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Improving Prospective Teachers’ Beliefs About a Double Discontinuity Between School Mathematics and University Mathematics
(2022-05-12)
Prospective teachers often perceive a “double discontinuity” between school mathematics and university mathematics. The first discontinuity can be described as the belief that there is no coherence between school mathematics and university mathematics, which forms part of the notoriously problematic transition from school to university. The second discontinuity can be described as a belief about the lack of relevance of university mathematics for the later professional practice of prospective teachers. Beliefs about ...
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An extendable key space integer image-cipher using 4-bit piece-wise linear cat map
(2022-09-30)
This paper presents a multiplierless image-cipher, with extendable 2048-bit key-space, based on a 4-dimensional (4D) quantized piece-wise linear cat map (PWLCM). The quantized PWLCM exhibits limit-cycles of 4-bit encoded integers with periods greater than 10⁷. The synthesis of the PWLCM in a finite state space allows to eliminate the undesirable finite precision effect due to the hardware realization. The proposed image-cipher combines chaos, modular arithmetic, and lattice-based cryptography to encrypt a color image ...
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On the implementation of large period piece-wise linear Arnold cat map
(2022-04-26)
This paper presents a piece-wise linear cat map (PWLCM) obtained by perturbing the conventional quantized Arnold cat map (QACM) with a nonlinear term. The effect of the nonlinear term on the dynamics of the QACM is investigated. We show that the eigenvalues, hence the Lyapunov exponents of the PWLCM depend on the initial conditions, which is not the case for the QACM. As a result, the proposed PWLCM is a generalized form of the QACM, whose the period exponentially increases with respect to the precision, thus taking ...
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Control of Jump Markov Uncertain Linear Systems With General Probability Distributions
(2022-02-25)
This paper introduces a method to control a class of jump Markov linear systems with uncertain initialization of the continuous state and affected by disturbances. Both types of uncertainties are modeled as stochastic processes with arbitrarily chosen probability distributions, for which however, the expected values and (co-)variances are known. The paper elaborates on the control task of steering the uncertain system into a target set by use of continuous controls, while chance constraints have to be satisfied for ...
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Using heuristic worked examples to promote solving of reality-based tasks in mathematics in lower secondary school
(2022-04-19)
This study examined whether learning with heuristic worked examples can improve students’ competency in solving reality-based tasks in mathematics (mathematical modeling competency). We randomly assigned 134 students in Grade 5 and 180 students in Grade 7 to one of three conditions: control condition (students worked on reality-based tasks), worked example condition (students studied worked examples representing a realistic process of problem-solving by fictitious students negotiating solutions to the tasks), and ...
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An 8-bit precision cipher for fast image encryption
(2022-04-21)
Implementing chaos based ciphers usually involves 32-bit floating-point arithmetics that is hardware resources costly. The limitation of the computational precision is hardware imposed and transforms chaotic orbits into limit cycles with short periods, hence alters their randomness. In cryptographic applications, short period dynamics and weak randomness result in security issues. In order to address this concern, we propose an 8-bit precision cipher that can be implemented with low-end microprocessors running 8-bit ...
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30 Years with Vladimir Gerdt
(2022-11-12)
My first meeting with Vladimir was in March 1990. I had started two months before as a Ph.D. student of Jacques Calmet in the computer science department of Karlsruhe University and Vladimir was visiting our computer algebra group for a seminar talk on his research. For him, it was one of his first travels into the west, as under Soviet rule he was not considered as sufficiently politically reliable for travels outside of the eastern block. It turned out that after his visit we were both riding on the same train to ...
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Complementary decompositions of monomial ideals and involutive bases
(2022-06-25)
Complementary decompositions of monomial ideals - also known as Stanley decompositions - play an important role in many places in commutative algebra. In this article, we discuss and compare several algorithms for their computation. This includes a classical recursive one, an algorithm already proposed by Janet and a construction proposed by Hironaka in his work on idealistic exponents. We relate Janet’s algorithm to the Janet tree of the Janet basis and extend this idea to Janet-like bases to obtain an optimised ...
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On Non-linear Characterizations of Classical Orthogonal Polynomials
(2022-12-03)
Classical orthogonal polynomials are known to satisfy seven equivalent properties, namely the Pearson equation for the linear functional, the second-order differential/difference/q-differential/ divided-difference equation, the orthogonality of the derivatives, the Rodrigues formula, two types of structure relations, and the Riccati equation for the formal Stieltjes function. In this work, following previous work by Kil et al., we state and prove a non-linear characterization result for classical orthogonal polynomials ...
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A Critical Review on the Complex Potentials in Linear Elastic Fracture Mechanics
(2022-01-03)
Introducing a crack in an elastic plate is challenging from the mathematical point of view and relevant within an engineering context of evaluating strength and reliability of structures. Accordingly, a multitude of associated works is available to date, emanating from both applied mathematics and mechanics communities. Although considering the same problem, the given complex potentials prove to be different, revealing various inconsistencies in terms of resulting stresses and displacements. Essential information on ...