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The level of difficulty of mathematical test items is a critical aspect for evaluating test quality and educational outcomes. Accurately predicting item difficulty during test creation is thus significantly important for producing effective test papers. This study used more than ten years of content and score data from China's Henan Provincial College Entrance Examination in Mathematics as an evaluation criterion for test difficulty, and all data were obtained from the Henan Provincial Department of Education. Based on the framework established by the National Center for Education Statistics (NCES) for test item assessment methodology, this paper proposes a new framework containing eight features considering the uniqueness of mathematics. Next, this paper proposes an XGBoost-based SHAP model for analyzing the difficulty of mathematics tests. By coupling the XGBoost method with the SHAP method, the model not only evaluates the difficulty of mathematics tests but also analyzes the contribution of specific features to item difficulty, thereby increasing transparency and mitigating the "black box" nature of machine learning models. The model has a high prediction accuracy of 0.99 for the training set and 0.806 for the test set. With the model, we found that parameter-level features and reasoning-level features are significant factors influencing the difficulty of subjective items in the exam. In addition, we divided senior secondary mathematics knowledge into nine units based on Chinese curriculum standards and found significant differences in the distribution of the eight features across these different knowledge units, which can help teachers place different emphasis on different units during the teaching process. In summary, our proposed approach significantly improves the accuracy of item difficulty prediction, which is crucial for intelligent educational applications such as knowledge tracking, automatic test item generation, and intelligent paper generation. These results provide tools that are better aligned with and responsive to students' learning needs, thus effectively informing educational practice. [ABSTRACT FROM AUTHOR]

This correction notice provides updates and corrections to a research paper on deep machine learning model-based cyber-attacks detection in smart power systems. The paper proposes a deep learning-based attack detection model that addresses system disturbances caused by natural events and cyber-attacks. It discusses the use of principal component analysis (PCA) for feature selection and the potential of deep learning-based and decision tree classifiers for detecting cyber-attacks in intelligent energy grids. The notice also includes updates to references, clarifications on the experimental data set, and feature selection based on PCA. The study evaluates the effectiveness of the proposed model using various assessment indexes and compares it to conventional methods, concluding that the proposed model is effective in detecting cyber-attacks and demonstrates good performance in detecting destructive attacks. [Extracted from the article]

This review paper gives an overview of the method of multipliers for partial differential equations (PDEs). This method has made possible a lot of solutions to PDEs that are of interest in many areas such as applied mathematics, mathematical physics, engineering, etc. Looking at the history of the method and synthesizing the newest developments, we hope to give it the attention that it deserves to help develop the vast amount of work still needed to understand it and make the best use of it. It is also an interesting and a relevant method in itself that could possibly give interesting results in areas of mathematics such as modern algebra, group theory, topology, etc. The paper will be structured in such a manner that the last review known for this method will be presented to understand the theoretical framework of the method and then later work done will be presented. The information of four recent papers further developing the method will be synthesized and presented in such a manner that anyone interested in learning this method will have the most relevant information available and have all details cited for checking. [ABSTRACT FROM AUTHOR]

In this paper, I attempt to describe Mark Burgin's results in non-Diophantine mathematics, which are important for the foundation of mathematics and its application in quantum field theory. In particular, the elimination of divergences in quantum electrodynamics is described. [ABSTRACT FROM AUTHOR]

Accurate sky identification is one of the most important functions of an automated telescope mount. The more accurately the robotic telescope is navigated to the investigated part of the sky, the better the observations and discoveries made. In this paper, we present mathematical methods for accurate sky identification (celestial coordinates determination). They include the automatic selection of the reference stars, preliminary and full sky identification, as well as an interaction with international databases, which are a part of the astrometric calibration. All described methods help to receive accurately calculated astrometric data and use it for the positional calibration and better navigation of the automated telescope mount. The developed methods were successfully implemented in the Collection Light Technology (CoLiTec) software. Through its use, more than 1600 small solar system objects were discovered. It has been used in more than 700,000 observations and successful sky identifications, during which, five comets were discovered. Additionally, the accuracy indicators of the processing results of the CoLiTec software are provided in the paper, which shows benefits of the CoLiTec software and lower standard deviation of the sky identification in the case of low signal-to-noise ratios. [ABSTRACT FROM AUTHOR]

The study of symmetry is a fascinating and unifying subject that connects various areas of mathematics in the twenty-first century. Algebraic structures offer a framework for comprehending the symmetries of geometric objects in pure mathematics. This paper introduces new concepts in algebraic structures, concentrating on semimodules over semirings and analysing the neutrosophic structure in this context. We explore the properties of neutrosophic subsemimodules and neutrosophic ideals after defining them. We discuss, utilizing neutrosophic products, the representations of neutrosophic ideals and subsemimodules, as well as the relationship between neutrosophic products and intersections. Finally, we derive equivalent criteria in terms of neutrosophic structures for a semiring to be fully idempotent. [ABSTRACT FROM AUTHOR]

The present paper investigates the following inhomogeneous generalized Hartree equation i u ˙ + Δ u = ± | u | p − 2 | x | b (I α ∗ | u | p | · | b) u , where the wave function is u : = u (t , x) : R × R N → C , with N ≥ 2 . In addition, the exponent b > 0 gives an unbounded inhomogeneous term | x | b and I α ≈ | · | − (N − α) denotes the Riesz-potential for certain 0 < α < N . In this work, our aim is to establish the local existence of solutions in some radial Sobolev spaces, as well as the global existence for small data and the decay of energy sub-critical defocusing global solutions. Our results complement the recent work (Sharp threshold of global well-posedness versus finite time blow-up for a class of inhomogeneous Choquard equations, J. Math. Phys. 60 (2019), 081514). The main challenge in this work is to overcome the singularity of the unbounded inhomogeneous term | x | b for certain b > 0 . [ABSTRACT FROM AUTHOR]

This study provides a systematic literature review of research in the field of teaching and learning mathematics through Content and Language Integrated Learning (CLIL), Content-Based Instruction (CBI), and English Medium Instruction (EMI). The review aims to examine the most relevant literature with a focus on mathematics and CLIL, CBI, or EMI in Scopus and Web of Science per the Preferred Reporting Items for Systematic Reviews (PRISMA) guidelines. Based on 151 sources, 52 papers were selected according to predefined selection criteria. The papers were analysed and coded according to the following categories: (1) geographical productivity, (2) diachronic productivity growth, (3) main objectives, (4) methodology, and research design, (5) variables and measurement instruments, (6) context and sample details, and (7) main findings. The results reveal that most of the research analysed has been carried out in the Asian continent, followed by Europe. The analysis of diachronic productivity shows that the study of the application of CLIL, CBI, and EMI programs in the teaching-learning process of mathematics has notably increased in recent years, especially in the last triennium (2020–2022). Regarding the objectives of the selected corpus, the majority aimed at teaching practices and learning processes, and outcomes in mathematics and language proficiency in CLIL, CBI, and/or EMI classrooms. The corpus analysed fits within one or more of the following categories: empirical, qualitative, descriptive, correlational, and cross-sectional. The samples utilized in different studies differ significantly, both in terms of quantity –ranging from one person to 700– and in the educational level being studied, which would be primary, secondary, or tertiary education. The main variables studied in the corpus focus on mathematical competence, language proficiency, teaching practices, teacher training, science competence, and teachers' perceptions. The most widely used instruments have been objective tests, such as questionnaires, together with standardized tests to measure some aspects related to mathematical competence and language proficiency. They are followed by an analysis of documents (academic records, teaching materials, official documents...), participant or non-participant observation, interviews, and video and audio recordings. In summary, in the scientific literature analysed, a positive or neutral view predominates on the effects of the CLIL, CBI, and EMI approaches on the learning of mathematics and the L2. This can be due to methodological issues fundamentally related to the methodology, research design, sample, and measurement instruments. Thus, we must highlight that some of the results from the selected papers must be interpreted with caution. Taking this factor into consideration, further comparative studies on a wider scale are required to examine thoroughly the effects of CLIL, CBI, and EMI on the teaching and learning of mathematics in an L2. Besides, it is important to study in greater depth the different levels of language acquisition since the research analysed shows that these have not been sufficiently addressed in the mathematical field of knowledge. [ABSTRACT FROM AUTHOR]

With the rapid development of modern technologies, autonomous or robotic construction sites are becoming a new reality in civil engineering. Despite various potential benefits of the automation of construction sites, there is still a lack of understanding of their complex nature combining physical and cyber components in one system. A typical approach to describing complex system structures is to use tools of abstract mathematics, which provide a high level of abstraction, allowing a formal description of the entire system while omitting non-essential details. Therefore, in this paper, autonomous construction is formalised using categorical ontology logs enhanced by abstract definitions of individual components of an autonomous construction system. In this context, followed by a brief introduction to category theory and ologs, exemplary algebraic definitions are given as a basis for the olog-based conceptual modelling of autonomous construction systems. As a result, any automated construction system can be described without providing exhausting detailed definitions of the system components. Existing ologs can be extended, contracted or revised to fit the given system or situation. To illustrate the descriptive capacity of ologs, a lattice of representations is presented. The main advantage of using the conceptual modelling approach presented in this paper is that any given real-world or engineering problem could be modelled with a mathematically sound background. [ABSTRACT FROM AUTHOR]

This paper presents a survey of advanced concepts and research perspectives, of a philosophical-mathematical approach to describe the dynamics of systems of many interacting living entities. The first part introduces the general conceptual framework. Then, a critical analysis of the existing literature is developed and referred to a multiscale view of a mathematics of living organisms. This paper attempts to understand how far the present state-of-the-art is far from the achievement of such challenging objective. The overall study leads to identify research perspectives and possible hints to deal with them. [ABSTRACT FROM AUTHOR]

It is well-known that the weighted averages of two competing forecasts may reduce mean squared prediction errors (MSPE) and may also introduce certain inefficiencies. In this paper, we take an in-depth view of one particular type of inefficiency stemming from simple combination schemes: Mincer and Zarnowitz inefficiency or auto-inefficiency for short. Under mild assumptions, we show that linear convex forecast combinations are almost always auto-inefficient, and, therefore, greater reductions in MSPE are almost always possible. In particular, we show that the process of taking averages of forecasts may induce inefficiencies in the combination, even when individual forecasts are efficient. Furthermore, we show that the so-called "optimal weighted average" traditionally presented in the literature may indeed be inefficient as well. Finally, we illustrate our findings with simulations and an empirical application in the context of the combination of headline inflation forecasts for eight European economies. Overall, our results indicate that in situations in which a number of different forecasts are available, the combination of all of them should not be the last step taken in the search of forecast accuracy. Attempts to take advantage of potential inefficiencies stemming from the combination process should also be considered. [ABSTRACT FROM AUTHOR]

This paper introduces the concept of symmetric difference operators in terms of overlap and grouping functions, for which the associativity property is not strongly required. These symmetric difference operators are weaker than symmetric difference operators in terms of positive and continuous t-norms and t-conorms. Therefore, in the sense of the characters of mathematics, these operators do not necessarily satisfy certain properties, such as associativity and the neutrality principle. We analyze several related important properties based on two models of symmetric differences. [ABSTRACT FROM AUTHOR]

The allocation of seats in a legislative body to groups based on their size is a crucial issue in legal and political studies. However, recent findings suggest that an optimal allocation of seats may not be proportional to the size of the groups. For instance, the European Parliament (EP) utilizes a subproportional system known as degressive proportionality. Unfortunately, current apportionment methods for the EP lack a rigorous axiomatic analysis and fail to adequately address equality. Building upon recent research on equality in subproportional settings, this paper proposed a novel generalization of existing axioms and divisor methods for proportionality to encompass subproportionality with relative equality. Specifically, we consider a function f (p) = a + b p γ on the standard number of seats for a group of size p, where a, b and γ are given non-negative constants, and a is an integer. This theory is exemplified through an empirical study focused on the EP. [ABSTRACT FROM AUTHOR]

Simple Summary: Cancer is basically a tough condition on a patient's body where cell grows uncontrollably. Normal cells are affected, which destroys the health of the patient. The main problem in cancer is spreading from one part to another. Therefore, the mathematical modeling of cancerous tumors integrates to check overall stability. A novel approach is introduced such as Bernstein polynomial with combination of genetic algorithm, sliding mode controller, and synergetic control. The proposed solution has easily eliminated cancerous cells within five days using synergetic control. In addition, five cases are incorporated to evaluate error function. In addition, a brief comparative study is added to contrast the simulation results with theoretical modeling. Cancerous tumor cells divide uncontrollably, which results in either tumor or harm to the immune system of the body. Due to the destructive effects of chemotherapy, optimal medications are needed. Therefore, possible treatment methods should be controlled to maintain the constant/continuous dose for affecting the spreading of cancerous tumor cells. Rapid growth of cells is classified into primary and secondary types. In giving a proper response, the immune system plays an important role. This is considered a natural process while fighting against tumors. In recent days, achieving a better method to treat tumors is the prime focus of researchers. Mathematical modeling of tumors uses combined immune, vaccine, and chemotherapies to check performance stability. In this research paper, mathematical modeling is utilized with reference to cancerous tumor growth, the immune system, and normal cells, which are directly affected by the process of chemotherapy. This paper presents novel techniques, which include Bernstein polynomial (BSP) with genetic algorithm (GA), sliding mode controller (SMC), and synergetic control (SC), for giving a possible solution to the cancerous tumor cells (CCs) model. Through GA, random population is generated to evaluate fitness. SMC is used for the continuous exponential dose of chemotherapy to reduce CCs in about forty-five days. In addition, error function consists of five cases that include normal cells (NCs), immune cells (ICs), CCs, and chemotherapy. Furthermore, the drug control process is explained in all the cases. In simulation results, utilizing SC has completely eliminated CCs in nearly five days. The proposed approach reduces CCs as early as possible. [ABSTRACT FROM AUTHOR]

This document is a correction notice for a paper titled "An Investigation on Spiking Neural Networks Based on the Izhikevich Neuronal Model: Spiking Processing and Hardware Approach." The correction addresses an error in the affiliation of one of the authors, Ehsan Nazemi. The updated affiliation should include the Institute of Fundamental and Applied Sciences, Duy Tan University, Ho Chi Minh City, Vietnam. Additionally, there was an error in the Introduction section where a paragraph with unexplained references was mistakenly copied from one of the author's theses. This paragraph has been removed, and the order of references has been adjusted accordingly. The authors assure that these corrections do not affect the scientific conclusions of the paper. [Extracted from the article]

Nowadays, graph theory is one of the most exciting fields of mathematics due to the tremendous developments in modern technology, where it is used in many important applications. The orthogonal double cover (O D C) is a branch of graph theory and is considered as a special class of graph decomposition. In this paper, we decompose the complete bipartite graphs K x , x by caterpillar graphs using the method of O D C s. The article also deals with constructing the O D C s of K x , x by general symmetric starter vectors of caterpillar graphs such as stars–caterpillar, the disjoint copies of cycles–caterpillars, complete bipartite caterpillar graphs, and the disjoint copies of caterpillar paths. We decompose the complete bipartite graph by the complete bipartite subgraphs and by the disjoint copies of complete bipartite subgraphs using general symmetric starter vectors. The advantage of some of these new results is that they enable us to decompose the giant networks into large groups of small networks with the comprehensive coverage of all parts of the giant network by using the disjoint copies of symmetric starter subgraphs. The use case of applying the described theory for various applications is considered. [ABSTRACT FROM AUTHOR]

In recent years, the theory of convexity has influenced every field of mathematics due to its unique characteristics. Numerous generalizations, extensions, and refinements of convexity have been introduced, and one of them is set-valued convexity. Interval-valued convex mappings are a special type of set-valued maps. These have a close relationship with symmetry analysis. One of the important aspects of the relationship between convex and symmetric analysis is the ability to work on one field and apply its principles to another. In this paper, we introduce a novel class of interval-valued (I.V.) functions called CR - γ -convex functions based on a non-negative mapping γ and center-radius ordering relation. Due to its generic property, a set of new and known forms of convexity can be obtained. First, we derive new generalized discrete and integral forms of Jensen's inequalities using CR - γ -convex I.V. functions. We employ this definition and Riemann-Liouville fractional operators to develop new fractional versions of Hermite-Hadamard's, Hermite-Hadamard-Fejer, and Pachpatte's type integral inequalities. We examine various key properties of this class of functions by considering them as special cases. Finally, we support our findings with interesting examples and graphical representations. [ABSTRACT FROM AUTHOR]

Fractional systems have been widely utilized in various fields, such as mathematics, physics and finance, providing a versatile framework for precise measurements and calculations involving partial quantities. This paper aims to develop a novel polynomial controller for a power system (PS) with fractional-order (FO) dynamics. It begins by studying the practical stability of a general class of tempered fractional-order (TFO) nonlinear systems, with broad applicability and potential for expanding its applications. Afterward, a polynomial controller is designed to guarantee the practical stability of the PS, encompassing the standard constant controller as a specific instance. The design conditions for this controller are resolved using the sum of squares (SOS) approach, a powerful technique for guaranteeing stability and control design. To showcase the practical value of the analytical findings, simulations of the PS are conducted utilizing SOSTOOLS. [ABSTRACT FROM AUTHOR]

This paper presents a simplified model of the population migration problem, addressed to first-year engineering students in order to show them the use of linear algebra tools. The study consists of predicting the census in the city centre and in the suburbs, determining the city population equilibrium point, and making a sociological interpretation of population flows. This practical problem is part of the seminar "Applications of Linear Algebra in Engineering", which is being held at the Universitat Politècnica de Catalunya-BarcelonaTech (UPC). This seminar consists in the learning of linear algebra by the implementation of real applications where mathematical tools are required to resolve them. This paper presents an application of linear algebra to the population migration problem and analyses students' appreciation through anonymous surveys and personal interviews. The surveys assessed students' motivation towards the subject of linear algebra and their learning of mathematical concepts. Personal interviews were conducted for students in order to let them express in detail their opinion about the seminar. The results confirm that the introduction of real applications in the learning of mathematics increases students' motivation and involvement, which implies an improvement in students' performance in the first courses of STEM degrees. [ABSTRACT FROM AUTHOR]

Conversations of educational equity in mathematics necessitate a more deliberate, nuanced look at the mathematical processes of learning for students of color from historically marginalized communities. This paper describes the theoretical work of a research collaborative that seeks to develop understanding of the experiences around mathematical identity of Latinas labeled with Learning Disabilities in mathematics classrooms. Expanding the theory of Complex Embodiment from Disability Studies, we explore new interdisciplinary theoretical and methodological tools to analyze the emotional, embodied experience of learning mathematics in the social worlds of mathematics classrooms, using emotional discourse. We take up theoretical and methodological practices around intersectionality through analysis of how power and positioning operate in mathematics identity development. We find that the young woman whose narratives we explore in this paper is positioned through deficit discourses around disability and multilingual learners, yet she understands herself through a positive mathematical affinity she shares with her mother. Over time, we see her narratives shift emotionally away from mathematics, as well as away from this connection with her mother. Her narratives help us develop a theoretical perspective that understands emotion in mathematics learning as both embodied and socially constructed. [ABSTRACT FROM AUTHOR]

In order to find solutions to math word problems, some modules have been designed to check the generated expressions, but they neither take into account the symmetry between math word problems and their corresponding mathematical expressions, nor do they utilize the efficiency of pretrained language models in natural language understanding tasks. Anyway, designing fine-tuning tasks for pretrained language models that encourage cooperation with other modules to improve the performance of math word problem solvers is an unaddressed problem. To solve these problems, in this paper we propose a BERT-based model for solving math word problems with a supervision module. Based on pretrained language models, we present a fine-tuning task to predict the number of different operators in the expressions to learn the potential relationships between the problems and the expressions. Meanwhile, a supervision module is designed to check the incorrect expressions generated and improve the model's performance by optimizing the encoder. A series of experiments are conducted on three datasets, and the experimental results demonstrate the effectiveness of our model and its component's designs. [ABSTRACT FROM AUTHOR]

This paper is an overview of some recent and ongoing developments of formal logical systems designed for reasoning about systems of rational agents who act in pursuit of their individual and collective goals, explicitly specified in the language as arguments of the strategic operators, in a socially interactive context of collective objectives and attitudes which guide and constrain the agents' behavior. [ABSTRACT FROM AUTHOR]

By using the generalization of the neutrosophic q-Poisson distribution series, we introduce a new subclass of analytic and bi-univalent functions defined in the open unit disk. We then apply the q-Gegenbauer polynomials to investigate the estimates for the Taylor coefficients and Fekete–Szegö type inequalities of the functions belonging to this new subclass. In addition, we consider several corollaries and the consequences of the results presented in this paper. The neutrosophic q-Poisson distribution is expected to be significant in a number of areas of mathematics, science, and technology. [ABSTRACT FROM AUTHOR]

The aim of this paper was to introduce and investigate a new seminorm of operator tuples on a complex Hilbert space H when an additional semi-inner product structure defined by a positive (semi-definite) operator A on H is considered. We prove the equality between this new seminorm and the well-known A-joint seminorm in the case of A-doubly-commuting tuples of A-hyponormal operators. This study is an extension of a well-known result in [Results Math 75, 93(2020)] and allows us to show that the following equalities r A (T) = ω A (T) = ∥ T ∥ A hold for every A-doubly-commuting d-tuple of A-hyponormal operators T = (T 1 , ... , T d) . Here, r A (T) , ∥ T ∥ A , and ω A (T) denote the A-joint spectral radius, the A-joint operator seminorm, and the A-joint numerical radius of T , respectively. [ABSTRACT FROM AUTHOR]

Thanks to the efforts of our reviewers in 2022, the median time to first decision was 17 days and the median time to publication was 37 days. Regardless of whether the articles they examined were ultimately published, the editors would like to express their appreciation and thank the following reviewers for the time and dedication that they have shown I Mathematics i : those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). I Mathematics i was able to uphold its high standards for published papers due to the outstanding efforts of our reviewers. [Extracted from the article]

This correction notice addresses an omission in a previously published paper titled "Ricci Vector Fields Revisited." The authors apologize for not including acknowledgements in the original paper and express their gratitude for the support received from King Saud University in Riyadh, Saudi Arabia. The authors assure readers that this correction does not impact the scientific conclusions of the paper. The correction has been approved by the Academic Editor and the original publication has been updated. The authors of the paper are Hanan Alohali, Sharief Deshmukh, and Gabriel-Eduard Vîlcu. [Extracted from the article]

In this paper, we describe an educational experience in the context of the Master's degree that is compulsory in Spain to become a secondary education mathematics teacher. Master's students from two universities in Madrid (Spain) attended lectures that addressed—emphasizing the concourse of a dynamic geometry software package—some historical, didactic and mathematical issues related to linkage mechanisms, such as those arising in the 18th and 19th centuries during the development of the steam engine. Afterwards, participants were asked to provide three different kinds of feedback: (i) working on an assigned group task, (ii) individually answering a questionnaire, and (iii) proposing some classroom activity, imagining it would be addressed to their prospective pupils. All three issues focused on the specific topic of the attended lectures. In the framework of Mason's reflective discourse analysis, the information supplied by the participants has been analyzed. The objective was to explore what they have learned from the experience and what their perception is of the potential interest in linkages as a methodological instrument for their future professional activity as teachers. This analysis is then the basis upon which to reflect on the opportunities (and problems) that this particular bar-joint linkages methodological approach could bring towards providing future mathematics teachers with attractive tools that would contribute to enhancing a STEAM-oriented education. Finally, the students' answers allow us to conclude that the experience was beneficial for these pre-service teachers, both in improving their knowledge on linkages history, mathematics, industrial, technological and artistic applications, and in enhancing the use in the classroom of this very suitable STEAM context. [ABSTRACT FROM AUTHOR]

Nowadays, ICT play a fundamental role in education, as they are essential in all areas of knowledge at any educational stage. Specifically, in the area of mathematics, within the Primary Education stage, they are also very valid resources. This is demonstrated by the latest scientific studies. In recent times, the scientific literature has provided evidence of all this through different research studies. The aim of this paper is to analyze all those publications that deal with the teaching–learning processes of mathematics through ICT in primary education. The aim is to show the current state of the scientific literature and what elements and aspects to highlight are common to the documents analyzed. By means of the systematic literature review method, we analyzed 11 articles indexed in Scopus and the Web of Science, with the result that the use of ICT in this area is still scarce, as the volume of publications in this respect is very low. This could also indicate a prevalence of traditional methodologies associated with the teaching of mathematics that could hinder students' acquisition of mathematical competence. However, those studies that show the use of ICT demonstrate that its use leads to an improvement in performance, motivation, and problem solving. [ABSTRACT FROM AUTHOR]

A singular graph G, defined when its adjacency matrix is singular, has important applications in mathematics, natural sciences and engineering. The chemical importance of singular graphs lies in the fact that if the molecular graph is singular, the nullity (the number of the zero eigenvalue) is greater than 0, then the corresponding chemical compound is highly reactive or unstable. By this reasoning, chemists have a great interest in this problem. Thus, the problem of characterization singular graphs was proposed and raised extensive studies on this challenging problem thereafter. The graph obtained by conglutinating the starting vertices of three paths P s 1 , P s 2 , P s 3 into a vertex, and three end vertices into a vertex on the cycle C a 1 , C a 2 , C a 3 , respectively, is denoted as γ (a 1 , a 2 , a 3 , s 1 , s 2 , s 3) . Note that δ (a 1 , a 2 , a 3 , s 1 , s 2) = γ (a 1 , a 2 , a 3 , s 1 , 1 , s 2) , ζ (a 1 , a 2 , a 3 , s) = γ (a 1 , a 2 , a 3 , 1 , 1 , s) , φ (a 1 , a 2 , a 3) = γ (a 1 , a 2 , a 3 , 1 , 1 , 1) . In this paper, we give the necessity and sufficiency that the γ − graph, δ − graph, ζ − graph and φ − graph are singular and prove that the probability that a randomly given γ − graph, δ − graph, ζ − graph or φ − graph being singular is equal to 325 512 , 165 256 , 43 64 , 21 32 , respectively. From our main results, we can conclude that such a γ − graph( δ − graph, ζ − graph, φ − graph) is singular if at least one cycle is a multiple of 4 in length, and surprisingly, the theoretical probability of these graphs being singular is more than half. This result promotes the understanding of a singular graph and may be promising to propel the solutions to relevant application problems. [ABSTRACT FROM AUTHOR]

The paper under discussion has some shortcomings in regard to mathematical formulation of the problem related to acoustics, its justification, some references analyses and the text, and we feel that this study has not been finished to its full extent. From the point of view of acoustic jet formation, the mathematical model used by the authors has a number of significant limitations which are not taken into account. Moreover, the formal analogy between the parameters of electromagnetic waves and acoustic waves should be used with caution in the conditions of localized beams (acoustic jet), and the limits of their application should be justified. [ABSTRACT FROM AUTHOR]

It is not simple to consider the world from only one side, but analyzing all sides can cloud comprehension without reaching deep insight found at the core. In a word as a whole, there is potential for telling the whole world in one word, i.e., data, leading to interpretations as phenomena and paradigms at the core of this review. The tug of war between the two sides explains that data represent the world, or vice versa, and present a fundamental view that systems or subsystems frame the world, even though they are encoded and composed of culture, rules, or approaches such as the threshold of democracy. When the COVID-19 pandemic posed a threat, human efforts contributed to finding potentially answers to questions presented by the world: what, who, where, when, why, and how (5 wh); a calling in the form of a challenge, where facts show something. All these questions resulted in research, education, and service activities, with their respective data frameworks producing results. This paper aims to reveal the meaning of the outcomes through an observation from an outside perspective. Therefore, like COVID-19 and its vaccines, the assertion of convexity and concave contradictions in the treatment of data leads to a mutually conjugate treatment of data. In this regard, statistics and artificial intelligence play separate and complementary roles. [ABSTRACT FROM AUTHOR]

This paper reports on nine elementary, middle, and high school in-service teachers who participated in a series of workshops aimed at exploring the wonder, joy, and beauty of mathematics through the creation and application of digitally fabricated tools (i.e., laser-cut and 3D printed). Using the Technological Pedagogical and Content Knowledge (TPACK) framework to investigate technological, pedagogical, contextual, and content knowledge, researchers applied qualitative methods to uncover the affordances and constraints of teaching and learning math concepts with digitally fabricated tools and examined how the workshops supported broadening participation in mathematics by focusing on the connections between mathematical inquiry, nature, and the arts. Affordances include opportunities for hands-on learning, visual support at the secondary level, and real-world connections that go beyond the state standards. Barriers include purchasing a laser-cutter, ventilation and noise issues, time constraints, misalignment with school and district priorities, and a lack of administrative support. All participants indicated that they were interested in additional workshops focused on designing their own digitally fabricated mathematics tools that better align with their grade level(s) and standards. [ABSTRACT FROM AUTHOR]

In recent years, using micro-lectures (a short video of less than ten minutes) as a digital learning medium has become popular for integrating technology into mathematics lessons, providing new experiences to students and helping them to master mathematical concepts. It is important to note that the effectiveness of a micro-lecture depends on how teachers use it and on students' perceptions of learning support media. Therefore, this research aims to determine Junior High School students' achievements and satisfaction using micro-lectures on the topic of Direct and Inverse Proportions. Teachers' micro-lectures in this research were designed based on their teaching styles, learning objectives, and considering the needs of their students. A quasi-experiment with 3, 2, and 1 pretest-posttest design with experimental and control groups was also used. In the first and second experimental groups, teachers used the same version of the micro-lectures in their classrooms as the main learning medium and at home to preview and review Direct and Inverse Proportions materials. At the same time, the control group used conventional expository learning activities. Finally, a learning satisfaction questionnaire was used to determine students' perceptions of using micro-lectures in their mathematics lessons. Findings indicate that the use of micro-lectures in the experimental classes I and II is better than in the control class and led to improved learning achievements and student satisfaction. Meanwhile, there is no significant difference between using micro-lectures inside and outside classrooms. Students' report cards also showed that their learning satisfaction using this technique was higher than the control class. Practical implications and suggestions for further investigation are to be also discussed at the end of the paper. [ABSTRACT FROM AUTHOR]

There is a noticeable gap between results of cognitive neuroscientific research into basic mathematical abilities and philosophical and empirical investigations of mathematics as a distinct intellectual activity. The paper explores the relevance of a Wittgensteinian framework for dealing with this discrepancy. [ABSTRACT FROM AUTHOR]

This correction notice addresses a mistake in the original paper titled "Cyberbullying Detection on Twitter Using Deep Learning-Based Attention Mechanisms and Continuous Bag of Words Feature Extraction." The authors apologize for omitting Prince Sultan University (PSU) from the funding statement. PSU was one of the main funders for the research, which was a collaborative effort between King Saud University, Universiti Teknologi Petronas, and Prince Sultan University. The authors express their appreciation for the generous support from these universities and state that the scientific conclusions of the paper remain unaffected. The funding statement in the original publication has been updated to include PSU. [Extracted from the article]

Solving systems of linear equations is a fundamental problem in mathematics. Combining mean shift clustering (MS) with greedy techniques, a novel block version of the Kaczmarz–Motzkin method (BKMS), where the blocks are predetermined by MS clustering, is proposed in this paper. Using a greedy strategy, which collects the row indices with the almost maximum distance of the linear subsystem per iteration, can be considered an efficient extension of the sampling Kaczmarz–Motzkin algorithm (SKM). The new method linearly converges to the least-norm solution when the system is consistent. Several examples show that the BKMS algorithm is more efficient compared with other methods (for example, RK, Motzkin, GRK, SKM, RBK, and GRBK). [ABSTRACT FROM AUTHOR]

The Victorian Curriculum and Assessment Authority (VCAA) introduced the use of Computer Algebra System (CAS) technology (calculator and software) into the senior secondary mathematics curriculum and examination assessment in three phases, starting with a research-based pilot from 2000, followed by parallel implementation of CAS and non-CAS subjects from 2006 and culminating in transition to CAS-assumed subjects in 2010. This paper reports reflections on these developments over two decades from the perspectives of a researcher and the state mathematics manager (the authors) in consultation with four implementing teachers (the consultants). The authors critically examined the strategic design decisions that were made for the initiative over time. Then, with contributions from the four consultants, technical design issues relating to assessment and to teaching and the changes over a decade were investigated. A range of modifications have been made over the two decades, driven by changes in device capability and progressively increasing teaching expertise. The place of CAS in senior mathematics is now widely accepted, partly because an examination component not allowing any technology has been implemented. Examination questions have become more general, which may have added difficulty, but more questions involve setting up a real situation mathematically. [ABSTRACT FROM AUTHOR]

The paper deals with the application of concept mapping in teaching mathematics at the primary level of education (ISCED 1), as well as the experimental verification of its qualitative contribution to the process of education. The authors present results obtained in a sample survey aimed at monitoring possibilities of influencing pupils' relationship to teaching and mastery of mathematics using digital concept maps within the process of pedagogical intervention. The authors tested the degree to which digital concept maps influence changes in the pupils' opinions and attitudes to mathematics (subject popularity, interest in the subject, easiness of understanding, fear and apprehension of mathematics) while explaining and fixating the curriculum of selected mathematics topics in the third grade of a primary school. The results of the experiment did not confirm that teaching supported by concept maps contributes to reducing pupils' negative attitudes towards subjects, particularly in mathematics at the primary level (ISCED 1). Due to the pedagogical intervention of concept mapping, no statistically significant changes occurred in the pupils' attitudes to the observed factors of mathematics. However, the authors have come up with research findings that may be notable in the field of branch didactics of mathematics and digital technologies. [ABSTRACT FROM AUTHOR]

The current paper explores the potential of the areas between mathematics and poetry. We will first recall some definitions and results that are needed to construct solutions of the Yang–Baxter equation. A new duality principle is presented and Boolean coalgebras are introduced. A section on poetry dedicated to the Yang–Baxter equation is presented, and a discussion on a poem related to a mathematical formula follows. The final section presents our conclusions and further information on these topics. [ABSTRACT FROM AUTHOR]

The annual exam questions in China National College Entrance Examination (CNCEE, commonly known as Gaokao) have attracted wide attention from the entire society in China. This paper analyzes an important math exam question related to always holding inequalities with an unknown parameter from the 2020 CNCEE and builds different solutions via parameter separation or classified discussion, and then gives a systematic and effective design framework for questions related to problems of always holding inequalities and zeros of functions with an unknown parameter for the CNCEE. The framework reveals the internal mechanism of designing the math exam questions in the CNCEE. According to the design framework, two simulated math exam questions with reference solutions will be proposed. [ABSTRACT FROM AUTHOR]

In the context of the science, technology, engineering, arts and mathematics disciplines in education, subjects tend to use contextualized activities or projects. Educational robotics and computational thinking both have the potential to become subjects in their own right, though not all educational programs yet offer these. Despite the use of technology and programming platforms being widespread, it is not common practice to integrate computational thinking and educational robotics into the official curriculum in secondary education. That is why this paper continues an initial project of integrating computational thinking and educational robotics into a secondary school in Barcelona, Spain. This study presents a project-based learning approach where the main focus is the development of skills related to science, technology, engineering, arts and mathematics and the acquisition of computational thinking knowledge in the second year of pupils' studies using a block-based programming environment. The study develops several sessions in the context of project-based learning, with students using the block-programming platform ScratchTM. During these sessions and in small-group workshops, students will expand their knowledge of computational thinking and develop 21st-century skills. We demonstrate the superior improvement of these concepts and skills compared to other educational methodologies. [ABSTRACT FROM AUTHOR]

This paper presents research work in which an innovative didactic proposal was designed for the study of 3D metric geometry in the second year of A-level courses (secondary education) in the specialty of the sciences. The designed didactic proposal has the format of a workshop of geometry practices and is framed in the Didactic Situations Theory and in the Anthropological Theory of the Didactic. Although the mentioned theories, which the French school has developed since the seventies, are widely known and studied by secondary school mathematics teachers during their training in pedagogy, especially in Spanish- and French-speaking countries, their innovative approach has not been fully implemented in the field of algebraic geometry. Mathematics textbooks follow the traditional approach in which the teacher simply provides the contents and instructs the student, who captures these concepts and reproduces them as they have been supplied. In the presented didactic proposal, the approach proposed by Brousseau is followed, in which three fundamental elements take part: student, teacher, and the didactic environment. The teacher is the one who facilitates the environment in which the student builds his knowledge. In the proposal the didactic contract is stablished, the didactic situations are designed, and the means and didactic variables for the study of 3D metric geometry are chosen. The methodology followed in this study consisted of identifying educational problems; describing the theoretical framework; developing the didactic proposal; and analysis, reflection, and criticism of this training product. The contextualization of the geometry workshop in the field of the construction of structures and the use of deductive reasoning techniques that are applied in synthetic geometry and also in the building information modeling (BIM) methodology may be a way of showing the use of analytical geometry in the industries of engineering and architecture; they also serve as an opportunity for students to better understand this mathematical work. [ABSTRACT FROM AUTHOR]

The study of symmetry is one of the most important and beautiful themes uniting various areas of contemporary arithmetic. Algebraic structures are useful structures in pure mathematics for learning a geometrical object's symmetries. In this paper, we introduce new concepts in an algebraic structure called BCI-algebra, where we present the concepts of bipolar fuzzy (closed) BCI-positive implicative ideals and bipolar fuzzy (closed) BCI-commutative ideals of BCI-algebras. The relationship between bipolar fuzzy (closed) BCI-positive implicative ideals and bipolar fuzzy ideals is investigated, and various conditions are provided for a bipolar fuzzy ideal to be a bipolar fuzzy BCI-positive implicative ideal. Furthermore, conditions are presented for a bipolar fuzzy (closed) ideal to be a bipolar fuzzy BCI-commutative ideal. [ABSTRACT FROM AUTHOR]

The perception that mathematics is difficult has always persisted. Nevertheless, mathematics is such an essential component of STEM education. Quantum technologies are already having enormous effects on our society, with advantages seen across a broad variety of industries, including finance, aerospace, and energy. These innovations promise to transform our lives. Managers in the business and public sectors will need to learn quantum computing. Quantum algorithmic literacy may help increase mathematical understanding and enthusiasm. The current paper proposes that one possible approach is to present the information in a reasonably gentle but intelligible way, in order to excite individuals with the mathematics that they already know by extending them to acquiring quantum algorithmic literacy. A gentle introduction to the mathematics required to model quantum computing ideas, including linear transformations and matrix algebra, will be given. Quantum entanglement, linear transformations, quantum cryptography, and quantum teleportation will be used as examples to illustrate the usefulness of basic mathematical concepts in formulating quantum algorithms. These exemplars in quantum algorithmic literacy can help to invigorate people's interest in mathematics. Additionally, a qualitative comparative analysis (QCA) framework is provided that teachers can utilize to determine which students to approach for remediation. This assists the teachers in dispelling any pupils' uncertainty about mathematical concepts. [ABSTRACT FROM AUTHOR]

This paper reports on how 10 middle and high school preservice teachers (PSTs) designed a social justice focused lesson using the culturally responsive mathematics teaching (CRMT) tool. Results from our analysis indicate that most of the PSTs were able to select appropriate social justice topics, though not all the PSTs integrated mathematics and social justice throughout their lessons. The results show that most of the PSTs need more experience with mathematization, handling controversial discussions, and developing transformative student action. Our work also led to a modification of the tool (CRMT-M). We discuss the implications of the study for mathematics teacher preparation. [ABSTRACT FROM AUTHOR]

We give a family of counterexamples of a theorem on a new upper bound for the α-indices of graphs in the paper that is mentioned in the title. We also provide a new upper bound for corrigendum. [ABSTRACT FROM AUTHOR]

Parallel restoration is a way to accelerate the black-start procedure of power systems following a blackout. An efficient sectionalizing scheme can reduce the restoration time of a system, taking into account the black-start ability, generation-load balance of subsystems, restoration time of branches, start-up time of generating units, and effects of dispatchable loads and faulted devices. Solving the sectionalizing problem is challenging since it needs to handle a large number of Boolean variables corresponding to the branches and nonlinear constraints associated with system topology. This paper investigates power system sectionalizing problem for parallel restoration to minimize the system restoration time (SRT). A novel sectionalizing method considering the restoration of generating units and network branches is proposed. Firstly, the minimum spanning tree (MST) algorithm is used to determine the skeleton network of a power system. Secondly, the number of subsystems is determined according to the number of black-start units. Based on the skeleton network, candidate boundary lines among subsystems are identified. Then, constraints are evaluated to identify feasible sectionalizing schemes. Except commonly used constraints on power balancing and black-start units, this paper also considers using dispatchable loads to meet the minimum output requirements of generating units. Finally, the sectionalizing scheme with theminimumSRT is selected as the final solution. The effectiveness of the proposed method is validated by the IEEE 39-bus and 118-bus test systems. The simulation results indicate that the proposed method can balance the restoration time of subsystems and minimize the SRT. [ABSTRACT FROM AUTHOR]

A six-dimensional nonlinear hydropower system controlled by a nonlinear predictive control method is presented in this paper. In terms of the nonlinear predictive control method; the performance index with terminal penalty function is selected. A simple method to find an appropriate terminal penalty function is introduced and its effectiveness is proved. The input-to-state-stability of the controlled system is proved by using the Lyapunov function. Subsequently a six-dimensional model of the hydropower system is presented in the paper. Different with other hydropower system models; the above model includes the hydro-turbine system; the penstock system; the generator system; and the hydraulic servo system accurately describing the operational process of a hydropower plant. Furthermore, the numerical experiments show that the six-dimensional nonlinear hydropower system controlled by the method is stable. In addition, the numerical experiment also illustrates that the nonlinear predictive control method enjoys great advantages over a traditional control method in nonlinear systems. Finally, a strategy to combine the nonlinear predictive control method with other methods is proposed to further facilitate the application of the nonlinear predictive control method into practice. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and uniqueness of a fixed point of almost contractions via simulation functions in metric spaces. Moreover, some examples and an application to integral equations are given to support availability of the obtained results. [ABSTRACT FROM AUTHOR]

Molenaar'smanifesto on psychology as idiographic science (Molenaar, 2004) brought the N = 1 times series perspective firmly to the attention of developmental scientists. The rich intraindividual variation in complex developmental processes requires the study of these processes at the level of the individual. Yet, the idiographic approach is all but easy in practical research. One major limitation is the collection of short interval times series of high quality data on developmental processes. In this paper, we present a novel measurement approach to this problem. We developed an online practice and monitoring system which is now used by thousands of Dutch primary school children on a daily or weekly basis, providing a new window on cognitive development. We will introduce the origin of this new instrument, called Math Garden, explain its setup, and present and discuss ways to analyze children's individual developmental pathways. [ABSTRACT FROM AUTHOR]