K-12 schools with the ambition to contribute to the continued implementation, integration and use of digital technologies in Swedish K-12 schools that stems from the evidence-based knowledge produced within the activities of GRADE. The research within GRADE will be characterized by close cooperation with stakeholders from school practice, with the aim to contribute to concrete school development. In GRADE, a multi-level approach that involves multiple layers or levels of school activities will be encouraged. When possible, studies will be longitudinal. This will imply studies from an organizational and management perspective, e.g. studies of school leaders and other members of senior management positions responsible for digital technology use and implementation. Also implied are studies of teachers' teaching practices and didactical considerations, as well as studies of the students in classrooms and their learning using digital technologies. This will also imply that several issues with a bearing on the digitalization of education, for example regarding school policy, teaching, learning, assessment and professional development will be researched from different perspectives and with different methodological approaches. In this paper, these points of departure will be explored based on the expectations and experiences of the first twelve admitted doctoral students and their supervisors.
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| 8. |
- Hrastinski, Stefan, 1980-, et al.
(author)
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Critical Imaginaries and Reflections on Artificial Intelligence and Robots in Postdigital K-12 Education
- 2019
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In: Postdigital Science and Education. - : Springer Science and Business Media LLC. - 2524-485X .- 2524-4868 .- 2662-5326. ; 1:2, s. 427-445
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Journal article (peer-reviewed)abstract
- It is commonly suggested that emerging technologies will revolutionize education. In this paper, two such emerging technologies, artificial intelligence (AI) and educational robots (ER), are in focus. The aim of the paper is to explore how teachers, researchers and pedagogical developers critically imagine and reflect upon how AI and robots could be used in education. The empirical data were collected from discussion groups that were part of a symposium. For both AI and ERs, the need for more knowledge about these technologies, how they could preferably be used, and how the emergence of these technologies might affect the role of the teacher and the relationship between teachers and students, were outlined. Many participants saw more potential to use AI for individualization as compared with ERs. However, there were also more concerns, such as ethical issues and economic interests, when discussing AI. While the researchers/developers to a greater extent imagined ideal future technology-rich educational practices, the practitioners were more focused on imaginaries grounded in current practice.
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| 9. |
- Janson, Svante, 1955-, et al.
(author)
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A piecewise contractive dynamical system and Phragmèn's election method
- 2019
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In: Bulletin de la Société Mathématique de France. - : FRENCH MATHEMATICAL SOC. - 0037-9484 .- 2102-622X. ; 147:3, s. 395-441
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Journal article (peer-reviewed)abstract
- We prove some basic results for a dynamical system given by a piece-wise linear and contractive map on the unit interval that takes two possible values at a point of discontinuity. We prove that there exists a universal limit cycle in the non-exceptional cases, and that the exceptional parameter set is very tiny in terms of gauge functions. The exceptional two-dimensional parameter is shown to have Hausdorff-dimension one. We also study the invariant sets and the limit sets; these are sometimes different and there are several cases to consider. In addition, we prove the existence of a unique invariant measure. We apply some of our results for the dynamical system, involving a study of rational and irrational rotation numbers, to a combinatorial problem involving an election method suggested by Phragmen, and we show that the proportion of elected seats for each party converges to a limit, which is a rational number except for a very small exceptional set of parameters.
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| 10. |
- Öberg, Anders, 1968-
(author)
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Hilary Putnam on Meaning and Necessity
- 2011
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Doctoral thesis (other academic/artistic)abstract
- In this dissertation on Hilary Putnam's philosophy, I investigate his development regarding meaning and necessity, in particular mathematical necessity. Putnam has been a leading American philosopher since the end of the 1950s, becoming famous in the 1960s within the school of analytic philosophy, associated in particular with the philosophy of science and the philosophy of language. Under the influence of W.V. Quine, Putnam challenged the logical positivism/empiricism that had become strong in America after World War II, with influential exponents such as Rudolf Carnap and Hans Reichenbach. Putnam agreed with Quine that there are no absolute a priori truths. In particular, he was critical of the notion of truth by convention. Instead he developed a notion of relative a priori truth, that is, a notion of necessary truth with respect to a body of knowledge, or a conceptual scheme. Putnam's position on necessity has developed over the years and has always been connected to his important contributions to the philosophy of meaning. I study Hilary Putnam's development through an early phase of scientific realism, a middle phase of internal realism, and his later position of a natural or commonsense realism. I challenge some of Putnam’s ideas on mathematical necessity, although I have largely defended his views against some other contemporary major philosophers; for instance, I defend his conceptual relativism, his conceptual pluralism, as well as his analysis of the realism/anti-realism debate.
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