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This paper proposes a framework for constructing high-order linearly implicit exponential integrators that conserve a quadratic invariant. This is then applied to the scalar auxiliary variable (SAV) approach. Quadratic invariants are significant objects that are present in various physical equations and also in computationally efficient conservative schemes for general invariants. For instance, the SAV approach converts the invariant into a quadratic form by introducing scalar auxiliary variables, which have been intensively studied in recent years. In this vein, Sato et al. (Appl. Numer. Math. 187, 71-88 2023) proposed high-order linearly implicit schemes that conserve a quadratic invariant. In this study, it is shown that their method can be effectively merged with the Lawson transformation, a technique commonly utilized in the construction of exponential integrators. It is also demonstrated that combining the constructed exponential integrators and the SAV approach yields schemes that are computationally less expensive. Specifically, the main part of the computational cost is the product of several matrix exponentials and vectors, which are parallelizable. Moreover, we conduct some mathematical analyses on the proposed schemes. [ABSTRACT FROM AUTHOR]

Abstract In the present article, the Hyers–Ulam stability of the following inequality is analyzed: 0.1 { d ( f ( ı + ȷ ) , ( f ( ı ) + f ( ȷ ) ) ) ≤ d ( ρ 1 ( ( f ( ı + ȷ ) + f ( ı − ȷ ) , 2 f ( ı ) ) ) d ( f ( ı + ȷ ) , ( f ( ı ) + f ( ȷ ) ) ) ≤ + d ( ρ 2 ( 2 f ( ı + ȷ 2 ) , ( f ( ı ) + f ( ȷ ) ) ) ) $$ \textstyle\begin{cases} d (f(\imath +\jmath ), \ (f(\imath )+ \ f(\jmath )) )\leq d (\rho _{1}((f(\imath +\jmath )+ f(\imath - \jmath ),\ 2f(\imath )) ) \\ \hphantom{ d (f(\imath +\jmath ), \ (f(\imath )+ \ f(\jmath )) )\leq}{}+ d (\rho _{2} (2f (\frac{\imath +\jmath}{2} ), \ (f(\imath )+ f(\jmath )) ) ) \end{cases} $$ in the setting of digital metric space, where ρ 1 $\rho _{1}$ and ρ 2 $\rho _{2}$ are fixed nonzero complex numbers with 1 > 2 | ρ 1 | + | ρ 2 | $1>\sqrt{2}|\rho _{1}|+|\rho _{2}|$ by using fixed point and direct approach.

Switched positive Persidskii systems with distributed and unbounded delays are studied. Right-hand sides of these systems are linear combinations of nonlinearities of a sector type. Special constructions of diagonal Lyapunov--Krasovskii functionals are proposed and conditions are derived under which the absolute stability of the considered systems can be proved with the aid of such functionals. The developed approaches are applied to the stability analysis of a mechanical system with switched nonlinear positional forces and to a problem of mobile agent deployment. Results of numerical simulations are presented confirming the theoretical conclusions. [ABSTRACT FROM AUTHOR]

This special issue of the Journal Mathematical Methods in the Applied Sciences on the topic of Modelling, Analysis, and Applications is devoted to the development of mathematical modelling, analysis, and applications, from theoretical and numerical perspectives, involving different applied sciences and engineering. It collects invited contributions, from selected quality full papers submitted to the International Conference on Mathematical Analysis and Applications in Science and Engineering (ICMA2SC2022). The conference took place in Porto, Portugal in 27–29 June 2022 (https://www.isep.ipp.pt/Page/ViewPage/ICMASC). [ABSTRACT FROM AUTHOR]

In this paper, we study the Onsager-Machlup function and its relationship to the Freidlin-Wentzell function for measures equivalent to arbitrary infinite dimensional Gaussian measures. The Onsager-Machlup function can serve as a density on infinite dimensional spaces, where a uniform measure does not exist, and has been seen as the Lagrangian for the "most likely element". The Freidlin-Wentzell rate function is the large deviations rate function for small-noise limits and has also been identified as a Lagrangian for the "most likely element". This leads to a conundrum - what is the relationship between these two functions? We show both pointwise and G-convergence (which is essentially the convergence of minimizers) of the Onsager-Machlup function under the small-noise limit to the Freidlin-Wentzell function - and give an expression for both. That is, we show that the small-noise limit of the most likely element is the most likely element in the small noise limit for infinite dimensional measures that are equivalent to a Gaussian. Examples of measures include the law of solutions to path-dependent stochastic differential equations and the law of an infinite system of random algebraic equations. [ABSTRACT FROM AUTHOR]

In this article, we consider a nonlinear elliptic unilateral equation whose model is − ∑ i = 1 N ∂ i σ i (x , u , ∇ u) + L (x , u , ∇ u) + N (x , u , ∇ u) = μ − div ϕ (u) in Ω. We prove the existence of entropy solutions for the aforementioned equation in the anisotropic Sobolev space, under the hypotheses, μ = f − div F belongs to L 1 (Ω) + W − 1 , p ′ (Ω) . The nonlinear terms L (x , s , ∇ u) satisfy the sign and growth conditions, and N (x , s , ∇ u) verifies only the growth conditions. [ABSTRACT FROM AUTHOR]

"The Story of Proof: Logic and the History of Mathematics" by John Stillwell is a book that explores the development of mathematical proofs throughout history. The author examines the emergence of proofs in antiquity and the long-standing dominance of Euclid's foundations of geometry. Stillwell presents a historical narrative of mathematical achievements and subfields, highlighting the ways in which different areas of mathematics are interconnected. The book also discusses the influence of linguistic and conceptual developments on the evolution of mathematics, including the role of infinity and the impact of notation. While the text assumes some prior knowledge of mathematical proofs, it provides numerous examples and explanations to help readers understand the subject. The book concludes with a discussion of modern foundations of mathematics and the potential for computer-assisted proof and automated proof verification. [Extracted from the article]

Mathematics is the most logical and unbiased scientific field there is. What is true in mathematics today will forever stay true, and mathematical arguments will always clearly indicate which side "wins". These are some of the characteristics of mathematics that I have liked very much since I was young. In this article, I will try to demonstrate the beauty of mathematics and show you that it appears in many situations in our daily lives--including places we might not expect. Then, I will give you a taste of two studies I conducted in a mathematical field called combinatorics, and I will explain how they are related to everyday problems. One of these problems is how to perform accurate mathematical analyses of large streams of data that we cannot save in a computer's memory. Finally, I will try to give you an idea of my life as a mathematician--the challenges and difficulties, alongside the immense enjoyment and satisfaction I derive from them. Join me to an extended dive into my work on the wonders of combinatorics. [ABSTRACT FROM AUTHOR]

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Aims: To let students know that packaging design teaching is both a science and an art like any other course, only when students understand this, can they teach and educate people once they are engaged in this profession. Methods: Observation, experiment, investigation, and other methods were used to carry out a relatively long time of teaching practice in the course of Packaging Design, and the teaching art was explored. Results: Practice shows that if students want to truly realize that teaching is both a science and an art, they must adhere to the purpose of instilling in teaching for a long time, and fully understand the history of teaching artistry, its characteristics, and its functions. Conclusion: Through the mathematical statistical analysis and research of teaching art practice and effect, we believe that any course is not a simple knowledge teaching, it is a process of educating people. We must cultivate the next generation to be well-rounded individuals, ensuring they are intellectually enriched and nurtured in moral values, physical health, aesthetic appreciation, and work skills. This holistic approach ensures that while they acquire knowledge, they are also exposed to and trained in the arts and practical skills. Only when people are well-trained can society be harmonious and beautiful. Therefore, the research on teaching art in packaging design has analyzed the three most important artistic effects of teaching blackboard art, language art, and teaching preparation art. Those with good teaching art effects account for 16-18%, those with medium teaching art effects account for 70-80%, and those with poor teaching art effects account for 10-16%.

Several studies have discussed the internalization of Islamic values in mathematics, including Islam and mathematics, internalization of Islamic values in mathematics learning with analogy strategies, mathematical analysis of the philosophy of the Quran, the secret of numbers in the Qur'an. However it has never been discused abaout the internalization of Islamic moderation values in mathematical problem solving. So this research is needed. Moreover, the phenomenon of Islam in Indonesia is currently shifting away from the image of moderate Islam and into a harsher conservative ideology. This is more worrying because it has entered the realm of education. For that it is necessary to internalize the value of Ilamic moderation in mathematical problem solving. One of the strategies that can be done to instill religious moderation values in mathematical problem solving is the analogy strategy. Facts, concepts, and principles in mathematics can be used a analogy to instill religious moderation values. In this study, describe some examples of the application of analogy strategies for internalizing Islamic value. This study uses the library method. The data obtained were publications of research artcles in scientific journals. The analyed articles include three stages, organizing, synthesizing and identifying. The results of data analysis show that the value of tasamuh in mathematics learning can be done through open ended problem solving. The characteristics of open ended problem is problem that have more than one solution and answer tolerance, so each student may have different and equally correct solutions and answers. This is in accordance with the principle of tasamuh value in Islamic moderation. Therefore, on classroom learning, the values of tasamuh can be implemented through open ended problem. [ABSTRACT FROM AUTHOR]

This article introduces the development history of fractional calculus and expounds on the positive significance of fractional calculus in integrating higher mathematics teaching. The author introduces an ordinary differential equation model case appropriately. In this way, ordinary differential equations play an important role in improving college students’ mathematical thinking ability and mathematical application ability.

We give some remarks on the recent paper [J. Math. Inequal. 13 (4) (2019), 1235– 1244]. Some misleading conclusions of their stability results are carefully discussed and corrected. Moreover, we reestablish their results with a more general assumption and a stronger conclusion. [ABSTRACT FROM AUTHOR]

In this paper, we aim to solving the open question left in [Nie, Yuan: Nonlinear Anal 196 (2020); J. Math. Anal. Appl 505 (2022) and Xiao, Fei: J. Math. Anal. Appl 514 (2022)]. We prove that a multidimensional chemotaxis system is ill-posedness in B ˙ 2 d , r - 3 2 × ( B ˙ 2 d , r - 1 2 ) d when 1 ≤ r < d due to the lack of continuity of the solution. [ABSTRACT FROM AUTHOR]

Our study commemorates this event by presenting a retrospective of the publications related to the use of mathematical tools for the analysis of economic, ecological, and environmental phenomena. We analyzed 1257 scientific publications using bibliometric techniques to examine the most productive and influential authors and their contributions in the economic, ecological, and environmental fields. Co-authorship among the top authors and countries, co-occurrence of the keywords, bibliographic coupling of the documents and authors, and author co-citation were analyzed by applying network analysis techniques using VOSviewer software, identifying the intellectual structure of the research and the collaborative networks in the fields. The results show that mathematics has grown impressively in terms of publication and citation. The contributions come from all over the world, but the majority are from the People's Republic of China and Spain. The results also show themes and trends in the economic, environmental, and ecological fields and a predominant use of mathematical tools in optimization processes in order to rigorously substantiate the decisions of investors and policymakers. Thus, our study offers support for any researcher to understand the current state of the art and develop a comprehensive understanding of journal publications. [ABSTRACT FROM AUTHOR]

Mathematics is one of the areas of knowledge where there are greater difficulties for its learning, due to its complexity, and the little usefulness that is perceived of them in real life, which is why pedagogical alternatives have been developed ( ABP). The usefulness of this tool in learning mathematics, which is sought by solving real problems through the use of mathematics. In this sense, the use of the PBL was evaluated in a group of 60 students of the Software career of the Polytechnic High School of Chimborazo (ESPOCH), 30 before the use of APB and 30 after its use comparing the changes in the academic performance of the students after the implementation of the PBL, as well as the perception of the students in relation to the interest, value, motivation and utilities of mathematics by solving problems using differential equations, The results found show that the students had a greater achievement when understanding and solving the equations adequately using derivation rules and analyzing and adequately interpret the information, which was related to a greater interest (50.85%), assessment (57.36%), satisfaction 57.62 (%) and utility 66.07%, despite the fact that the use of the PBL achieved a significant increase in grades, even the students state that they make a great effort to carry out projects in the area of mathematics, which was expressed by more than half of the students approached, the findings found lead to the conclusion that the use of this learning tool translates into a significant improvement in the acquisition of mathematical skills since a greater understanding and importance of the use of the itself, increases the interest and motivation of students for their study. [ABSTRACT FROM AUTHOR]

In mathematics, the term "differential" refers to several related notions derived from the early days of mathematical analysis, which became rigorous later, such as infinitesimal differences and the derivatives of functions. The notion is used in various areas of mathematics such as algebraic geometry, algebraic topology, calculus, differential geometry etc. The term differential is used non rigorously in calculus referring to an infinitely small ("infinitesimal") variation change in a quantity. For example, if one considers x as a variable, then a "bigger" change in the value of x is often denoted by x. The differential dx is an infinitesimal change of the variable x. The concept of an infinitely small or infinitely slow change is very useful, and there are several of mathematical tools to make it precise. Using calculus, it is possible to relate the infinitesimal changes of several variables to each other using function derivatives. In this article we present the notions of Gateaux and Fréchet differentials of a multivariable function with their properties, geometric interpretation and applications to function approximations. [ABSTRACT FROM AUTHOR]

The mathematics used in physical chemistry has changed greatly in the past forty years and it will certainly continue to change more quickly. Theoretical chemists and physicists must have an acquaintance with abstract mathematics if they are to keep up with their field, as the mathematical language in which it is expressed changes. Thinking about it, in this article, we want to show some of the most important concepts of Mathematical Analysis involved in obtaining analytical functions to represent the potential energy interaction for diatomic systems. A basic guide for the construction of a potential based on Dunham's coefficients and an example of a new potential obtained from this methodology is also presented.

Fuzzy logic is becoming one of the most-influential fields of modern mathematics with applications that impact not only other sciences, but society in general. This newly found interest in fuzzy logic is in part due to the crucial role it plays in the development of artificial intelligence. As a result, new tools and practices for the development of the above-mentioned field are in high demand. This is one of the issues this paper was composed to address. To be more specific, a sizable part of fuzzy logic is the study of fuzzy connectives. However, the current method used to generalize them is restricted to the use of basic automorphisms, which hinders the creation of new fuzzy connectives. For this reason, in this paper, a new method of generalization is conceived of that aims to generalize the fuzzy connectives using polynomial automorphism functions instead. The creation of these automorphisms is achieved through numerical analysis, an endeavor that is supported with programming applications that, using mathematical modeling, validate and visualize the research. Furthermore, the automorphisms satisfy all the necessary criteria that have been established for use in the generalization process and, consequently, are used to successfully generalize fuzzy connectives. The result of the new generalization method is the creation of new usable and flexible fuzzy connectives, which is very promising for the future development of the field.

In addition to reducing R&D investment and changing senior managers or corporate name, the significant uncertainty and reputation shock brought by civil litigation may also have a vital impact on corporate governance of the sued company. We use A-share listed companies in Shanghai and Shenzhen stock exchanges from 2011 to 2018 to empirically examine the impact of civil litigation on corporate governance and its mechanism. It is found that, civil litigation is negatively correlated with corporate governance, and financing liabilities (FL) play an intermediary role in the negative correlation. Furthermore, judicial local protectionism weakens the negative correlation between civil litigation and corporate governance. Finally we close with directions for future research.

In the article we investigated the problem of forming in students' minds an integrated image of the way of solving mathematical problems. There were considered the possibilities of realization of intrasubject integration in the process of studying math. The methodical conditions under which due to the integrated image of the method of solving the problem the realization of the continuity of teaching mathematical disciplines during the transition from school to university was revealed. We have chosen the method of completing the square to describe the authors' methods of teaching math. Curricula and school textbooks currently in force in Ukraine were analyzed. We also considered universities' curricula of Mathematical Analysis. An experimental study with students on the formation of an integrated image of the method of solving problems, which showed positive dynamics and proved the effectiveness of the developed methodology. It was concluded that the formation of an integrated image of the method of solving problems during a long study of mathematics was possible with the correct choice of problems that are solved in a given way; application of the system of didactic principles, in particular systematicity, consistency and perspective; constant generalization and systematization of knowledge, skills and abilities of students. [ABSTRACT FROM AUTHOR]

Creative thinking is the ability to generate new ideas. The affective aspect related to creativity is self-confidence. This study aims to describe the characteristics of students' mathematical creative thinking skills based on self-confidence. Involved as many as 60 students who took the Differential Calculus course. The method used is descriptive qualitative. Data collection was obtained through questionnaires, tests, and interviews. Data analysis techniques include data reduction, data presentation, and concluding. Subjects were grouped into three self-confidence categories: low, moderate, and high, based on the questionnaire results. Using the purposive sampling technique, two people were taken as subjects in each category. The study results explain that students with low, moderate, and high self-confidence can write down several different answers and review them in detail. Students in the high self-confidence category can find relationships between concepts and answer questions in various or unique solving algorithms. Selfconfidence is directly proportional to the achievement of creative thinking skills. [ABSTRACT FROM AUTHOR]

The rhythmic and stable operation of trolleybuses and autonomous trolleybuses or urban electric buses, depends to a large extent on the reliability of the equipment installed on the trolleybus. The actual operational reliability of trolleybus electrical equipment (EE) depends on its technical condition. Under the influence of external factors and specific operating modes, the technical condition of the equipment is continuously deteriorating, reliability indicators are decreasing, and the number of failures is increasing. Using the mathematical theory of reliability, probability theory and mathematical statistics, numerical methods of solving nonlinear and transcendental equations, this article defines the conditions of diagnostics depending on the intensity of failures and the given probability of failure-free operation of the equipment. Additionally, the inverse problem of determining the current reliability of electrical engineering systems depends on the terms of diagnostics and the intensity of failures being solved. As a result of the processing of statistical information on failures it is established that for the electrical equipment of a trolleybus, after a number of repair measures, the maximum density of failures occurs at a lower mileage, and the probability of failure-free operation can vary depending on the degree of wear of the equipment, i.e., on the number of previous failures. It is theoretically substantiated and experimentally confirmed that the reliability of trolleybus electrical equipment changes according to the exponential law of distribution of a random variable. It has been established that the real averaged diagnostic terms regulated by instructions are not optimal in most cases and differ several times from those defined in this paper. The dependence of switching equipment run-in on time has been clarified, which served as a prerequisite for specifying the inter-repair period for various types of trolleybus electrical equipment. A method of adjustment of the inter-repair time for the electrical equipment of trolleybuses is proposed.

To understand the reasons for the decline in the birth rate of our country, the effects of the decline in the birth rate and the countermeasures needed to reduce the birth rate, as well as the urgency of the reform of education, especially art education.

The article discusses about developing expressive mathematical language in young children. Topics discussed include that teachers can help develop children's expressive mathematical language by giving them opportunities to speak and write as they use hands-on materials alongside, discusses types of mathematical writing including exploratory writing, argumentative writing and mathematically creative writing; and it also includes samples in which teachers notice the math in what a child is doing.

The purpose of this paper is twofold. The first goal is to provide a simple and constructive proof of Korn inequalities in half-space with weighted norms. The proof leads to explicit values of the constants. The second objective is to use these inequalities to show that the linear elasticity system in half-space admits a coercive variational formulation. This formulation corresponds to the physical case in which the solution is evanescent at infinity. [ABSTRACT FROM AUTHOR]

We consider the non-resonance with one-sided superlinear growth conditions for the indefinite planar system z' = f (t, z) from a rotation number viewpoint, and obtain the existence of 2π- periodic solutions by applying a rotation number approach together with the Poincar´e-Bohl theorem. We allow that the angular velocity of solutions of z' = f (t, z) is controlled by the angular velocity of solutions of two positively homogeneous and oddly symmetric systems z' = Li(t, z), i = 1, 2 on the left half-plane, which have rotation numbers that satisfy ρ(L1) > n/2 and ρ(L2) < (n + 1)/2, and allow f (t, z) to grow superlinearly on the right half-plane. In order to estimate the rotation angle difference of solutions, we develop a system methodology of "tracking" the angle difference of solutions of the system z' = f (t, z) on each small interval on the given side under sign-varying conditions. [ABSTRACT FROM AUTHOR]

In this paper, a new class of Hermite-Hadamard type integral inequalities using a strong type of convexity, known as n-polynomial exponential type s-convex function, is studied. This class is established by utilizing the Hölder's inequality, which has several applications in optimization theory. Some existing results of the literature are obtained from newly explored consequences. Finally, some novel limits for specific means of positive real numbers are shown as applications. [ABSTRACT FROM AUTHOR]