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A recommendation algorithm is a specific application of data mining technology, which is an important means to expand the dissemination of Civics education in cyberspace. The elbow method is employed in this paper to determine the best number of clusters to cluster the Civics course evaluation dataset and finish preprocessing large-scale data. Aiming at the problem of different importance of rating bias between items, the traditional Slope One algorithm is optimized by comprehensively considering the relationship between the number of common ratings and the similarity between items and the prediction results. The optimized Slope One algorithm is used to perform one round of prediction and fill the scoring matrix to solve the data sparsity problem and improve prediction accuracy and stability. The optimized algorithm reduces by 8.56% compared to the user-based collaborative filtering algorithm, 5.83% compared to the item-based collaborative filtering algorithm, and 5.25% compared to the content-based recommendation algorithm, with a significant improvement in accuracy, recall, and F1 value. The four hypothesis validations indicate that the diversity, personalization, and accuracy of resource recommendations all have positive effects on the dissemination effect of civic education. The recommendation algorithm’s dissemination effect score in this paper is 3.781, which is significantly superior to the dissemination effect of Civic Education.

In order to realize the quantitative analysis of the digital literacy education strategies of Chinese and foreign youth, this paper constructs the evaluation index system with the help of a hierarchical analysis algorithm and comprehensive fuzzy evaluation to explore the differences between the digital literacy of Chinese and foreign youth, and to compare the differences between the digital literacy education strategies of Chinese and foreign youth. After analyzing and discussing, this paper identifies 10 expert panelists, conducts two rounds of the Delphi method survey on the preliminary expert consultation questionnaire, and finally determines the evaluation index system based on the standard of arithmetic mean and coefficient of variation. The evaluation indexes are divided into a hierarchical analysis structure. The judgment matrix of evaluation indexes is constructed according to the 1-9 scale method. The hierarchical single sorting and consistency tests are carried out to derive the weight value of each evaluation index. Based on the system of evaluation indexes, the evaluation factors are composed into a collection of factors, and the semi-trapezoidal distribution function is used to determine the subordination degree of the indexes. The principle of calculation of fuzzy comprehensive evaluation is given to realize the fuzzy comprehensive evaluation model. Construction. Using the above model to evaluate the digital literacy of Chinese and foreign youths, the evaluation result of the digital literacy of youths in country M is strong, with an affiliation degree of 3.9444, and the evaluation result of the digital literacy of youths in China is medium, with an affiliation degree of 3.1581. The difference between the digital literacy of Chinese youths and that of youths in country M is large. The education strategy determines the digital literacy of youths to a large extent. The results of the analysis of the evaluation are summarized in terms of three dimensions of distance education, feedback and communication, and evaluation system to explore the difference between Chinese and foreign youth digital literacy education strategies.

This paper proposes a data-based gray preprocessing neural network optimization model based on the principle of the BP neural network model and its optimization method. SPSS statistical software was used for statistical analysis to compare the student physical education test data of S college students from 2019 to 2023. The GA-BP optimization prediction model is applied to generate and process the raw data of 5-year sports tests to find the law of systematic changes and generate the data sequence with strong regularity so as to predict the condition of the future development trend of things. The results show that the measured and actual numbers have a high rate of conformity, and the maximum error is 0.67%, which is high accuracy. The overall speed and strength of boys and girls in sports is on the rise. The overall endurance level of boys is increasing while the overall endurance level of girls is decreasing. In the future, endurance training should be strengthened as a way to improve the continuing trend of low endurance quality among students.

Traditional art has a deep cultural heritage and rich connotation, combining artificial intelligence technology with traditional aesthetics as a way to realize the innovation of traditional painting education mode. In this paper, starting from the performance of traditional painting’s aesthetic value, a personalized intelligent painting teaching mode is constructed based on the visual art teaching method and combined with artificial intelligence. The CCME-GAN model is introduced for the construction of a personalized painting learning generation model, and the BERT-LDA model is established by combining the BERT model and the LDA model for the intelligent evaluation of personalized painting works. Xinjiang folk artworks were selected as the research data source to carry out the validation of personalized painting work generation and assessment, and teaching comparison experiments were designed to analyze the teaching effect of the personalized intelligent teaching mode. The subjective evaluation score of Xinjiang folk art paintings generated by the CCME-GAN model was 9.11, and among the nine clustering results of painting work assessment, the eigenvalues of the theme of painting evaluation of Cluster 1 range between [0,5.5]. At the end of the teaching experiment, students’ creativity scores increased from 2.74±0.37 before the experiment to 4.63±0.26. Relying on artificial intelligence technology can enhance the teaching effect of traditional painting methods and improve students’ painting levels.

By integrating a heart rate monitoring module and a step frequency monitoring module, this study aims to use informatization technology to monitor the exercise level of students in public physical education courses and to adjust the teaching methods of public physical education courses based on informatization feedback. A school had 332 students divided into an experimental and control group for comparative experiments. Based on the results of students’ exercise data monitoring by heart rate oximetry module and acceleration sensor, it was found that the exercise density of students in the experimental group (73.3%) was much higher than that of students in the control group (54.4%), which means that the students’ exercise status was better under the informatization teaching mode. During the experimental period, there were significant differences in the average heart rate and MVPA percentage time between the experimental group and the control group students (p

In order to build an education information platform to support “industry-education dialogue” and at the same time serve the teaching and research of industry-education integration. In this paper, we use big data technology to construct a system of education management and resource optimization, supporting education construction in the context of industrial transformation and upgrading. For larger-scale WSNs, an energy-efficient data collection algorithm based on non-uniform clustering is proposed. To deal with large-scale optimization problems, a hybrid algorithm of genetic algorithms and particle optimization hierarchical collaboration is introduced. Then, the algorithm is applied to cluster WSNs, an adaptation function is constructed with the goal of equalizing energy consumption, and a data collection algorithm based on non-uniform clustering is proposed. The Apriori algorithm, which is based on the interest degree model, is used to mine the course evaluation data for the system design in the university industry-teaching fusion model. This system design does both course evaluation and resource optimization. The system’s total evaluation score is high, at 90.84 points. It also improves the efficiency of cooperation between the education industry and schools. It proves that the system’s design is effective and high-level and has a positive effect on teaching management and resource optimization in education.

With the development of ESL education in recent years, real-time language translation has gradually become a research hotspot in the field of natural language communication. In this paper, the data processing technology of real-time translation is introduced, and then a Transducer (RNN-T) model based on real-time language translation is constructed based on the BPE subword segmentation method under the condition of probability gradient calculation algorithm and transcription network. Secondly, this model determines and tests the content involved in the process of real-time language translation and finally obtains results to improve the quality of real-time language translation. The experimental results show that the accuracy of character translation among the six language models is 85.8%, 86.2%, 88%, 93.4%, 94.03%, and 94.36%, respectively. In addition, the error rate of inserting, deleting, and replacing characters is < 20%. Compared with the other four models, the word error rate of the RNN-Transducer algorithm is reduced by 1.59%, 3.94%, 3.86%, and 9.01%, and the RTF is reduced by 3.91%, 10.96%, 10.09%, and 7.94%, respectively, which reflects the superiority of the RNN-Transducer method proposed in this paper.

In this paper, we first digitally design the urban management resource base and then construct a dynamic bargaining resource allocation model based on super-marginal analysis with corner point equilibrium and dynamic bargaining. In the context of sustainable development, an intelligent optimization model is used to analyze the problem of water resource allocation and optimize urban traffic infrastructure paths in order to achieve carbon reduction targets. Finally, six recommendations are made for optimizing and managing urban resources in the context of sustainable development. The results of the prioritization coefficients of each subarea are 0.186, 0.164, 0.153, 0.158, 0.153, 0.0769, and 0.111, respectively, and the results of the allocation of the optimal solution for the social objective, the optimal solution for the economic benefit, the optimal solution for the ecological benefit, and the optimal solution for the comprehensive benefit are very close to each other. In addition, the rates of water shortage for agricultural, industrial, and other uses. are 3.29%, 4.85%, 3.94%, 2.92%, and 3.87%, respectively. The variation range of the average passenger load factor and departure frequency under the carbon reduction target is between ±40%, the average vehicle area varies in the range of ±20%, and the variation range of the energy consumption factor and CO2 emission factor is -40%-0%. The intelligent optimization model for urban resource management proposed in this paper has practical significance for urban water resource planning and transportation structure optimization.

Tourism practice teaching is an important means to cultivate high-quality skilled tourism talents. When virtual reality is in tourism practice, teaching undoubtedly shows a beautiful blueprint for modern tourism education. The article demonstrates the advantages of the application of VR technology, designs a VR practical training system for tourism education, and develops VR resources and interface interaction design for tourism education. Based on the binocular stereo imaging model to obtain maximum and minimum parallax, and combined with the adaptive weight PSO algorithm for optimizing tourism virtual scene viewpoints. Then, a comparison test for teaching was designed using the VR training system for tourism education. By analyzing students’ eye movement data to understand the viewpoint optimization effect of the AW-PSO algorithm, we also verified their learning performance, learning experience, and academic mood changes. The edge strength and clarity of the virtual scene obtained by the AW-PSO algorithm were 67.06 and 14.71, respectively, and the theoretical score of the experimental class students was 83.15, which was 9.06% higher than that of the control class students. Based on tourism education, VR practical training systems can effectively enhance students’ learning interest and academic mood for tourism education courses and help develop high-quality tourism education courses in colleges and universities.

The cultivation of talents in interdisciplinary education mode is increasingly important, both in terms of policy support and industry demand. In this paper, we identify the core requirements for interdisciplinary talent cultivation and establish the structure of the interdisciplinary education course system. The implementation path of higher vocational interdisciplinary talent cultivation is divided into four directions: building talent cultivation mode, strengthening the teacher team, innovating school-enterprise cooperation, and building capacity mechanisms. For the personalized education of students, the XDeepFM algorithm is used to establish the course classification model and realize the personalized course resource integration recommendation. Evaluating the effect of talent cultivation under the interdisciplinary education model, there is a significant difference in learning psychology between students with different development plans. In the evaluation of the effect of learning activities, the results of the second round of interdisciplinary education action research are higher than those of the first round, and the difference between the two actions is 7.515. The interdisciplinary education talent cultivation model designed in this paper has shown its initial success.

The selection of college grants is affected by many factors, but some of the influencing factors cannot be quantitatively judged at present. In order to be more fair and reasonable to the students in need of financial assistance for the review and evaluation, this paper introduces the fuzzy comprehensive evaluation method into the process to establish a fuzzy hierarchical analysis based on the college national scholarship evaluation model, the article finally through a specific example of the application of this paper based on the fuzzy comprehensive evaluation of the optimization model of the bursary system. The affiliation degree of the fuzzy comprehensive evaluation-based scholarship system constructed in this paper is (0.536,0.323,0.130,0.014), and the article concludes that Zhang’s moral education status is very good, and he is qualified to receive the scholarship.

The rapid development of information technology and multimedia education in today’s world has led to important changes in the delivery of ideological values in the process of art education. In this paper, according to the form of multimedia presentation of knowledge, a multimedia learning cognitive model is used to study the process of multimedia transmission of Civic and political values. A diverse and efficient recommendation algorithm is proposed, and a recommendation system for Civics and Politics courses is constructed based on the algorithm. The method mines the set of high-efficiency items related to the relationship between courses and categories through a conceptual hierarchical tree and introduces a diversity upper bound function to optimize the mining process. Thus, rules are efficiently generated in the efficiently used generalized itemsets. A collaborative filtering algorithm between users and course types is used to find the neighbors that best match the target users. The results show that the four indicators of this paper’s recommender system are better than other recommender methods. The F-measure of this paper’s system is 0.45, which indicates that this paper’s method can produce the best recommendation effect of the Civics courses. The students in the process of art education can get the Civics course recommendation that is more suitable for them, which improves the efficiency of the Civics value transmission among students.

The advancement of information technology in the digital era has raised the standards for the content and methods of civic and political education in colleges and universities. This paper utilizes knowledge mapping technology as its foundation, integrates resources from civic and political education in colleges and universities, and builds a knowledge mapping system for civic and political courses. This system enables the digital and intelligent development of civic and political education in these institutions. The entities of Civics and Political Science courses are obtained through the Hidden Markov Model and BiLSTM-CRF model; semi-structured and unstructured data are crawled from web pages to constitute the entity relations of the Civics and Political Science domain, and the knowledge of Civics and Political Science courses is fused by using the double-layer filtering method. In addition, this paper constructs a Civics knowledge mapping system, which includes data modules and application modules, to visualize the complexity of the Civics knowledge system. This system is then applied to the practice of Civics teaching in colleges and universities. Based on the empirical analysis of the TAM model, it is concluded that the mean value of five potential variables (PEU, PUA, TU, BI, and SU) of the Civic Political Knowledge Mapping System constructed in this paper is high, and the acceptance of the system by students is long. The implementation of this paper’s system in actual teaching can significantly improve the overall civics learning objectives of the class.

In the context of comprehensively promoting information-based teaching, the construction of school classrooms is also developing in the direction of specialization, connotation as well as intelligence. The article designs a learning system for augmented technology in the context of augmented reality technology. The AR image-matching algorithm is proposed through the study of the system, and then the traditional KCF algorithm is improved, and a target tracking algorithm based on fusion matching is obtained by fusing the improved KCF algorithm with the image-matching algorithm. Then, the target is judged whether the target is lost or not by comparing the magnitude of the peak value (PSR) and the empirical value, and the test results show that the improved KCF has stronger target tracking utility. Subsequently, target tracking experiments were conducted using the fusion matching-based target tracking algorithm on students’ expressions in the classroom as well as head-up rate and concentration, while the accuracy of both experiments ranged from 83% to 95%. Thus, the algorithm performed better. Finally, a goal-tracking system was conducted for civics teaching with 75 students from University A. The results were based on the mean changes before and after the seven dimensions of the Civics course. It was found that after using the goal tracking system, the students’ post-test means all changed, of which the largest change in the score mean was historicism, with a difference of 21.04 between the before and after meals and the most obvious educational effect. It is favorable for teachers to conduct civics lessons to improve students’ learning status and performance.

Photovoltaic solar thermal integration is a new idea for buildings to fully utilize clean energy for energy saving and emission reduction. The digital modeling of photovoltaic solar thermal integration in buildings is finished by creating the energy balance equation of the energy supply system and the physical model of the main equipment. The objective function is established and the constraints are set to achieve optimization of the operation model through planning and solving problems. The TRNSYS simulation platform carries the integrated operation model and optimization scheme, and the optimization effect is simulated and validated. The results of computer simulation effect analysis show that the RMS of the parameters is 2.74%, 11.71%, 8.55%, and 2.47%, respectively, and the simulated and experimental values are consistent. The average system thermal efficiency and average water tank thermal efficiency of the optimized scheme are better than those of the PV/T-FPC system, and the cumulative cost savings in the whole life cycle after replacing the proposed scheme in Beijing and other five cities are 8432 yuan, 5937 yuan, 3513 yuan, 31744 yuan, 34893 yuan, and 31877 yuan, respectively, and the optimization effect is significant.

Artificial intelligence empowers education and promotes its integration and innovation, enabling the use of learning analytics to accurately diagnose the learning situation and implement precise teaching. In this paper, we have leveraged artificial intelligence to innovate traditional teaching methods, developed a wisdom teaching model specifically for college civic education, and concentrated on the precise application of wisdom teaching in this context. First, it creates student profiles based on academic data and then provides personalized recommendations for learning resources by combining a collaborative filtering algorithm and an improved K-means algorithm. Finally, we analyze the actual impact of the teaching intervention by comparing the student portraits before and after the intervention. When the number of neighbors K takes the same value, the improved cosine similarity calculation method outperforms other calculation methods, and the RMSE, MAE, accuracy, recall, and F1 values are 0.14, 0.24, 0.99, 0.74, and 0.85 when K = 60, respectively, which indicates that the algorithm improvement in this paper is practical. This paper categorizes the sample students into four groups based on their learning behavior and outcomes: potential, excellent, marginal, and hardworking. All these student types have demonstrated some progress following the teaching intervention, confirming the efficacy of this paper’s artificial intelligence-based teaching approach for college civic education. Simultaneously, the created profiles of these student types establish the groundwork for future personalized and accurate instruction.

With the continuous development of the information society, the information flow model makes the counselor able to transfer some civic and political knowledge to the students more quickly, while the students can get the latest information more feverishly. This paper, recognizing the impact of information flow on civics education in colleges and universities, introduces an enhanced information flow model. It specifically builds an information flow and optimization model using network information technology and information analysis. The article employs an open Civics class at a university as an empirical research object, delving into the proposed information flow model to examine the real impact of Civics teaching on teacher-student interaction. The analysis of students’ learning of the six Civics objectives reveals an improvement in their overall literacy, as evidenced by the quantitative scores of the pre-test and post-test, which are 3.051 and 3.999, respectively, with a 0.948 difference in value pursuit. The analysis of classroom organizational differences reveals that the experimental group has a behavior management score of M = 5.62, while the control group has a score of M = 4.73. This indicates that the teachers of civics courses in the experimental group are capable of providing clear and forward-looking instruction in behavior management, ensuring that students’ behavior management is in place.

At present, with the adjustment of industrial structure and industrial restructuring and upgrading, the demand for high-end skilled personnel is increasing, and the state has timely launched the integration of industry and education and the school-enterprise cooperation policy to strengthen the “double subject” status of enterprises and universities. With the help of evolutionary game theory, the study constructs a two-party evolutionary game model of talent cultivation under school-enterprise cooperation between enterprises and colleges and universities, constructs the replication dynamic equations of the two main parties of the game, and explores the stability of the Nash equilibrium point. In order to verify the reliability of the game model results, numerical simulation is carried out, which more intuitively demonstrates the dynamic process of the evolutionary game of talent cultivation under school-enterprise collaboration. When the proportion of enterprises and colleges adopting the strategy of not actively participating in collaborative talent cultivation gradually decreases in the benefits of the two groups, the enterprises choose to deeply participate, and the colleges choose to actively participate in collaborative talent cultivation is an evolutionarily stable strategy and the optimal solution of the model (1,1). It can realize the innovation of university talent cultivation mode through university-enterprise collaboration by establishing a complete information channel, incentive, and constraint mechanism to protect the interests of the two main parties of the game.

To provide new context-creation tools in architectural education to give students a full sense of immersion and presence. This paper discusses the integration of artificial intelligence and virtual reality technology in architectural education, and then the integration of artificial intelligence and virtual reality technology proposed architecture education virtual learning environment human-computer interaction system. Through the experimental study, in the application effect satisfaction analysis, the student satisfaction in the virtual reality learning environment in stimulating students’ learning interest is 80%, and 72% of the students agree to learn architectural expertise in the virtual reality learning environment.

Based on the current practice progress and the realistic dilemma of teacher evaluation reform in colleges and universities, the teacher evaluation reform in colleges and universities in the new era should adhere to the main principles of moral education, systematic and pluralistic, scientific and effective, and talent cultivation. We built a teacher quality evaluation index system in this study to follow the new rules for how teachers are evaluated in colleges and universities. We used fuzzy hierarchical analysis and fuzzy comprehensive evaluation to look at the multivariate structured data. It also combines natural language processing technology to achieve multivariate evaluations for college teachers. The results demonstrate that the construction of a multivariate evaluation model enables a diversified assessment of teachers’ teaching ability, thereby assisting teachers in achieving a more objective and accurate evaluation. The innovative path of teacher evaluation in colleges and universities should be based on establishing moral character, implementing the mission of teaching as the starting point, taking the sound evaluation framework system as the focus point, taking the rational use of assessment results as the key point, and promoting teacher growth and development as the ending point.

It is well known that the set of algebraic numbers (let us call it A) is countable. In this paper, instead of the usage of the classical terminology of cardinals proposed by Cantor, a recently introduced methodology using 1 -based infinite numbers is applied to measure the set A (where the number 1 is called grossone). Our interest to this methodology is explained by the fact that in certain cases where cardinals allow one to say only whether a set is countable or it has the cardinality of the continuum, the 1 -based methodology can provide a more accurate measurement of infinite sets. In this article, lower and upper estimates of the number of elements of A are obtained. Both estimates are expressed in 1 -based numbers. [ABSTRACT FROM AUTHOR]

Mathematical proof is the primary form of justification for mathematical knowledge, but in order to count as a proper justification for a piece of mathematical knowledge, a mathematical proof must be rigorous. What does it mean then for a mathematical proof to be rigorous? According to what I shall call the standard view, a mathematical proof is rigorous if and only if it can be routinely translated into a formal proof. The standard view is almost an orthodoxy among contemporary mathematicians, and is endorsed by many logicians and philosophers, but it has also been heavily criticized in the philosophy of mathematics literature. Progress on the debate between the proponents and opponents of the standard view is, however, currently blocked by a major obstacle, namely, the absence of a precise formulation of it. To remedy this deficiency, I undertake in this paper to provide a precise formulation and a thorough evaluation of the standard view of mathematical rigor. The upshot of this study is that the standard view is more robust to criticisms than it transpires from the various arguments advanced against it, but that it also requires a certain conception of how mathematical proofs are judged to be rigorous in mathematical practice, a conception that can be challenged on empirical grounds by exhibiting rigor judgments of mathematical proofs in mathematical practice conflicting with it. [ABSTRACT FROM AUTHOR]

In practice, mathematical proofs are most often the result of careful planning by the agents who produced them. As a consequence, each mathematical proof inherits a plan in virtue of the way it is produced, a plan which underlies its "architecture" or "unity." This paper provides an account of plans and planning in the context of mathematical proofs. The approach adopted here consists in looking for these notions not in mathematical proofs themselves, but in the agents who produced them. The starting point is to recognize that to each mathematical proof corresponds a proof activity which consists of a sequence of deductive inferences—i.e., a sequence of epistemic actions—and that any written mathematical proof is only a report of its corresponding proof activity. The main idea to be developed is that the plan of a mathematical proof is to be conceived and analyzed as the plan of the agent(s) who carried out the corresponding proof activity. The core of the paper is thus devoted to the development of an account of plans and planning in the context of proof activities. The account is based on the theory of planning agency developed by Michael Bratman in the philosophy of action. It is fleshed out by providing an analysis of the notions of intention—the elementary components of plans—and practical reasoning—the process by which plans are constructed—in the context of proof activities. These two notions are then used to offer a precise characterization of the desired notion of plan for proof activities. A fruitful connection can then be established between the resulting framework and the recent theme of modularity in mathematics introduced by Jeremy Avigad. This connection is exploited to yield the concept of modular presentations of mathematical proofs which has direct implications for how to write and present mathematical proofs so as to deliver various epistemic benefits. The account is finally compared to the technique of proof planning developed by Alan Bundy and colleagues in the field of automated theorem proving. The paper concludes with some remarks on how the framework can be used to provide an analysis of understanding and explanation in the context of mathematical proofs. [ABSTRACT FROM AUTHOR]

This paper shows an application of plausible reasoning methods (mainly, specialization and analogous) in mathematical modeling. Our attention is how a practitioner to determine analogously a more balanced scientific model to assist the desire during solving the entire problem. Taking interval samples modeling as a problem, we exemplify (with consideration paid to the motivation and course of discovering) how to discover, based on the classical corresponding methods, three linear regression models and two linear-like interpolation models relying on n-variable interval input-1-variable interval output samples. The rationality of these recommended models are proved, and applications of them are illuminated in detail by examples. Strategies to model further interval samples towards satisfactory are also exposed. [ABSTRACT FROM AUTHOR]

We describe in the form of a dialogue a development of various reflections on the combinatorics of set partitions; among the topics we pursue are the number of ways of partitioning a finite set into a fixed number d of subsets of odd or even size, into a parts of odd and b parts of even size, and into d parts, each of which has a size in a certain congruence class modulo some natural number m . To this end, pattern guessing, recursion and induction, combinatorial interpretation and generating functions are employed. The participants of the dialogue represent different perspectives on and approaches to mathematics. [ABSTRACT FROM AUTHOR]

In a wide range of fields, the word "modular" is used to describe complex systems that can be decomposed into smaller systems with limited interactions between them. This essay argues that mathematical knowledge can fruitfully be understood as having a modular structure and explores the ways in which modularity in mathematics is epistemically advantageous. [ABSTRACT FROM AUTHOR]

Inspired by the Ryōan-ji garden in Kyoto, we formulate and solve some problems on the arrangement of two sets of points in the plane. We imagine a Zen Master and a Zen Monk that want to create a dry garden: they will discover the importance of convexity and collinearity, as well as incidence and betweenness. We present these problems with a didactic exposition and we suggest some generalizations which require other mathematical tools. [ABSTRACT FROM AUTHOR]

The Einstein initial foundations of general relativity have to do with his great intuition and they are not clear as it is for special relativity. As has been widely emphasized, for example, in the book of Ohanian and Ruffini, the very name of the theory indicates a misconception. Despite this, the high school textbooks (at least the Italian ones) and books of scientific divulgation introduce the Einstein gravitational theory still following the initial approach leading, in our opinion, to misinterpretations. A careful student, for example, immediately asks: it is not true that the Earth rotates because I can consider it at rest thanks to general relativity theory. The relativity of motion is trivial in mathematics while it has deep meaning in physics and it is not sufficiently analyzed. Similarly, the arbitrary choice of the origin and the perfect equivalence between all coordinate systems are mathematical properties satisfied by general relativity and students can confuse it with physical equivalence between reference systems that have a deeper meaning and the Einstein theory does not verify it. Gravitational force does not exist and this is the real core of Einsteinian revolution. As happens in Newtonian physics and Special Relativity, also in general relativity the relative motions are the geodesic motions with the difference that spacetime can be curved and geodesics may not be a straight line. Similarly, in general relativity forces cause non-geodesic motion. Geodesic and non-geodesic are tensorial properties and for this reason they are absolute. [ABSTRACT FROM AUTHOR]

The general purpose of this article is to shed some light on the understanding of real numbers, particularly with regard to two issues: the real number as the limit of a sequence of rational numbers and the development of irrational numbers as a continued fraction. By generalizing the expression of the golden ratio in the form of the limit of two particular sequences, a new characterization of this number will appear. In that process, an infinite sum of iterated radicals is obtained. Based on that result, this article will then proceed to analyse that sum. The conditions under which the infinite sum yields an integer will be inspected, thereby calculating the value of the sum. After that, a method is established to develop some algebraic irrationals as a continued fraction. Finally, the results will be applied to an infinite difference of iterated radicals. [ABSTRACT FROM AUTHOR]