Splitting authentication codes with perfect secrecy: New results, constructions and connections with algebraic manipulation detection codes

<p style='text-indent:20px;'>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 <i>group generated</i> 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.</p><p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ (k, c) $\end{document}</tex-math></inline-formula>, we determine necessary and sufficient (or almost sufficient) conditions for the existence of <inline-formula><tex-math id="M2">\begin{document}$ (v, k \times c, 1) $\end{document}</tex-math></inline-formula>-splitting BIBDs that can be equitably ordered. The pairs for which we can solve this problem are <inline-formula><tex-math id="M3">\begin{document}$ (k, c) = (3, 2), (4, 2), (3, 3) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ (3, 4) $\end{document}</tex-math></inline-formula>, as well as all cases with <inline-formula><tex-math id="M5">\begin{document}$ k = 2 $\end{document}</tex-math></inline-formula>.</p>

Analisis Kemampuan Siswa dalam Pembuktian Kesebangunan Dua Segitiga

Mathematics learning in junior high is inseparable from proof, including of congruence of two triangles. This study analyzes the ability to prove the similarity of two triangles. The procedure used (1) grouped the answers of 51 students based on categories of able, underprivileged and unable, (2) analyzing the ability of evidence-based basic mathematical knowledge, representation of evidence, and assumptions used. The result (1). students who can prove have basic knowledge of the Pythagoras theorem, algebra operations, and the principle of equality, the representations used are symbolic and formal evidence, the assumptions used are logical. (2) Students who are unable to prove to have basic knowledge of congruence, comparison, and the principle of equality, the representation used is visual and formal evidence, the assumptions used are logical. (3) Students who have not been able to prove to have the basic knowledge that is not relevant in supporting evidence, the representation used is symbolic evidence but is wrong in algebraic manipulation and informal evidence, students' assumptions are lacking or illogical.Abstrak:Pembelajaran matematika di SMP tidak terlepas dari pembuktian termasuk kesebangunan dua segitiga. Penelitian ini menganalisis kemampuan siswa dalam membuktikan kesebangunan dua segitiga. Prosedur yang digunakan (1) mengelompokkan jawaban 51 siswa berdasarkan kategori mampu, kurang mampu dan belum mampu, (2) menganalisis kemampuan pembuktian berdasarkan pengetahuan matematis dasar, representasi bukti, serta asumsi yang digunakan. Diperoleh hasil bahwa (1). siswa yang mampu membuktikan memiliki pengetahuan dasar teorema Pythagoras, operasi aljabar, dan prinsip kesetaraan, representasi yang digunakan berupa bukti simbolis dan formal, asumsi yang digunakan adalah logis. (2) Siswa yang kurang mampu membuktikan memiliki pengetahuan dasar kesebangunan, perbandingan, dan prinsip kesetaraan, representasi yang digunakan adalah bukti visual dan formal, asumsi yang digunakan adalah logis. (3) Siswa yang belum mampu membuktikan memiliki pengetahuan dasar yang tidak relevan dalam mendukung pembuktian, representasi yang digunakan adalah bukti simbolis, tetapi salah dalam manipulasi aljabar dan bukti tidak formal, asumsi siswa kurang atau tidak logis.

Comfort for the Computationally Crippled

The difference between engineering and science, and all other human activity, is the fact that engineers and scientists make quantitative predictions about measurable outcomes and can specify their uncertainty in such predictions. Because those predictions are quantitative, they must employ mathematics. This chapter is intended as review of some of the more useful mathematical concepts, strategies, and techniques that are employed in the description of vibrational and acoustical systems and in the calculation of their behavior. Topics in this review include techniques such as Taylor series expansions, integration by parts, and logarithmic differentiation. Equilibrium and stability considerations lead to relations between potential energies and forces. The concept of linearity leads to superposition and Fourier analysis. Complex numbers and phasors are introduced along with the techniques for their algebraic manipulation. The discussion of physical units is extended to include their use for predicting functional dependencies of resonance frequencies, quality factors, propagation speeds, flow noise, and other system behaviors using similitude and the Buckingham Π-theorem to form dimensionless variables. Linearized least-squares fitting is introduced as a method for extraction of experimental parameters and their uncertainties and error propagation is presented to allow those uncertainties to be combined.

Utilizing Graphical Elements for Concept Map Analysis to Design Teaching and Learning Assessment

The graphical elements as parts of concept map construction are employed to assess both learning and teaching. Augmenting the use of concept maps, this study examines the graphical elements, such as, nodes, edges, cliques, diameters, travelling paths and structures of the graphs to relate to ones’ understanding to a topic, in this case, polynomials for middle school. In the aspect of teaching assessment, the teacher’ concept map drawn according to the lesson plan is served as the master map, which echoes the teacher’s expectation of students’ learning. On the other hand, students’ maps also reveal their understanding through the nodal relationship, which can be the definitions of terms, related examples, graph representation and algebraic manipulation. Data collection includes a focus group of 10 students and 1 teacher undergoing the concept map assessment task with restricted node terms. Graphically analyzed, students’ concept maps reveal some common elements as in the teacher’s map. In addition, the interview with the teacher also suggests that concept map as the assessment tool is an effective teaching reflection for which the teacher can see what to fulfill for future classes.

A unified framework for analysis of individual-based models in ecology and beyond

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.

STUDENTS’ DIFFICULTIES AND MISCONCEPTIONS OF THE FUNCTION CONCEPT

This study explores understanding of function concept amongst 310 grade 11 science stream students in one administrative zone of Ethiopia. A test that included tasks given in different representations, about definition, about examples of functions in word description and applications of properties of functions was administered. Lesson observation and interview was also used for triangulation. Results have shown that limited mental image of approach to functions, fragmented conceptions and dependence on ordered pairs, limitation in algebraic manipulation, limitation on converting word expression into mathematical expressions, confusing combination and composition, unnecessary interchanging order of operations during algebraic manipulations and drawing graph without considering sufficient points were observed difficulties. Whereas, a relation is a function if it has algebraic expression, overgeneralization that a representation is a functions if it is symbolized as an ordered pairs, and considering every point of discontinuity as an asymptote were identified misconceptions. Thus, special attention should be given in the teaching-learning to overcome identified difficulties and misconceptions.

Kesalahan Mahasiswa dalam Pembuktian Matematik

Proofs are the key component in mathematics and mathematics learning. But in reality, there are still many students who make errors when constructing mathematical proofs. Therefore this study aimed to identify common errors when the students are constructing mathematical proofs. The participant of this study was 51 of 3rd year students of Mathematics Education Department who enrolled in Real Analysis course in the second semester of the 2017/2018 academic year. The data of the study were obtained by conducting a test consisting of five questions and interview guidelines. The errors identified in this study were (1) proving general statements using specific examples, (2) inappropriate algebraic manipulation in mathematical induction, (3) incorrect reasoning and assumptions in proving with contradictions, and (4) reasoning errors involving natural numbers in mathematical induction. Hence, further study can be developed learning models that promote the conceptual understanding, logical reasoning, and mastery of mathematical proof techniques.

SPEKTRUM SYMBOL DAN STRUCTURE SENSE MATEMATIKA SISWA MADRASAH TSANAWIYAH

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.