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From Computer Assisted to Computer Driven Mathematics?
Alexei Semenov (math-net.ru)
Federal Research Center "Computer Science and Control" of Russian Academy of Sciences, Moscow, Russia
Lomonosov Moscow State University, Moscow, Russia
Moscow Institute of Physics and Technology (National Research University), Moscow, Russia
Abstract
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How Computers Have Changed Mathematics, аnd Why It Is Fateful for Teaching
Nikolai Vavilov(researchgate.net)
St. Petersburg State University, St. Petersburg, Russia
Abstract
I plan to discuss the use of computers in performing standard calculations, mathematical experiment and high-end human-computer mathematical research. I plan to illustrate in several real-life examples how computers have already changed mathematical way of thinking, and the way professional mathematicians perceive mathematics. [None of these will be about formalisation of mathematics, in my view formalised proofs are a completely different activity, outside of of the scope of mathematics]. But in the first place, computers have changed applications of mathematics, and made most of the skills we are trying to cultivate in our students obsolete. We cannot persist in retarded ways of teaching mathematics, and should nurture big ideas and conceptual understanding instead, relegating all routine work to the computing machines.
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The “Donkey-and-Carrot” Technique in Dynamic Mathematics
Vladimir Dubrovsky(istina.msu.ru)
AESC MSU – Kolmogorov boarding school, Moscow, Russia
Abstract
There are two interactive mathematical systems,MathKit and The Geometer's Sketchpad, that support a special type of animation in which a point follows a target running away from it according to a certain rule. The trace of such a point is, in fact, a numerical solution by Euler’smethod toa certain differential equation. A nice feature of this technique is that the righthand side of the equation can be specified without formulas, geometrically. Thetechnique makes it possible to illustrate a number of interesting problemsin a graphicway, easily understandable by school students, and sometimes even to solve them. Various examples of its application are given: the study of the shape of the suspension bridgecables, satellite dishes, planetary orbits, etc.
Keywords: MathKit, Euler’s method, Differential equation, Mathematical modeling, Interactive mathematical systems
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Using Technology in Teaching Introductory Statistics at SGU
Aleksandr Mylläri 1) (researchgate.net)
Tatiana Mylläri 1)
Dishon Edwards 1,2)
Stefan Hypolite 1)
1) School of Arts and Sciences, SGU, Grenada, West Indies
2) TA Marryshow Community College, Grenada, West Indies
Abstract
We discuss the use of technology in teaching introductory statistics. Traditionally, taking a course in statistics involved a lot of cumbersome and painstaking (as well as rather boring) manual calculations, which didn't leave students too much time to think about the ideas and methods involved. Using computers in the course saves time by doing most of the work on the computer and leaves more time in class for explanations and discussions. It also prepares the student for the real life – no one does statistical analysis by hand today. We encourage students to use RCommander. It was chosen because RCommander can be installed on Windows as well as on MAC OS, it is free, fairly easy to use, and contains many tools and methods for statistical analysis. It can also be used as an intermediate step towards using R. For those students who are having trouble installing or using RCommander, JASP is an easy alternative. JASP is open-source, free and has very nice GUI. Both RCommander and JASP are good to use for in-class demonstrations as well as for students to do statistics labs.
Keywords: Technology in Education
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Computer Tool and Crisis in School Mathematical Education
Sergey Ivanov 1)
Valery Ryzhik 2)
1) St. Petersburg Electrotechnical University "LETI"
2) Lyceum "Physical-Technical High School" named after Zh.I. Alferov, St. Petersburg
Federal Research Center "Computer Science and Control" of Russian Academy of Sciences, Moscow, Russia
Lomonosov Moscow State University, Moscow, Russia
Moscow Institute of Physics and Technology (National Research University), Moscow, Russia
Abstract
It is proposed to use computer tools to mitigate the crisis in school mathematics education based on:
1) interpretation of school mathematics as an introduction to science
2) introduction of experimental mathematics
3) study of planimetry based on the axiomatics of A.D. Alexandrov on a limited area of the plane
4) use of dynamic geometry
5) inclusion of research activities.
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Number Theory for Mathematics Instruction of Future Teachers in the Digital Era
Sergei Abramovich(researchgate.net)
School of Education and Professional Studies State University of New York at Potsdam, USA
Abstract
It has been almost a century since Godfrey Harold Hardy, a prominent British number theorist, recommended topics of elementary number theory for early mathematical instruction [1]. In the digital era, this recommendation was reflected in similar statements by mathematicians and mathematics educators extending instruction from yearly grades to the entire K-16 context [2, 3]. The paper will present technology-enhanced activities with triangular, square, and other polygonal numbers arranged in basic geometric shapes – equilateral and isosceles triangles and squares.Computational algorithms for the summation of such numbers within each geometric structure will be discussed. The activities, supported by Wolfram Alpha and Maple, are recommended for the use by instructors of technology-motivated mathematics teacher education courses. The paper emphasizes the value of technology-immune/technology-enabled mathematical problem solving [4] in the modern-day teaching topics of elementary number theory across multiple grade levels and educational programs. The paper will argue that the power of digital tools allows future teachers of mathematics, in the context of elementary number theory, to appreciate the use of simple algorithms in achieving sophisticated computational outcomes.
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Interaction of Algebraic Thinking and Computational Thinking
Sergei Pozdniakov (researchgate.net)
Elena Tolkacheva
St. Petersburg Electrotechnical University "LETI"
Abstract
The report discusses the problems associated with teaching algebra in a rich computer environment. The technique based on the convolution of algorithms, analyzed in detail in the work of I.M. Shapiro “From Algorithms to Judgments”, and the application of the APOS theory by Ed Dubinsky to form the concepts of group theory are analyzed. On the example of using dynamic geometry to introduce the concept of linear transformations, the concept of connecting different cognitive schemes is presented for a method of forming new concepts, alternative to the convolution of algorithms.
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An Elementary Proof of the Theorem on the Asymptotic Normality of Estimating the Variance of a Random Variable with Visualization of the Results Using Examples of Samples from Different Distributions
Nikolay Sosnovsky
St. Petersburg Electrotechnical University "LETI"
Abstract
The article offers an elementary proof of the asymptotic normality of the sample variance for distributions with the first four moments. The proof of this well-known and important fact in mathematical statistics is often omitted in elementary textbooks on probability theory and mathematical statistics due to some cumbersome reasoning. Note that this is a case of the validity of the central limit theorem, when the random variables forming the sum are equally distributed, but are dependent. In order to provide greater visibility and persuasiveness of the statement, the presentation shows histograms and graphs of the marginal distribution of the sample variance, made for various probabilistic laws of the original general aggregates.
Keywords: asymptotic, distribution, normality, variance, moment.
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Does 1 Plus 1 Always Make 2? Emergence of Binary Logic as Precursor for the Invention of Electronic Computing Devices
Viktor Freiman
Université de Moncton, Moncton, Canada
Abstract
The paper I will pursue the account of our ongoing work on the history of computations and computing devices in mathematics education before the advent of electronic calculators (Freiman and Volkov, 2018). More specifically, Freiman and Robichaud (2018) have analyzed the emergence of and evolution of ideas and computing technologies from Schickard to de Colmar. There appears to be two lines of development: one related to the invention of first mechanical calculators (e. g., Pascaline) eventually leading to different types of arithmometer. The second, going back to Leibniz and his work on binary arithmetic and invention of Boolean Logic, might be seen as a precursor of modern computers. In my presentation I will discuss this second line.
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Mathematical Education in an Era of Change
Sergey Karakozov
Moscow Pedagogical State University (MPGU)
Abstract
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Среды компьютерного моделирования в образовании
(Computer Modelling Environments in Education)
Yuri Senichenkov
Peter the Great St. Petersburg Polytechnic University, Russia
Abstract
Современные среды визуального моделирования используются повсеместно и их число растет с каждым годом, потому что среды - это основной инструмент изучения сложных явлений, проектирования новых видов техники и обучения. К созданию новых сред моделирования подключились ведущие разработчики математического обеспечения: компании Wolfram Research (System Modeler), Maplesoft (MapleSim). В нашей стране Правительство включило «математическое моделирование для решения задач технических вычислений» в приоритетные направления поддержки (гранты на разработку и внедрение российских цифровых решений - https://ит-гранты.рф/pnp). Министерство обороны Российской Федерации поставило задачу создать собственную среду визуального моделирования (https://mil.ru/scientific_competitions.htm).
Первые среды визуального моделирования были созданы в конце 20-го века одновременно во многих странах мира (Simulink, Ptolemy II, Dymola), в том числе и в России (МАРС, программный комплекс «МВТУ», MvStudium, AnyLogic). Среды моделирования, как и любые инструменты, должны обеспечивать нужное качество изготавливаемых с их помощью конечных продуктов, компьютерных моделей. Информации об отдельных средах сейчас достаточно, но их сравнением и оценкой качества моделирования, как выясняется, практически никто не занимается. Современные модели – сложные программно-аппаратные комплексы, и выбор среды моделирования становится важной проблемой, учитывая финансовые затраты на создание и сопровождение моделей. Сравнительный анализ позволит реально оценить положение дел в нашей стране в области компьютерного моделирования, дать рекомендации для потребителей по выбору среды для решения практических задач (наука, образование, промышленность), и наметить направления дальнейшего развития отечественных визуальных сред компьютерного моделирования. Однако важнейшей задачей является обучение специалистов любых областей использовать среды компьютерного моделирования для решения повседневных задач, и начинать нужно с пропаганды. В статье рассматриваются отечественные среды моделирования сложных динамических систем, сравниваются их возможности с аналогичными зарубежными средами, обсуждаются проблемы обучения компьютерному моделированию школьников и студентов
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The Algorithmic Verifier-Falsifier Games and Game Semantics
Sergei Soloviev
L'Institut de recherche en informatique de Toulouse, Toulouse, France
Abstract
Our goal is to study how the asymmetrical constraints imposed on Verifier and Falsifier may influence Game Semantics. These constraints may concern computational power (complexity of strategies) of the players, their access to information such as history (previous moves), the right to backtrack, etc. We have already studied certain cases where semantics has been modified, e.g., if Verifier is able to compute a universal function for the set of the strategies of Falsifier, then under certain conditions Verifier may win in semantic games based on formulas that are false. Further goal is to develop a more uniform view on semantic modifications induced by asymmetry (in particular, computational asymmetry), using Type Theory as a background
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Algebraic Approaches to the Construction and Analysis of Decision Making Mathematical Models with Preference Relations
Victor V. Rozen
St. Petersburg Electrotechnical University "LETI"
Abstract
The purpose of this report is to demonstrate algebraic and set-theoretic approaches to the construction and analysis of mathematical models of decision-making problems in which the goal structure is formalized by preference relations. Mathematical models of decision-making of the following types are considered in the report: models of decision-making under uncertainty; antagonistic and non-antagonistic games in normal form; games on graphs; models of multi-criteria optimization with a partially ordered set of criteria. Some problems related to mathematical models of decision-making in which the goal structure is formalized by binary preference relations are formulated.
Keywords: preference relation; optimal alternative; equilibrium situation; acceptable outcome.
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On Computer Modeling of Dynamical Systems
Natalia Ampilova
Igor Soloviev
St. Petersburg State University, Russia
Abstract
We discuss the application of computer methods for several problems in the area of dynamical systems, and the using dynamical systems ideas in other subject areas. The main technics is the method of applied symbolic dynamics – symbolic image, which is an oriented graph constructed by a system f and a finite partition {M_i} of the phase space. Vertices of the graph correspond to cells of partition, and there is an edge from i to j ifff(M_i)∩Mj≠∅. This method allowed to solve the problem of localization of invariant sets, estimation of the Morse spectrum and an approximation of an invariant measure of a system.
Approximation of a measure means the construction of a stationary distribution for a Markov chain. As it turned out, this method may be used in digital image analysis. We may construct the stationary flow on the oriented graph for an image, and this flow maximizes so called weighted entropy. This value is considered as a classifying sign.
We also consider probabilistic chains as a model of the distribution of resources and discuss well known Takens' method for reconstruction of attractors.
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Noise-Filtering Numerical Differentiation Schemes: Uniform Probabilistic Error Estimates
Sergey Kolonitskii
St. Petersburg Electrotechnical University "LETI"
Abstract
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Mathematical Education in an Era of Change
Sergei Vakulenko 1,3)
Dmitry Grigoriev 2,3)
John Reinitz 4)
1) Institute of Problems of Mechanical Engineering, Russian Academy of Sciences
2) CNRS, University of Lille, Lille, France
3) St. Petersburg Electrotechnical University "LETI"
4) University of Chicago, United States
Abstract
21-th century will be a century of complexity. One of the biggest challenges for us is to explain the complexity of Biosystems, in particular, evolution to multicellular organisms.
The first steps were done still by E. Schrodinger (1943) and on a mathematical level, by A.M. Turing (1952), he could explain spontaneous emergence of periodical structures like to zebra patterns, however, now we need a much more complete theory, for example, a theory explaining how the Hox genes work.
In this paper, we are focused on this problem. Our approach is based on the following ingredients:
(i) Idea of M. Gromov-A.Carbone on replicative stability in systems supporting homeostasis;
(ii) Model of morphogenesis proposed by J. Reinitz et al.;
(iii) Universal approximations by deep genetic networks;
(iv) Theory of large deviations and stochastic transitions in random dynamical systems
(M. Freidlin, Yu. Kifer et al.).
The mathematical model is a chemical reactor, described as a set of compartments. Inside each compartment, chemical kinetics is described by Michaelis-Menten model with random effects. The structure of this reactor is prescribed by gene networks, which work according to J. Reinitz model. These networks can very slowly evolve.
The main mathematical result is that the probability of spontaneous emergence of complex reactors is not vanishingly small.
In more philosophical words, evolution is a mindless, purposeless mechanism for generating complexity.
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Versions of the “Onion Husk” Algorithm in the Pseudo-Geometric Traveling Salesman Problem with Small Variance
Boris Melnikov 1,2)
Elena Melnikova 3)
1) Shenzhen MSU – BIT University, Shenzhen, China
2) Center for Information Technologies and Systems of Executive Authorities, Presnensky Val street, 19-1, Moscow, 123557
3) Russian State Social University, Moscow
Abstract
We continue to consider the pseudo-geometric traveling salesman problem. Specifically, we are considering several auxiliary algorithms needed to implement different versions of the onion husk algorithm. We have not found in the literature an accurate description of specific versions of algorithms for the geometric version (however, this is not necessary, since it is necessary to implement the original versions for the pseudogeometric version), so we start with the geometric version.
The dimension is indicated in the table of calculation results for rows. All other results (in each cell of the table) are averaged as follows: a certain number of computational experiments were carried out with randomly generated data, then the smallest and largest values obtained were discarded, the arithmetic mean was calculated for the remaining values. At the same time, in the last two columns, the result is given as a percentage: for example, the value 4.17 corresponds to an increase of 1.0417 times.
The "K" column shows the average number of resulting contours for the geometric variant. Column "R" is the ratio of the solution with contours to the optimal solution; in this column, the dimension is considered no more than 79, because exact solutions are required for the table, they are calculated using the method of branches and boundaries (the variant of it that was described in previous publications), because of this, restrictions occur. The last columns are marked with the values of the standard deviation for the pseudo-geometric version. In them we place the calculated ratio of the “fitted to the answer” solution of the pseudo-geometric version (i.e., the solution corresponding to the order of points of the geometric version) to the geometric solution. It is clear that in real algorithms for solving the pseudo-geometric version of the traveling salesman problem, we cannot get such a value; however, we can get it by knowing the generation algorithm, and these values are interesting for describing the solution algorithms themselves.
The obtained results of computational experiments in general approximately correspond to the expected values
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Applications of Tropical Mathematics to Neural Network Architecture
Anna Petrova
Viktoria Kazakevich
St. Petersburg Electrotechnical University "LETI"
Abstract
Using the methods provided by tropical mathematics we can simplify the structure of a neural network, which increases its explainability, without decreasing its accuracy. This paper aims to explore the use of tropical functions in neural networks and compare their efficiency with classical ones. Theoretical framework of tropical mathematics is a semiring with idempotent addition, which is a natural approach to piecewise-linear neural networks, e.g. networks with ReLU activation. Within this approach, piecewiselinear convex function is a tropical polynomial, and general piecewise-linear functions are tropical rational functions. Thus a layer of a neural network with linear preactivation and ReLU activation can be viewed as a vector-valued tropical rational function, which in turn can be represented by two tropical layers. Two tropical layers were implemented, and five tropical architectures were constructed. The models were trained on a heart disease dataset, aiming to determine the presence of heart disease. All models had the same hyperparameters. Each of the models was trained for 100 epochs using Adam and SGD optimizers. The results of the comparison showed that the best accuracy was achieved by a mixed-architecture model using two linear layers. The comparison results showed that the best accuracy was achieved by a mixed-architecture model with two linear layers with a min-layer and a max-layer in between. This accuracy was achieved by using an Adam optimizer. The classical model scored 77.3% and the tropical 77.7%.
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Various Phases of Chaos in Models Described by Equations of Nonlinear Dynamics
Alexandr Liaptsev
Herzen University, St. Petersburg
Abstract
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The Thrive for Simplicity in Science Teaching
Semjon Adlaj
Department of Mathematical Methods for Ensuring Systems Security of Gubkin University, Moscow
A.A. Dorodnicyn Computing Center of the Russian Academy of Sciences, Moscow
Abstract
Mathematics since ancient times “has stood on three whales”, namely,
1) on astronomy, or rather then on astrology, which investigated the path of the Sun across the starry sky (divided into 12 “zodiac signs”) during the sidereal year, the duration of which (366 days, 6 hours and 9 minutes) is 20 minutes longer than the duration of the tropical year (365 days, 5 hours and 49 minutes), divided by “right angles” into four seasons,
2) on geometry as the science of Earth measuring and, in particular, measuring the area of a triangle and a disk on a sphere, and
3) on music theory, which naturally joined number theory and harmonic analysis.
Natural difficulties arise on the path of conveying the fundamental concepts of mathematics and physics. We shall try to establish their causes and thereby make transparent, although not easy, the means of solving modern, quite common troubles of teaching natural sciences. Our contemplations will be supported by illustrative examples, available for broad discussion and suitable for subsequent dissemination, since improving the quality of education, in particular school education, is possible with the collaborative coordination of active conscious actions, partly based on the effective use of computer technology.
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Groupiod and semigroup of finite automaton PRI defined for the only finite language
Meng Lingqian
Lomonosov Moscow State University
Abstract
In order to study the important equivalence relation on a set of formal languages, i. e., the equivalence relation in infinity, we considered the binary coverage relation and used infinite iterative morphism trees, for each of which, after combining its equivalent vertices, a deterministic finite automaton is obtained. Among some similar variants of automata corresponding to an infinite iterative morphism tree, the greatest interest is induced by the so-called primary automaton (PRI automaton), which is defined in a special way for a given pair of finite languages. It is defined on subsets of the set of prefixes of one of these languages, and we consider these subsets as states of the defined automaton. To continue to consider the primary automaton constructed for any pair of finite languages, we define new objects: an automaton constructed for only one finite language and the groupoid and semigroup of transformations corresponding to this automaton. If we “combine by line” the structures arising in the PRI automaton, then we get the result of the transition of each state in the case when the input is some language. As a result, we can study a primary automaton built on the basis of only one language, and this language can be called the basic one. For both the set of states and the input alphabet, we shall use a subset of the set of proper prefixes of the base language, or its prototype obtained using some inverse morphism. Constructed groupoid is researched and a theorem is proved, which is about possiblity to define a semigroup of transformations. We also discuss some algebraic properties of the described semigroup, including simple examples.
Keywords: Algorithmic mathematics, Discrete analysis, Theoretical computer science, Symbolic computations, Formal language, Groupoid, Semigroup, Morphism, Deterministic finite automaton.
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Exact solution of the equilibrium problem of a nonlinear pendulum
Egor Moschjonok
The National University of Science and Technology MISIS, Moscow
Abstract
The simple pendulum has two equilibrium positions, which correspond to two non-linear differential equations.
The purpose of the work is to demonstrate the exact solutions of these differential equations, exhibit their duality and calculate the periods, corresponding to these solutions.
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Review of Expert Systems for Identifying Pathologies of Pregnancy Using Machine Learning Technologies
Darina Ripka
St. Petersburg Electrotechnical University "LETI"
Abstract
Significant progress has been made in the field of medical diagnostics due to the introduction of machine learning methods. This article presents a comprehensive review of studies in which scientists have developed expert systems for diagnosing pregnancy pathologies using various machine learning methods. Emphasizing the critical role of data for training and its preprocessing, the articles reviewed compare the accuracy of various machine learning algorithms in this area. The studies under consideration are primarily focused on building expert systems for diagnosing pathologies and complications associated with pregnancy in order to improve the results of treatment of expectant mothers and their future children. Using a wide range of machine learning approaches, including decision trees, support vector machines, random forest, artificial neural networks, and others, the authors investigated the effectiveness of each algorithm in accurately predicting pregnancy-related problems.
One of the key points of this review is the focus on the quality and diversity of the training data. The researchers emphasized the importance of carefully selected and complete datasets allowing machine learning algorithms to achieve higher diagnostic accuracy. In their work, the authors use for training those parameters that are recognized as indicators of various pathologies by the medical community. In addition, data pre-processing to handle missing values, noise, and class imbalance has played a key role in improving the performance of the developed expert systems. By critically evaluating the methodologies and algorithms used in the reviewed studies, this article provides valuable information for future research and development in this area.
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Research of entropy and redundancy of artificial languages on the example of the language of predicate logic of the first order
Egor Malyutin
St. Petersburg Electrotechnical University "LETI"
Abstract
Currently, there is an urgent question of the optimality of data storage and the efficiency of computing systems. Due to the limited computing resources, it is worth thinking about how to optimize these tasks.
One of the ways to solve the problem may be to optimize the methods of data records, i.e. for memory efficiency, it is necessary to eliminate redundant characters. Entropy helps us determine the informativeness of data, i.e. how much information each data object carries. The smaller the entropy, the more informative the objects are.
It is proposed to investigate the question posed by the example of calculating entropy and redundancy of the first-order predicate logic language in different definitions.
The aim of the work is to develop a method for studying the entropy and redundancy of artificial languages and its application on the example of a standard language of first-order predicate logic.
The following issues are touched upon in the work.
1. Calculation of entropy and redundancy of languages with an unfixed alphabet size;
2. Comparison of entropy and redundancy of predicate languages with natural languages.
According to the above questions, the theory of the construction of the entropy function, based on the works of Shannon K. and Fadeev D.K; the implementation of pseudo-random generation of formulas of the predicate logic language in different definitions; the method of calculating entropy and redundancy of artificial languages on the example of the predicate logic language, are considered.
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WiseTasks. Graphs
Dmitry Zaykov
St. Petersburg Electrotechnical University "LETI"
Abstract
The article discusses various approaches to the use of computer technologies in education. Such technologies are used both for the development and for the control of students’ knowledge. With classical approaches to such computer systems, the student’s research activity is rarely encouraged. The teacher creates a set of tasks, and the student must solve them, while the teacher cannot follow the student’s train of thought. The article also presents the concept of the wise-taskgraph system, which provides the following opportunities for various groups:
• students who develop this it system;
• students who study mathematics and algorithms and create implementations of system modules in the java programming language, from which tasks are compiled. Such students study both programming and algorithms;
• students who use the resource for research activities related to selfformulation and problem solving;
• teachers who are interested in creating constructive problems in graph theory, as well as monitoring the work of students on them;
• students and schoolchildren studying graph theory;
The article also describes the practical experience of using such a system, including the teamwork of students when creating algorithmic modules for it. In conclusion, it tells about further plans for the development of the system and conducting research based on it.
Keywords: digital technologies; mathematics education; pedagogical technologies
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Integrative plots as a way of smoothing the barrier between school and university mathematics
Andrey Lavrenov
St. Petersburg Electrotechnical University "LETI"
Abstract
The repetition of the school course through the study of new mathematical concepts is considered, making the most of the basic ideas that all students have. At the same time, the task of formalizing new concepts is not set, as opposed to studying them on well-chosen basic examples that do not go beyond the school curriculum. Using the example of one lesson conducted as part of a pedagogical experiment, it is shown how a student can see the material being studied in a new way and eliminate the barrier between his school and university ideas.
Keywords: computer tool; adaptation course; integrative story.
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Various Phases of Chaos in Models Described by Equations of Nonlinear Dynamics
Michael Zhegalin
Vladislav Yandrinsky
St. Petersburg Electrotechnical University "LETI"
Abstract
The aim of this study is to create a laboratory for comparing the grammar of an Olympiad participant with a reference grammar.
In this laboratory, the Olympiad compiler should be able to input the reference grammar and create sample exercises based on it. The Olympiad participant should be able to input their own grammar, create sample exercises based on it, and also check any examples for their adherence to the grammars.
Description of the laboratory: The implemented module works as follows: after the user inputs a context-free grammar, control examples are generated using a tree structure. It should be noted that input in Latin characters, numbers, and special symbols is allowed. The module includes input validation, so the user will be notified if an incorrect attempt is made. Each level of the tree is constructed entirely, and then the final words are assembled from the individual levels. Duplicate words are not included in the final array of control examples. All generated examples from both grammars are compared to each other (each with each). If all generated examples match, the comparison ends and the comparison result is displayed on the screen. If not all examples match, the unmatched examples are checked using the Earley algorithm, which allows you to determine whether a word is output in a given context-free grammar.
Conclusions: The module yields a comparison result of the grammars in percentages. The percentage of words generated by the Olympiad compiler's grammar that belong to the Olympiad participant's grammar, and the percentage of words generated by the Olympiad participant's grammar that belong to the Olympiad compiler's grammar.
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Mathematics for Cryptography A Guide to Mathematical Fundamentals of Different Classes of Cryptography Algorithms
Zahra Dorostkar
Skolkovo Institute of Science and Technology, Moscow
Abstract
Cryptography is the practice of securing communication and protecting sensitive data, and understanding the mathematical concepts behind these algorithms is crucial for working with them effectively. Each class of cryptography algorithms is associated with specific mathematical areas that are essential to be familiar with. For example, number theory plays a vital role in public-key cryptography, as it involves the difficulty of factoring large prime numbers. Algebraic structures such as groups, rings, and fields are fundamental in constructing cryptographic systems. Probability theory is used to analyze the security aspects of these systems. Additionally, knowledge of computer science topics like data structures, algorithms, and programming is important for implementing and utilizing cryptography algorithms. By exploring the related mathematical areas for each class of cryptography, individuals can gain the necessary prerequisites to engage effectively with these algorithms and contribute to the ongoing research and development in this important field. This article aims to provide insight into the mathematical foundations underlying different classes of cryptography algorithms. There are numerous materials about different cryptographic algorithms and systems and surely there are infinite materials for different areas of mathematics. However, while you want to work on some cryptographic algorithm, you are surrounded by many materials that may or may not related to your goal algorithm and you will never know until you spend lots of time on some unnecessary ones. So in this work, the most related topics in mathematics that an individual would be better to be familiar with for working on each algorithm are mentioned and shortly discussed.
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Computing of Tropical Sequences Somos-4 and Somos-5 in Gfan package
Farid Mikhailov
St. Petersburg Electrotechnical University "LETI"
Abstract
The main objective of this work is to study tropical recurrent sequences Somos-4 and Somos-5. For a set of tropical recurrent sequences, D. Grigoriev put forward a hypothesis of stabilization of the maximum dimensions of solutions to systems of tropical equations given by polynomials, which depend on the length of the sequence under consideration. The validity of such a hypothesis would make it possible to calculate the dimensions of these solutions for systems of arbitrary length. The main purpose of this work is to compute finite tropical sequences Somos-4 and Somos-5 using the Gfan package and to test the Grigoriev hypothesis.
Keywords: tropical recurrent sequence, tropical prevariety, Gfan package, Somos sequence
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Competitiveness Function in the Generalized Model Abrams-Strogatti in the Case of an Inconstant Community Numbers
Alexander Medvedev
Lobachevsky State University of Nizhny Novgorod
Abstract
The purpose of this work is to construct competitiveness functions for the modified Abrams-Strogatti model describing the competition of several languages. Materials and methods. The paper uses a modified Abrams-Strogatti model for the case of a non-constant community size, in which languages are characterized and differ in terms of prestige, volatility, and the initial number of native speakers. To restore the fitness function, a ranking method is used, which boils down to dividing the language choice strategies into two classes “the first language displaces the second” and “the second displaces the first”. Results. The article presents a technique in which the competitiveness function is first restored in a general form, as a function depending on the parameters of the language. In this case, the identification ofthe values of the coefficients of the function is carried out based on the processing of the available data on the dynamics of the model. Then the final formulas of the competitiveness function for the language dynamics model with known parameters are derived. An analysis of the values of the competitiveness functions is carried out and, on its basis, a forecast is made for the further development of dynamics. Conclusion. The proposed method for constructing the competitiveness function is quite general and can be applied to a wide range of models that describe the dynamics of populations. The forecast made on the basis of the constructed competitiveness functions is in good agreement with the empirical data.
Keywords: language competition, language dynamics, selection, language preservation, competitiveness function, fitness function, selection criterion, selection processes, mathematical model, ordinary differential equations.
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