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In this paper, we present the results of a comparative study on the teaching of algebra in France and in Chili. Within the Anthropological Theory of the Didactic, we studied and compared the mathematical knowledge related to the distributive property, taught within each school system. The analysis of programs and textbooks reveals significant differences in the knowledge to be taught or taught. We also analyzed the knowledge of students, through a questionnaire given to French and Chilean students (aged 14-15). The analysis of the answers allows us to establish differences between the mathematical praxeologies students have learnt. It also reveals similarities that reveal the difficulties of (Chilean or French) students in learning algebra. Finally, we conducted a brief experimentation in a French class that introduced a type of tasks present in the Chilean textbook. This experimentation allows us to examine the role of geometric framework in the study of the distributive property, more represented in the knowledge to be taught and taught in Chile than in France. Dans cet article, nous présentons les résultats d’une étude comparative sur l’enseignement du calcul algébrique en France et au Chili. Nous situant dans le cadre de la Théorie Anthropologique du Didactique, il s’est agi d’analyser et de comparer les organisations de savoirs mathématiques enseignées en lien avec la propriété de distributivité, au sein de chaque institution scolaire. L’analyse des programmes et de manuels scolaires révèle des différences significatives dans les savoirs à enseigner et enseignés. Nous avons également mené une analyse des praxéologies apprises, au moyen d’un questionnaire proposé à des élèves français et chiliens (élèves de 14-15 ans). L’analyse des réponses nous permet d’établir des différences entre les praxéologies mathématiques apprises au sein des deux institutions, mais aussi des points communs qui renvoient entre autres à des difficultés similaires dans l’apprentissage du calcul algébrique. Enfin, nous avons conduit une brève expérimentation dans une classe de seconde française visant à « importer » un type de tâches présent dans le manuel chilien et absents des manuels français. Cette expérimentation nous permet d’interroger le rôle du cadre géométrique dans l’étude de la propriété de distributivité, plus représenté dans les savoirs à enseigner et enseignés au Chili qu’en France.

In this paper, we study the relationship between algebraic manipulation detection (AMD) codes and highly nonlinear functions. As applications, on one hand, a generic construction for systematic AMD codes is introduced based on highly nonlinear functions. Systematic AMD codes with new parameters can be generated from known highly nonlinear functions. Especially, several infinite classes of optimal systematic AMD codes, some with asymptotically optimal tag size, can be constructed. On the other hand, systematic AMD codes are used to construct highly nonlinear functions. The known construction by Cramer et al. (2008) for systematic AMD codes turns out to be based on a special kind of functions with high nonlinearity.

This paper provides a combinatorial characterization of weak algebraic manipulation detection (AMD) codes via a kind of generalized external difference families called bounded standard weighted external difference families (BSWEDFs). By means of this characterization, we improve a known lower bound on the maximum probability of successful tampering for the adversary's all possible strategies in weak AMD codes. We clarify the relationship between weak AMD codes and BSWEDFs with various properties. We also propose several explicit constructions for BSWEDFs, some of which can generate new optimal weak AMD codes.

The study was conducted due to the unsatisfied results of students’answers and interviews on Differential Calculus material in which the studentsgraphically did not understand the interpretation of a function that was being solved and were not proficient in manipulating algebra, therefore, no solution was found. Based on these findings, a geogebra-assisted worksheet for smartphones was designed to improve students' understanding onthe topic of inequality. This study aimedto develop student worksheets in understanding the topic of inaccuracy in a function. This worksheet consistedof two parts that help students buildingtheir own understanding onthe graphic form of an equation and its solution both graphically and procedurally through algebra. To see the feasibility of the worksheets, expert validation and practicalitytests were conducted to students. The results of the revision were then distributed to 10 Untirta Mathematics Education students who were taking the MK Differential Calculus. Students spent 60 minutes to complete it. They tried to find thesolutions to the problems given with the help of geogebra for smartphones, by firstlysolving them graphically and finding the points referred to in the Cartesian diagram. Then,they solved the problemswith algebraic manipulation as a form of confirmation of the answersobtained. After completing the whole worksheet, it was shownthat the students couldsolve the problems graphically well.points referred to in the Cartesian diagram. Then,they solved the problemswith algebraic manipulation as a form of confirmation of the answersobtained. After completing the whole worksheet, it was shownthat the students couldsolve the problems graphically well.

Individual-based models, ‘IBMs’, describe naturally the dynamics of interacting organisms or social or financial agents. They are considered too complex for mathematical analysis, but computer simulations of them cannot give the general insights required. Here, we resolve this problem with a general mathematical framework for IBMs containing interactions of an unlimited level of complexity, and derive equations that reliably approximate the effects of space and stochasticity. We provide software, specified in an accessible and intuitive graphical way, so any researcher can obtain analytical and simulation results for any particular IBM without algebraic manipulation. We illustrate the framework with examples from movement ecology, conservation biology, and evolutionary ecology. This framework will provide unprecedented insights into a hitherto intractable panoply of complex models across many scientific fields., Individual-based models are widely used to simulate complex systems of interacting agents. Here the authors provide a mathematical framework that automates the analysis of any model in a wide class, facilitating a deeper understanding of the scientific questions these models are used to address.

A collection of disjoint subsets ${\cal A}=\{A_1,A_2,\dotsc,A_m\}$ of a finite abelian group is said to have the \emph{bimodal} property if, for any non-zero group element $\delta$, either $\delta$ never occurs as a difference between an element of $A_i$ and an element of some other set $A_j$, or else for every element $a_i$ in $A_i$ there is an element $a_j\in A_j$ for some $j\neq i$ such that $a_i-a_j=\delta$. This property arises in various familiar situations, such as the cosets of a fixed subgroup or in a group partition, and has applications to the construction of optimal algebraic manipulation detection (AMD) codes. In this paper, we obtain a structural characterisation for bimodal collections of sets.

The purpose of this study is to determine the symbol sense and structure of mathematical sense in terms of students' ability to solve algebraic problems or other mathematical problems that require symbol expression or structure. Difficulties experienced by students in solving mathematical or algebraic problems may be due to the ability of symbol sense and low structure sense or didactic design that the teacher conveyed is not in accordance with the category of symbols and structure sense that students have. A student with good symbols and structure sense is able to appreciate the power of symbols, knowing when to use the right symbols and being able to manipulate and understand symbols in various contexts. The method used in the study used a qualitative descriptive method. The population of this study was seventh grade students of the Islamic junior high school, the instruments used were symbol and structure sense tests, questionnaires, and interview forms. The results of the study indicate that the ability of symbol sense and student structure sense is still low because of a lack of conceptual knowledge and algebraic manipulation, for this reason it is necessary to have an appropriate learning model to improve both of these abilities.

This study aims to give a comprehensive account of pre-service secondary mathematics teachers’ understanding of absolute value. Thirty two-item absolute value understanding test was developed and administered to thirty-eight students attending mathematics education department at one private university in Jakarta City, Indonesia. Five of them were selected purposively and interviewed to gain deep information and confirm their written responses in the test. We find that most participants struggled with the absolute value task. There are inconsistencies of the definition of absolute value expressed by them. Besides, typical mistakes made are: (a) removal of absolute value bars; (b) focus heavily on rules; (c) conversion of absolute value bars to parentheses; (d) exclusion of number inside absolute value bars; (e) poor algebraic manipulation; and (f) inability to draw absolute value graph. Based on the findings, the most common cause of mistakes made by the participants is didactical contract in mathematics teaching and learning. Limitation and implications of the study are presented.

A splitting BIBD is a type of combinatorial design that can be used to construct splitting authentication codes with good properties. In this paper we show that a design-theoretic approach is useful in the analysis of more general splitting authentication codes. Motivated by the study of algebraic manipulation detection (AMD) codes, we define the concept of a group generated splitting authentication code. We show that all group-generated authentication codes have perfect secrecy, which allows us to demonstrate that algebraic manipulation detection codes can be considered to be a special case of an authentication code with perfect secrecy.We also investigate splitting BIBDs that can be "equitably ordered". These splitting BIBDs yield authentication codes with splitting that also have perfect secrecy. We show that, while group generated BIBDs are inherently equitably ordered, the concept is applicable to more general splitting BIBDs. For various pairs \begin{document}$ (k, c) $\end{document}, we determine necessary and sufficient (or almost sufficient) conditions for the existence of \begin{document}$ (v, k \times c, 1) $\end{document}-splitting BIBDs that can be equitably ordered. The pairs for which we can solve this problem are \begin{document}$ (k, c) = (3, 2), (4, 2), (3, 3) $\end{document} and \begin{document}$ (3, 4) $\end{document}, as well as all cases with \begin{document}$ k = 2 $\end{document}.

Early modern mathematical culture used the written word in highly distinctive ways. The various kinds of mathematical activity were all experienced in the early modern period as special kinds of performance: arithmetical calculation, proportional reasoning, algebraic manipulation and geometrical proving. The normal way they were transmitted—taught and learned—involved demonstration by a teacher, private rehearsal and finally specimen performance by the student. Slates, waste paper, and the margins of printed texts all bear the traces of learners’ rehearsals. This chapter examines the evidence for this mathematical culture and the influences on it of ancient and early modern examples.

The paper describes some findings from a programming task book project. The book contains tasks on arithmetic (operations, calculating the value of numeric expressions, common fractions etc), number theory (factors, prime numbers, GCD, LCM, Euclid algorithm), constructions in planar geometry, algebra (linear equations and inequalities, systems of linear equations, polynomials), some types of nonroutine tasks. The paper also brings up the necessary information processing that is not explicit when mathematical tasks are solved by paper-and-pencil method: parsing of algebraic expressions, finding the coordinates of intersection points by planar constructions, programming of algorithms for the tasks that appear in textbooks only in the form of one single numerical example. Some warnings are given about the impact of ”brute force“ computer solutions of reasoning-oriented tasks., Proceedings of the 14th International Conference on Technology in Mathematics Teaching – ICTMT 14: Essen, Germany, 22nd to 25th of July 2019, p. 109

In the scheme of reconstruction, non-polynomial predictors improve the forecast from chaotic time series. The algebraic manipulation in the Maple environment is the basis for obtaining of accurate predictors. Beyond the different times of prediction, the optional arguments of the computational routines optimize the running and the analysis of global mappings.

Algebraic manipulation detection codes were introduced in 2008 to protect data against a special type of a modification: an algebraic manipulation. One of the most effective ways of constructing such codes is based on polynomial encoding functions. In this paper a new family of polynomials is proposed, which may lead to higher detection capabilities and lower computational complexity of encoding and decoding procedures.

In the study of cryptography and information security, combinatorial structures arise in a natural and essential way, especially in the context of unconditional security (also termed information-theoretic security). In this chapter, the author discusses several interesting examples of interactions between cryptography and combinatorics. In 1993, he gave an invited talk at the Fourteenth British Combinatorial Conference. The paper he wrote to accompany his talk was entitled “Combinatorial Designs and Cryptography.” The purpose of this paper is to elucidate some of these connections. This is not intended to be an exhaustive survey, but rather a sampling of some research topics in which he have a personal interest. The author also discusses the connections between the One-time Pad, perfect secrecy and latin square. He focuses on three topics of continuing and/or recent interest, namely, threshold and ramp schemes, all-or-nothing transforms, and algebraic manipulation codes.

This paper provides an achievable rate and performance analysis between OFDM and cyclic prefixed SC-DFE when no channel knowledge is available at the transmitter. Through some algebraic manipulation of the OFDM and SC-DFE capacities and using the Jensen's inequality, we prove that the SC-DFE achievable rate is always superior to that of an OFDM scheme for 4- and 16-QAM for any given channel. For higher-order modulations, however, the advantage of the ideal SC-DFE is much smaller and also the results indicate that OFDM may surpass the ideal SC-DFE achievable rate by a small amount in some specific scenarios. Finally, we provide simulation results in order to assess, corroborate and illustrate the analytical results in terms of achievable rate and bit-error rate, where we conclude that a well designed OFDM system can be, in practical context, as effective as an ideal SC-DFE but with much smaller complexity when compared to a SC-DFE implementation with error-propagation mitigation.

In this chapter, we show how the problem of controller synthesis can be posed as a form of convex optimization in an operator-theoretic framework. Furthermore, we show how to: (a) Parameterize the integral and multiplier operator-valued decision variables using finite-dimensional vectors; (b) Verify and enforce positivity and negativity of multiplier and integral operators using positive matrices; (c) Invert positive integral and multiplier operators through the use of a new formula based on algebraic manipulation. Finally, we show how these 3 parts can be combined into a computational procedure for finding a stabilizing state-feedback controller for systems defined by differential-difference equations—a class which includes differential systems with discrete delays. Finally, a numerical example is used to illustrate the form of the resulting stabilizing controller.

Mathematical literacy is a person's ability to formulate, apply, and interpret mathematics in various contexts. This study is a qualitative descriptive that aims to describe the ability of mathematics literacy of students in thesubject of exponents and logarithms in Vocational High School. Data analysis is done by collecting data, reducing data, and verifying data. The results obtained in this study are subjects with values above the MEC cansolve the problem by using information, performing representations based on concepts, using procedural knowledge in the form of algebraic manipulation in accordance with the nature of exponents and logarithms,and can connect it with the real world to determine the outcome of completion, according to level 4 PISA. Subjects with the same value as MEC can solve problems, interpret by using the properties of exponents andlogarithms, and implement procedures according to the 3rd level of PISA. Subjects below the MEC can only solve commonly resolved problems using simple properties, corresponding to the PISA 1st level.

Reactive transport modeling plays a critical role in predicting and quantifying impacts of potential CO2 leakage into aquifers at geological carbon sequestration sites. However, a numerical approach generally requires significant computation. This study presents a semi-analytical approach to reactive transport of CO2 leakage into an aquifer through decoupling transport equations of reactive aqueous species with algebraic manipulation and then solving the algebraic equation sets representing the mass action law of geochemical reactions with the Newton-Raphson method. The semi-analytical approach was implemented in a simulation tool, SASCO2M, and verified against a numerical approach for 2-D synthetic cases. Verification shows that the semi-analytical approach matched reasonably well the results simulated with the numerical approach. The semi-analytical approach was applied to simulate pulse-like CO2 release tests which were used to demonstrate CO2 leakage detection in a shallow aquifer. The semi-analytical approach reproduced the overall trends of groundwater pH, dissolved inorganic carbon, alkalinity, and concentrations of Ca and Br observed in the testing well. The semi-analytical approach was further applied to assess the efficiency of a groundwater monitoring network for CO2 leakage detection in a shallow aquifer at a CO2-enhanced oil recovery site. This study demonstrates that the semi-analytical approach is simple and efficient and can be followed as a strategic procedure for assessing risks of CO2 leakage on groundwater quality and efficiency of groundwater monitoring networks for leakage detection at geological CO2 sequestration sites. © 2015 Society of Chemical Industry and John Wiley & Sons, Ltd

Algebraic manipulation detection codes were introduced in 2008 to protect data against a special type of its modification – an algebraic manipulation. There are three classes of codes: weak, strong and stronger ones. Weak codes detect only weak algebraic manipulations, while strong and stronger are capable to detect both weak and strong manipulations. In this paper, a method to transform codes from weak to stronger is described resulting in a construction of generalized robust codes. The proposed construction forms the second known family of stronger AMD codes. Codes provide simple procedures of error detectiona and correction, and allow using multi-code codec design that makes it applicable in fault-tolerant computing, storage and cryptographic devices.