Mathematical Examples Research Articles

The Mathematical Safe Problem and Its Solution (Part 2)

Introduction. The problem of mathematical safe arises in the theory of computer games and cryptographic applications. The article considers numerous variations of the mathematical safe problem and examples of its solution using systems of linear Diophantine equations in finite rings and fields. The purpose of the article. To present methods for solving the problem of a mathematical safe for its various variations, which are related both to the domain over which the problem is considered and to the structure of systems of linear equations over these domains. To consider the problem of a mathematical safe (in matrix and graph forms) in different variations over different finite domains and to demonstrate the work of methods for solving this problem and their efficiency (systems over finite simple fields, finite fields, ghost rings and finite associative-commutative rings). Results. Examples of solving the problem of a mathematical safe, the conditions for the existence of solutions in different areas, over which this problem is considered. The choice of the appropriate area over which the problem of the mathematical safe is considered, and the appropriate algorithm for solving it depends on the number of positions of the latches of the safe. All these algorithms are accompanied by estimates of their time complexity, which were considered in the first part of this paper. Conclusions. The considered methods and algorithms for solving linear equations and systems of linear equations in finite rings and fields allow to solve the problem of a mathematical safe in a large number of variations of its formulation (over finite prime field, finite field, primary associative-commutative ring and finite associative-commutative ring with unit). Keywords: mathematical safe, finite rings, finite fields, method, algorithm.

A novel interval linear programming based on probabilistic dominance

This study investigates a novel interval linear programming based on probabilistic dominance. Firstly, the definition of interval linear programming is briefly reviewed. Then a new interval linear programming model is presented based on probabilistic dominance. The probabilistic dominance index treats the intervals as uniformly distributed variables and the interval inequality relation is further defined by probability. To deal with non-linearity in probabilistic dominance index, sequential quadratic programming is used to solve the problem and the performance measure approach is proposed to overcome the convergence difficulties. The determination and sensitivity analysis of the target performance measure are discussed to assess the sequential quadratic programming algorithm. Meanwhile, the extension of the proposed method to fuzzy interval linear programming is discussed. Furthermore, the proposed method is applied to the design of the plane truss structure with interval parameters. Finally, the effectiveness and rationality of the developed method are demonstrated by two mathematical examples and one interval parametric plane truss structure optimization example.

Numerical analysis of nonlinear variable fractional viscoelastic arch based on shifted Legendre polynomials

An innovative numerical procedure for solving the viscoelastic arch problem based on variable fractional rheological models, directly in time domain, is proposed and investigated. First, the nonlinear integral‐differential governing equation is established according to the variable fractional constitutive relation and geometrical relationship. Second, the nonlinear integral‐differential governing equation is transformed into algebraic equations and solved by using the shifted Legendre polynomials. Furthermore, the accuracy and effectiveness of the algorithm are verified according to the mathematical example. A small value of the absolute error between numerical and accurate solution is obtained. Finally, the dynamic analysis of viscoelastic arch is investigated to determine the displacement at different times and positions. The displacement of the viscoelastic arch is compared under various loading (uniformly distributed load and linear load). The displacement of the viscoelastic arch of different materials under the same load conditions is also investigated. The results in the paper show the efficiency of the proposed numerical algorithm in the dynamical analysis of the viscoelastic arch.

An Efficient Strategy for Accurate Detection and Localization of UAV Swarms

Unmanned aerial vehicle (UAV) swarms have shown great potential for Internet of Things (IoT). Meantime, its malicious use may cause huge threat to the national security. UAV swarms show the characteristic of high density which poses formidable challenges to radar resolution in the defense of critical areas. In this article, we consider a radar equipped with the coprime array, and then, use the coherent long-time integration (LTI) technique and gridless sparse technique to detect and localize UAVs in a swarm. This strategy takes full account of advantages of the coprime array, coherent LTI technique, and gridless sparse technique, i.e.: 1) the coprime array can provide a larger array aperture than the uniform linear array with the same number of array elements to relieve the stress of the gridless sparse technique and 2) the combination of coherent LTI technique and gridless sparse technique can maximize their advantages and make up for their shortcomings. By mathematical analyses and extensive numerical examples, we show the superiority of the proposed strategy in terms of accurate detection and localization of UAV swarms.

Reliability-based design optimisation of geotechnical systems using a decoupled approach based on adaptive metamodels

ABSTRACT Geotechnical engineering analysis and design involves considerable uncertainties. Reliability-based design optimisation (RBDO) intends to minimise an objective function such as material quantity or cost subject to design requirements specified by probabilistic constraints. In this work, the RBDO of geotechnical systems is studied using a decoupled approach based on adaptive metamodels. To improve the traditional double-loop structure of RBDO, the proposed approach decouples the numerical reliability analysis loop from the design optimisation loop. A metamodeling method based on augmented radial basis function (ARBF) is adopted to create approximate functions of the reliability indices, so that the design optimisation is performed based on the metamodels of the probabilistic constraints. To improve the accuracy of predicting the reliability indices, an adaptive technique of the ARBF metamodels is applied. The optimal point in one design iteration is treated as an additional sample point for updating the metamodels of the reliability indices. The accuracy of the metamodels is progressively enhanced through this dynamic process, especially in the neighbourhood of the optimal point. Mathematical and geotechnical engineering examples are solved and numerical results are presented. The proposed design optimisation framework works well and is a useful alternative for solving RBDO of geotechnical engineering problems.

Adaptive Multi-Level Search for Global Optimization: An Integrated Swarm Intelligence-Metamodelling Technique

Over the last decade, metaheuristic algorithms have emerged as a powerful paradigm for global optimization of multimodal functions formulated by nonlinear problems arising from various engineering subjects. However, numerical analyses of many complex engineering design problems may be performed using finite element method (FEM) or computational fluid dynamics (CFD), by which function evaluations of population-based algorithms are repetitively computed to seek a global optimum. It is noted that these simulations become computationally prohibitive for design optimization of complex structures. To efficiently and effectively address this class of problems, an adaptively integrated swarm intelligence-metamodelling (ASIM) technique enabling multi-level search and model management for the optimal solution is proposed in this paper. The developed technique comprises two steps: in the first step, a global-level exploration for near optimal solution is performed by adaptive swarm-intelligence algorithm, and in the second step, a local-level exploitation for the fine optimal solution is studied on adaptive metamodels, which are constructed by the multipoint approximation method (MAM). To demonstrate the superiority of the proposed technique over other methods, such as conventional MAM, particle swarm optimization, hybrid cuckoo search, and water cycle algorithm in terms of computational expense associated with solving complex optimization problems, one benchmark mathematical example and two real-world complex design problems are examined. In particular, the key factors responsible for the balance between exploration and exploitation are discussed as well.

Understanding the choice and use of examples in mathematics teacher education multilingual classrooms

Understanding the choice and use of examples in mathematics teacher education multilingual classrooms

Normalized neighborhood component feature selection and feasible-improved weight allocation for input variable selection

In design optimization, as the number of input variables increases, the convergence rate of optimization tends to decrease, and the number of function calls and design change costs tend to increase. Neighborhood component feature selection (NCFS) was adopted to select significant input variables. However, the parameter determination process of the NCFS incurs a high computational cost and weakens robustness. Therefore, this study proposes a normalized NCFS (nNCFS) by normalizing scales between mean loss and regularization terms via the initial dataset Additionally, in the case of a multi-response system, complex decision-making processes that involve the allocation of weights for multiple responses are required. It is possible to allocate weights by using conventional methods such as the analytic hierarchy process and entropy methods. However, the analytic hierarchy process method is highly influenced by the designer’s subjectivity, and the entropy method is unable to consider a design optimization problem. Accordingly, the feasible-improved weight allocation (FIWA) method is now proposed by considering a design optimization problem objectively. Comparing the NCFS with the nNCFS through mathematical examples, we found that the nNCFS significantly improved the computational cost and robustness. Moreover, the FIWA method selected significant input variables that yielded feasible and improved designs. Then, the nNCFS and the FIWA methods were applied to the design of the body-in-white of a vehicle. The significance of input variables was analyzed using the nNCFS, and feasible and improved designs were obtained on the basis of the significant input variables selected using the FIWA method.

Optimal extraction of echelle spectra: Getting the most out of observations

Context. The price of instruments and observing time on modern telescopes is quickly increasing. Therefore, it is worth revisiting the data reduction algorithms to extract every bit of scientific information from available observations. Echelle spectrographs are typical instruments used in high-resolution spectroscopy, but attempts to improve the wavelength coverage and versatility of these instruments has resulted in a complicated and variable footprint of the entrance slit projection onto the science detector. Traditional spectral extraction methods generally fail to perform a truly optimal extraction when the slit image is not aligned with the detector columns and, instead, is tilted or even curved. Aims. Here, we present the mathematical algorithms and examples of their application to the optimal extraction and the following reduction steps for echelle spectrometers equipped with an entrance slit that is imaged with various distortions. The new method minimises the loss of spectral resolution, maximises the signal-to-noise ratio, and efficiently identifies local outliers. In addition to the new optimal extraction, we present order splicing and a more robust continuum normalisation algorithm. Methods. We developed and implemented new algorithms that create a continuum-normalised spectrum. In the process, we account for the (variable) tilt or curvature of the slit image on the detector and achieve optimal extraction without prior assumptions about the slit illumination. Thus, the new method can handle arbitrary image slicers, slit scanning, and other observational techniques aimed at increasing the throughput or dynamic range. Results. We compare our methods with other techniques for different instruments to illustrate the superior performance of the new algorithms compared to commonly used procedures. Conclusions. Advanced modelling of the focal plane requires significant computational effort but it has proven worthwhile thanks to the retrieval of a greater store of science information from every observation. The described algorithms and tools are freely available as part of our PyReduce package.

PEDAGOGICAL SKILLS AND (E-)TEACHING - THE EXAMPLE OF MATHEMATICS AND READING

Problems faced by students with learning difficulties vary considerably for each student per subject and per educational level, having a common distinctive feature: they relate to areas in the handling of mathematical calculations and in the production of written speech. This brief report emphasises pedagogical skills, techniques and behaviour management as a cross-learning process in mathematics and reading in classroom. The paper can provide highlights in (e-)teaching through the example of mathematics and reading.
 
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System reliability-based design optimization with interval parameters by sequential moving asymptote method

Reliability-based design optimization (RBDO) offers a powerful tool to deal with the structural design with heterogeneous interval parameters concurrently. However, it is time-consuming in the practical engineering design. Therefore, a novel sequential moving asymptote method (SMAM) is proposed to improve the computational efficiency for convex model in this study, in which the nested double-loop optimization problem is decoupled to a sequence of deterministic suboptimization problems based on the method of moving asymptotes. In addition, the sensitivity of reliability index is derived, so the finite difference for the nested optimization loop can be avoided to tremendously improve the computational efficiency. Then, the accuracy of the SMAM is proved based on the error analysis. Furthermore, the Kreisselmeier-Steinhauser (KS) function is used to assemble the multiple constraints to deal with the parallel and series RBDO problems. One benchmark mathematical example, three numerical examples, and one complex civil engineering example, i.e., tower crane, are tested to demonstrate the efficiency of the proposed method by comparison with other existing methods, and the results indicate that SMAM offers a general and effective tool for non-probabilistic reliability analysis and optimization.

Guiding Questions for Selecting Mathematical Examples

Examples are an essential part of mathematics teaching and learning, used on a daily basis to teach and practice content. Yet, selecting good examples for teaching is complex and challenging. This article presents ideas to consider when selecting examples, drawn from a research study with algebra 2 teachers.

Psychophysiological features of solving mathematical examples with fractions as adaptability markers to the cognitive load

This study is devoted to identifying a solution algorithm for standard fractions as one of the tasks that allow investigating the level of human adaptability to the cognitive load. The influential factor for a successful solution was the number of stages, and for an unsuccessful one - their duration. It was revealed that the solution success/failure correlated with the spectral power values and ratio in the theta- and alpha-diapasons of the EEG. The successful solution is accompanied by the maintenance of a stable level of theta-diapason and desynchronization in the alpha-diapason. The unsuccessful solution is characterized by an increase in the theta-diapason power, its shift to the frontal zones, and a lack of alpha-desynchronization.

ИНТЕГРАЦИЯ В ОБУЧЕНИИ МАТЕМАТИКЕ И ИНОСТРАННОМУ ЯЗЫКУ НА УРОКАХ В СРЕДНЕЙ ШКОЛЕ

This work considers the integrated teaching of mathematics and a foreign language as a process of strengthening interdisciplinary connections. Modern education presupposes the formation of a comprehensively developed personality capable of competing in conditions of varying complexity. Inter disciplinary approach allows you to create such conditions for the development of a multifaceted and successful personality. The main task of the work is the selection of a certain mathematical terminology (according to the themes of the school curriculum), its translation into English and the creation of guidelines for teachers, which reflect the basic principles, methods and techniques of organizing integrated activities. The practical part contains mathematical problems and examples that allow you to work out the lexical material in practical application. It is assumed that the study of mathematics in a foreign language will deepen the knowledge of students in both subject areas and will contribute to the development of thinking, mental flexibility and creativity

Ideological and political assessment system based on data visualization

Data visualization is a very important research topic of current computer science and technology. It is a theoretical method and technology that uses the technology of computer processing massive and complex data to transform the data results produced in daily life and production into graphics or images to show to the information receiver, and carry out human-computer interaction. Data visualization is a research hotspot in the field of Computer Science in recent years. Data visualization technology is mainly used in Applied Mathematics. With the continuous development of data visualization technology, the application of data visualization technology will be widely used in daily life. Higher vocational colleges, as educational institutions to cultivate high-level skilled and application-oriented talents, should strengthen the foundation, broaden the occupation, improve the specialty, and strengthen the ideological and political construction. Therefore, the ideological and Political Theory Teachers in higher vocational colleges should practice the education policies of the party and the state, ensure the correct direction of Ideological and political education, and actively explore and re select the ideological and political theory line through a variety of evaluation methods. With the continuous development and replacement of high and new technology, the teaching of higher vocational colleges should also keep up with the times. Data visualization technology should also be integrated into the ideological and political assessment of students. This paper introduces the data visualization technology and methods, puts forward some key and difficult points and breakthrough points in the application of data visualization in the ideological and political assessment system, and introduces several mathematical model algorithm evaluation examples, hoping to help the data visualization of the ideological and political assessment system in higher vocational colleges.

Development of moral and ethical qualities of primary schoolchildren through a reflective environment (through the example of mathematics and music lessons)

Introduction. The development of pedagogical science requires the introduction of new technologies, forms, and methods of influencing children’s microcommunity. At the present stage of social development, the need for moral and ethical education of the younger generation increasingly arises as the assimilation of external requirements for behavior in accordance with their own convictions, which are actively developing in primary school age. In this regard, it is relevant to study the possibilities of using the reflective environment in multidirectional mathematics and music lessons. The purpose of the study was to study the development of moral and ethical qualities as a result of the organization of a reflective environment at mathematics and music lessons for children of primary school age. Materials and methods. The experimental study was conducted from September 2016 to May 2020. The study was conducted in the Municipal Educational Institution “General Secondary School No. 22” and the Municipal Budgetary General Education Institution “General Secondary School No. 28” in Orekhovo-Zuevo, Moscow Region. Two hundred and twelve schoolchildren studying at these schools took part in the study. The χ2-Pearson statistical test was used for statistical processing. Results. The results obtained during the ascertaining stage of the experiment made it possible to identify and characterize a number of shortcomings in the cognitive, emotional, and behavioral components of the moral and ethical development of primary schoolchildren. In accordance with the need to develop moral and ethical qualities, an appropriate reflective environment was organized in the experimental classes of both schools. The development of moral and ethical qualities of primary schoolchildren through the reflective environment was successful, which is confirmed by statistically significant differences at the control stage of the experiment (χ2 = 42.27>χ20.05). Discussion of results and conclusion. The need for the formation of reflexive abilities in primary school age necessitates the creation of a special environment, the use of effective author’s techniques. It is advisable to include the use of reflective practice-oriented situations in the structure of mathematics and music lessons.

Reservoir history matching by ensemble smoother with principle component and sensitivity analysis for heterogeneous formations

Ensemble-based data assimilation methods have been widely investigated and applied for inverse problems of fluid flow in porous media during the past decades. Among these methods, the ensemble Kalman filter and the ensemble smoother are probably the most popular in history-matching applications. For large-scale problems, the ensemble size is limited to the computational resources in running the forward simulation, and is usually much smaller than the total number of gridblocks. In this case, the ensemble forms a reduced-order subspace, and principle component analysis is suggested by retaining the leading eigenvectors after truncation.In this paper, we propose to improve the ensemble smoother by proper orthogonal decomposition with optimal parameter selection. The motivation behind this method is that the estimation would be more accurate if the subspace spanned by the ensemble favors the dominant components. Specifically, the principle component analysis is first applied to obtain the eigenvalues and orthogonal eigenvectors. Then, a number of eigenvectors are selected according to importance from sensitivity analysis, rather than the eigenvalues. Finally, the ensemble smoother is implemented using these eigenvectors.The proposed method is illustrated by a simple mathematical example, and tested in a 2D single-phase and a 3D multiphase flow problem. The numerical results show that the traditional ensemble smoother may provide poor results in some ill-posed inverse problems, because some important features are lost in the randomly generated realizations. The principle component analysis alleviates this issue by sorting the eigenvectors based on energy, but it may still produce unsatisfactory results. The combination of sensitivity analysis and principle component analysis provides more accurate matching and prediction than those obtained from the previous two methods. The local sensitivity analysis is preferred since it is much more efficient than the global sensitivity analysis, and its accuracy is validated in the case studies.

Are There Mathematical Hinges?

Abstract In this paper I argue that, contrary to what several prominent scholars of On Certainty have claimed, Wittgenstein did not maintain that simple mathematical propositions like “2 × 2 = 4” or “12 × 12 = 144,” much like G. E. Moore’s truisms, could be examples of hinge propositions. In particular, given his overall conception of mathematics, it was impossible for him to single out these simpler mathematical propositions from the rest of mathematical statements, to reserve only to them a normative function. I then maintain that these mathematical examples were introduced merely as objects of comparison to bring out some peculiar features of the only hinges he countenanced in On Certainty, which were all outside the realm of mathematics. I then close by gesturing at how the distinction between mathematical hinges and non-hinges could be exemplified and by exploring its consequences with respect to (Wittgenstein’s) philosophy of mathematics.

Adam Smith as an Example of Samuelson’s 1952 Point That Mathematics Can Be Written Out in the English Language: On Smith’s Anti-Utilitarianism Based on the Differences Between Precise and Imprecise Probability Presented in 1776 in the Wealth of Nations

In the Wealth of Nations in 1776, Smith gave two clearly worked out mathematical examples involving a comparison- contrast examining the concepts of precise probability (exact, definite, linear, numerical) and imprecise probability(inexact, indefinite, nonlinear, non numerical) that must incorporate uncertainty, which means there is missing or unavailable evidence that is not available to the decision maker at the time that he must make a choice between two or more different alternative options or alternatives. Smith’s analysis is carefully presented on pp. 106-113 and pp. 419-423 of the Modern Library edition of the Wealth of Nations edited by Cannon with the foreword by Max Lerner. It is interesting that there has not been a single academic economist, philosopher, historian, sociologist, psychologist, political scientist, social scientist or decision theorist in the 244 years since Smith published the Wealth of Nations in 1776 to note this fact. The fact that Smith believed that the use of precise probability, as advocated by Jeremy Bentham, who was Smith’s great intellectual opponent and adversary, was possible only under very special conditions, explains why Smith rejected utilitarianism as an ethical system and foundation for the science of economics-the requirement for precise probabilities and precise outcomes was, in general, not possible, due to the fact of missing or unavailable relevant information data ,knowledge or evidence that the decision maker would need to estimate the consequences in the future of his present decision to act. An example of this severe misunderstanding and confusion of Smith’s approach to decision making can be seen, for just one instance, in the 2016 paper by Hollander Thus, discussions about whether Smith was a utilitarian ,partly a utilitarian, whatever that may mean, not a utilitarian or anti-utilitarian are all besides the point once it is realized that Smith completely rejected the additivity and linearity of the probability calculus upon which Bentham based his utilitarianism on, that all men can calculate. Smith realized that Bentham’s belief in the ability to calculate future consequences was extremely limited. Apparently, economists can’t read the English Language that Smith used to express his mathematical analysis of his approach to decision making in the Wealth of Nations. The belief that Smith did not use mathematical analysis in the Wealth of Nations can only be a conclusion reached by economists who are themselves mathematically illiterate, inept, innumerate or severely confused about how mathematical arguments and analysis can be presented. This leads to the conclusion that the M. Friedman, G. Becker, and G. Stigler school of economics, that is taught at the University of Chicago, can have nothing to do at all with Adam Smith’s Wealth of Nations because they base all of their economic analysis on precise probability, which is an approach that is identical to that expressed in the original work of Jeremy Bentham in chapter IV of the Principles of Morals and Legislation (1787).

Undesired Cross Terms [Lecture Notes

This lecture note briefly describes two examples in mathematics on how and why cross terms are disliked: orthogonal space-time block code (OSTBC) from orthogonal designs and free probability theory. The first example shows that the elimination of cross terms can significantly simplify the optimization that has many practical applications in, for instance, multiple antenna systems in wireless communications. The second example gives an intuitive and/or fundamental understanding of free probability theory, which has important applications in random matrix theory.