The Experts below are selected from a list of 12834 Experts worldwide ranked by ideXlab platform
Abbas Edalat - One of the best experts on this subject based on the ideXlab platform.
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Extensions of Domain Maps in Differential and Integral Calculus
2015 30th Annual ACM IEEE Symposium on Logic in Computer Science, 2015Co-Authors: Abbas EdalatAbstract:We introduce in the context of differential and Integral Calculus several key extensions of higher order maps from a dense subset of a topological space into a continuous Scott domain. These higher order maps include the classical derivative operator and the Riemann integration operator. Using a sequence of test functions, we prove that the subspace of real-valued continuously differentiable functions on a finite dimensional Euclidean space is dense in the space of Lipschitz maps equipped with the L-topology. This provides a new result in basic mathematical analysis, which characterises the L-topology in terms of the limsup of the sequence of derivatives of a sequence of C1 maps that converges to a Lipschitz map. Using this result, it is also shown that the generalised (Clarke) gradient on Lipschitz maps is the extension of the derivative operator on C1 maps. We show that the generalised Riemann Integral (R-Integral) of a real-valued continuous function on a compact metric space with respect to a Borel measure can be extended to the Integral of interval-valued functions on the metric space with respect to valuations on the probabilistic power domain of the space of non-empty and compact sets of the metric space. We also prove that the Lebesgue Integral operator on integrable functions is the extension of the R-Integral operator on continuous functions. We finally illustrate an application of these results by deriving a simple proof of Green's theorem for interval-valued vector fields.
Kirsti Hemmi - One of the best experts on this subject based on the ideXlab platform.
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the state of proof in finnish and swedish mathematics textbooks capturing differences in approaches to upper secondary Integral Calculus
Mathematical Thinking and Learning, 2017Co-Authors: Andreas Bergwall, Kirsti HemmiAbstract:ABSTRACTStudents’ difficulties with proof, scholars’ calls for proof to be a consistent part of K-12 mathematics, and the extensive use of textbooks in mathematics classrooms motivate investigations on how proof-related items are addressed in mathematics textbooks. We contribute to textbook research by focusing on opportunities to learn proof-related reasoning in Integral Calculus, a key subject in transitioning from secondary to tertiary education. We analyze expository sections and nearly 2000 students’ exercises in the four most frequently used Finnish and Swedish textbook series. Results indicate that Finnish textbooks offer more opportunities for learning proof than do Swedish textbooks. Proofs are also more visible in Finnish textbooks than in Swedish materials, but the tasks in the latter reflect a higher variation in nature of proof-related reasoning. Our results are compared with methodologically similar U.S. studies. Consequences for learning and transition to university mathematics, as well as ...
Giorgio Kaniadakis - One of the best experts on this subject based on the ideXlab platform.
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theoretical foundations and mathematical formalism of the power law tailed statistical distributions
Entropy, 2013Co-Authors: Giorgio KaniadakisAbstract:We present the main features of the mathematical theory generated by the κ-deformed exponential function exp k (x) = ( 1 + k 2 x 2 + kx) 1 k , with 0 ≤ κ < 1, developed in the last twelve years, which turns out to be a continuous one parameter deformation of the ordinary mathematics generated by the Euler exponential function. The κ-mathematics has its roots in special relativity and furnishes the theoretical foundations of the κ-statistical mechanics predicting power law tailed statistical distributions, which have been observed experimentally in many physical, natural and artificial systems. After introducing the κ-algebra, we present the associated κ-differential and κ-Integral Calculus. Then, we obtain the corresponding κ-exponential and κ-logarithm functions and give the κ-version of the main functions of the ordinary mathematics.
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theoretical foundations and mathematical formalism of the power law tailed statistical distributions
Entropy, 2013Co-Authors: Giorgio KaniadakisAbstract:We present the main features of the mathematical theory generated by the κ-deformed exponential function exp k (x) = ( 1 + k 2 x 2 + kx) 1 k , with 0 ≤ κ < 1, developed in the last twelve years, which turns out to be a continuous one parameter deformation of the ordinary mathematics generated by the Euler exponential function. The κ-mathematics has its roots in special relativity and furnishes the theoretical foundations of the κ-statistical mechanics predicting power law tailed statistical distributions, which have been observed experimentally in many physical, natural and artificial systems. After introducing the κ-algebra, we present the associated κ-differential and κ-Integral Calculus. Then, we obtain the corresponding κ-exponential and κ-logarithm functions and give the κ-version of the main functions of the ordinary mathematics.
Reinhard Hochmuth - One of the best experts on this subject based on the ideXlab platform.
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bridging probability and Calculus the case of continuous distributions and Integrals at the secondary tertiary transition
arXiv: History and Overview, 2018Co-Authors: Charlotte Derouet, Gaetan Planchon, Thomas Hausberger, Reinhard HochmuthAbstract:This paper focuses on two mathematical topics, namely continuous probability distributions (CPD) and Integral Calculus (IC). These two sectors that are linked by a formula are quite compartmented in teaching classes in France. The main objective is to study whether French students can mobilize the sector of IC to solve tasks in CPD and vice versa at the transition from high school to higher education. Applying the theoretical framework of the Anthropological Theory of the Didactic (ATD), we describe a reference epistemological model (REM) and use it to elaborate a questionnaire in order to test the capacity of students to bridge CPD and IC at the onset of university. The analysis of the data essentially confirms the compartmentalisation of CPD and IC.
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bridging probability and Calculus the case of continuous distributions and Integrals at the secondary tertiary transition
INDRUM 2018, 2018Co-Authors: Charlotte Derouet, Gaetan Planchon, Thomas Hausberger, Reinhard HochmuthAbstract:This paper focuses on two mathematical topics, namely continuous probability distributions (CPD) and Integral Calculus (IC). These two sectors that are linked by the formula P(a<=X<=b)=int_a^b f(x)dx are quite compartmented in teaching classes in France. The main objective is to study whether French students can mobilize the sector of IC to solve tasks in CPD and vice versa at the transition from high school to higher education. Applying the theoretical framework of the Anthropological Theory of the Didactic (ATD), we describe a reference epistemological model (REM) and use it to elaborate a questionnaire in order to test the capacity of students to bridge CPD and IC at the onset of university. The analysis of the data essentially confirms the compartmentalisation of CPD and IC.
Irene Penesis - One of the best experts on this subject based on the ideXlab platform.
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Transforming learning with computers: Calculus for kids
Education and Information Technologies, 2020Co-Authors: Andrew E. Fluck, Irene Penesis, Dev Ranmuthugala, C. K. H. Chin, Jacky Chong, Yang Yang, Asim GhousAbstract:The Calculus for Kids project was deliberately designed to use computers in the transformation of curriculum. The intervention used multi-media learning materials to assist teachers and Year 6 (aged 11–12 years) students understand the principles of Integral Calculus. They used Maple mathematics software to solve real-world problems using these principles and by employing conventional mathematics notation on their individual computers. Between June 2010 and April 2016, it was implemented in 23 classes at 19 schools involving 434 students. Two methods were used to calculate effect sizes of 22.19 (pre-test/post-test Cohen’s d ) and 1.17 (age-maturation). Positive gains were also found in students’ attitudes, particularly in Technology confidence. This article discusses methods for calculating effect sizes for transformational education with computers and recommends further research in the field.
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Calculus in elementary school an example of ict based curriculum transformation
The Journal of Computers in Mathematics and Science Teaching, 2012Co-Authors: A Fluck, D Ranmuthugala, Christopher Chin, Irene PenesisAbstract:Integral Calculus is generally regarded as a fundamental but advanced aspect of mathematics, and it is not generally studied until students are aged about fifteen or older. Understanding the transformative potential of information and communication technology, this project undertook an investigation in four Australian schools to train pupils aged ten to twelve years how to solve problems using Integral Calculus with computer algebra system software. After eleven lessons the students completed a test constructed from items at the level of a first year engineering degree Calculus examination. The average achievement was at the credit level, and students showed good understanding of the applications of Integral Calculus. This leads to a discussion about new ways of understanding curriculum in the light of new technology.
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Calculus in elementary school an example of ict based curriculum transformation
22nd International Conference of the Society for Information Technology & Teacher Education (SITE 2011), 2011Co-Authors: A Fluck, D Ranmuthugala, Christopher Chin, Irene PenesisAbstract:Integral Calculus is generally regarded as a fundamental but advanced aspect of mathematics, and it is not generally studied until students are aged about fifteen or older. Understanding the transformative potential of information and communication technology, this project undertook an investigation in four Australian schools to train pupils aged ten to twelve years how to solve problems using Integral Calculus with computer algebra system software. After eleven lessons the students completed a test constructed from items at the level of a first year engineering degree Calculus examination. The average achievement was at the credit level, and students showed good understanding of the applications of Integral Calculus.
