This paper investigates the qualitative behavior and explicit solutions of a system of rational difference equations of the form with arbitrary initial conditions u−2, u−1, u0, v−2, v−1, v0 ∈ \(\mathbb{R}\). The constants ai ∈ {−1, 1} for i = 1, 2, 3, 4. Four special cases of the system are examined based on the sign combinations in the denominators. For each case, we prove that all solutions are periodic with period twelve and derive explicit closed-form formulas for the solutions in terms of the initial values. The proofs are established using mathematical induction. Numerical examples are provided to verify and illustrate the theoretical results, demonstrating the periodic nature of the solutions. This work contributes to the understanding of higher-order rational difference equations and their periodic behavior.

This paper systematically investigates the dynamical properties of a class of max-type fuzzy difference equation. The study first establishes the existence and uniqueness of the solution sequence under given initial conditions with positive fuzzy numbers. Subsequently, by applying the cut-set theory, the fuzzy equation is transformed into a system coupled by two ordinary difference equations. Through a combination of case analysis and mathematical induction, the study rigorously demonstrates that the solutions of this system exhibit global periodicity with a period of 4, while also deriving the exact closed-form expressions of the periodic solutions. Based on the periodic solutions obtained from the ordinary difference system, the research successfully reveals the periodic characteristics of the solutions to the original fuzzy difference equation and rigorously analyzes their boundedness and persistence. Finally, numerical simulations conducted with Matlab 2016 provide robust data support for the theoretical conclusions and the effectiveness of the methodology.

Matching algorithms are fundamental tools in combinatorial problem-solving and network optimization. Acyclic matching is a variant of matching that requires cycle-free induced subgraphs. This constraint creates unique computational challenges while making it essential for applications requiring strict hierarchical structures, that prohibit feedback loops. In this paper, comprehensive algorithmic techniques for solving the acyclic matching problem for certain well-studied graph structures such as triangular, corona and friendship graphs are developed. We present efficient linear time algorithms based on the exact value of acyclic matching derived through mathematical induction. We prove that even though these classes of graphs have very different structures, they all follow the same predictable rules when it comes to acyclic matchings. This consistency allows us to design efficient algorithms that work across all these graph types. The developed algorithms not only improve the understanding of acyclic matching behavior but also support the design of efficient, cycle-free and fault-tolerant network architectures. Such structures are essential for modern applications in communication, scheduling and molecular modeling, contributing to the creation of sustainable, intelligent and resilient systems that promote long-term connectivity and efficient resource use.

This paper presents an exploration of linear piecewise fractional delayed systems, aiming to derive an explicit solution with the help of mathematical induction. We establish the existence and uniqueness of global solutions in the semilinear context through the application of the Banach fixed-point theorem, while also addressing the stability of these semilinear systems. Furthermore, we outline both sufficient and necessary conditions for the controllability of linear control systems utilising the Gramian matrix, and we identify sufficient conditions for the controllability of semilinear control systems through the Schaefer fixed-point theorem. Numerical examples are presented to support our findings.

Abstract We present two recurrence relations involving the central Delannoy numbers D n , the small Schröder numbers s n , and the large Schröder numbers S n using combinatorial arguments in terms of lattice paths. The first relation expresses D n in terms of smaller central Delannoy numbers and large Schröder numbers, and it is not so well known. The second relation expresses s n in terms of smaller small Schröder numbers and large Schröder numbers, and it is a completely new. Using the known recurrence for S n , together with the second relation and the mathematical induction, we provide the new proof of the identity S n = 2 s n , for all natural numbers n .

Three multi-level mixed finite element methods for the steady Boussinesq equations are analyzed and discussed in this paper. The nonlinear and multi-variables coupled problem on a coarse mesh with the mesh size $h_0$ is solved firstly, and then, a series of decoupled and linear subproblems with the Stokes, Oseen and Newton iterations are solved on the successive and refined grids with the mesh sizes $h_j$, $j$ = 1, 2, . . . , $J$. The computational scales are reduced and the computational costs are saved. Furthermore, the uniform stability and convergence results in both $L^2$ - and $H^1$ -norms of are derived under some uniqueness conditions by using the mathematical induction and constructing the dual problems. Theoretical results show that the multi-level methods have the same order of numerical solutions in the $H^1$ -norm as the one level method with the mesh sizes $h_j$ = $h^2_j$−1, $j$ = 1, 2, . . . , $J$. Finally, some numerical results are provided to investigate and compare the effectiveness of the multi-level mixed finite element methods.

This study aims to describe students’ critical thinking skills. Critical thinking is essential for students, especially in 21st-century learning, where many students use technology to explore answers without fully understanding the questions and the solutions. This study focuses on students’ understanding of set operations in mathematics and identifies the difficulties they face. The participants were 27 ninth-grade students at Al-Manar Junior High School. The study employed a qualitative approach with a descriptive design to analyze students’ abilities to solve problems, decide and implement solution steps, perform operations, and apply mathematical induction related to set operations. The findings are expected to reveal students’ critical thinking skills and their understanding of basic mathematical concepts, particularly in set operations, as well as to identify factors influencing their understanding, thereby improving the quality of mathematics instruction at the primary and secondary education levels. The instrument used to collect data was a test administered to students on the topic of set operations. Data analysis followed these steps: data reduction, data display, and drawing conclusions. The results indicate that students’ ability to solve mathematical problems on set operations is uneven. Based on the critical thinking categories used in this study, no students reached the “very good” category; 80% were in the “good” category, 5% in the “fair” category, and 15% in the “low” category. This distribution was evident in students’ responses, suggesting that their overall problem-solving ability can still be considered good. Therefore, the researcher recommends that teachers pay greater attention to and further develop students’ critical thinking skills and their understanding of basic mathematical concepts.

The classical method of proving imperative programs uses Hoor logic, applies Dijks-tra's weakest precondition predicate, and involves specifying the program with first-order predicates as the so-called precondition and postcondition. The classical method of proving functional programs uses the method of mathematical induction and involves specifying the program with recursive functions. This work explores the automated formal verification of imperative programs using recursive specifications. This study appeals to both academics and industry professionals engaged in algorithmic research and software engineering, illuminating the intricate process of formally substantiating the correctness of complex algorithms. The work specifically focuses on developing formal correctness proofs for the Merge Sort algorithm and the Knapsack Problem implementations. By employing contem-porary tools like the Dafny verification system and leveraging the power of recursive speci-fications, the research demonstrates the development of detailed specifications and precise program proofs, successfully verifying an array-based implementation of Merge Sort and requiring the formulation of custom conditions for index handling. The findings of this research hold practical implications, particularly in optimizing software systems where the assurance of algorithmic correctness is paramount and establishes recursive specifications as an effective approach for verifying imperative programs.

Abstract We prove a nearly optimal error bound on the exponential wave integrator Fourier spectral (EWI-FS) method for the logarithmic Schrödinger equation (LogSE) under the assumption of an $H^{2}$-solution, which is theoretically guaranteed. Subject to a Courant–Friedrichs–Lewy (CFL)-type time step size restriction $\tau |\!\ln \tau | \lesssim h^{2}/|\!\ln h|$ for obtaining the stability of the numerical scheme affected by the singularity of the logarithmic nonlinearity, an $L^{2}$-norm error bound of order $O(\tau |\!\ln \tau |^{2} + h^{2} |\!\ln h|)$ is established, where $\tau $ is the time step size and $h$ is the mesh size. Compared to the error estimates of the LogSE in the literature, our error bound either greatly improves the convergence rate under the same regularity assumptions or significantly weakens the regularity requirement to obtain the same convergence rate. Moreover, our result can be directly applied to the LogSE with low regularity $L^\infty $-potential, which is not allowed in the existing error estimates. Two main ingredients are adopted in the proof: (i) an $H^{2}$-conditional $L^{2}$-stability estimate, which is established using the energy method to avoid singularity of the logarithmic nonlinearity and (ii) mathematical induction with inverse inequalities to control the $H^{2}$-norm of the numerical solution. Numerical results are reported to confirm our error estimates and demonstrate the necessity of the time step size restriction imposed. We also apply the EWI-FS method to investigate soliton collisions in one dimension and vortex dipole dynamics in two dimensions.

This paper considers the number of limit cycles of a piecewise cubic Hamiltonian system with general polynomial perturbations. We express the first-order Melnikov function as a linear combination of several generating integrals with polynomial coefficients, and prove that the coefficients of these polynomials are independent using mathematical induction. Then we obtain lower bounds for the number of limit cycles near the centre and heteroclinic loop by applying asymptotic expansions.

Objective: To investigate the effectiveness of project-based learning in developing mathematical thinking skills in mathematics among gifted intermediate-level students. Design: The researcher followed a quasi-experimental approach. The sample consisted of 24 gifted intermediate-level students in Al-Ahsa Governorate who had passed the Mawhiba Scale. The researcher designed a project-based educational program, based on the models of (Krajcik et al., 1999; Krauss & Boss, 2013). A mathematical thinking skills test tool was administered to the sample. Results: The study results revealed statistically significant differences between the mean scores of the experimental group students in the pre- and post-tests of the skills [logical thinking (Z = 2.67, p = < 0.05) - symbols and semantics (Z = 2.41, p = < 0.05) - mathematical deduction (Z = 2.43, p = < 0.05) - mathematical induction (Z = 2.39, p = < 0.05) - mathematical thinking skills as a whole (Z = 2.61, p = < 0.05)] in favor of the experimental group. Conclusion: Employing project-based learning in teaching gifted students aligns with the Kingdom of Saudi Arabia's Vision 2030, which emphasizes the importance of equipping students with the knowledge and skills of this era, and discovering their abilities and talents to nurture and support them, transforming them from an idea in the mind into a tangible object (tool) that can be used as a product that serves the nation and its people.

State bounding for discrete-time switched genetic regulatory networks with time delay and exogenous disturbances.

Problem. Construction of houses, structures, and road embankments in conditions of dense urban development requires taking into account the mutual influence of settlements of the foundations of nearby houses and structures on the stress-strain state (hereinafter referred to as the SSS) of each other. The problem is that the settlements of foundations, taking into account their mutual influence, are determined using the scheme of joint calculation of the SSS of the system "base - foundations - buildings structures" (hereinafter referred to as BFBS) and boundary and finite element techniques, may differ significantly from those calculated using the methodology of Ukrainian state building codes (hereinafter referred to as DBN). At the same time, the "corner points" method recommended in DBN V.2.1-10-2018 and DBN V.2.1-10-2010 for determining foundation settlements, taking into account their mutual influence, can be used only when using a separate calculation scheme. In addition, the "corner point method" is incorrect (or completely impossible) with different depths of the footing of neighboring foundations. Therefore, there is a problem of jointly calculating the SSS of the BFBS system with the use of the methods recommended in the DBN for determining foundation settlements. The research materials presented in this article are aimed at solving this problem. Methodology: Two models of BFBS were developed to conduct numerical analysis using the iteration process and the application of the finite element method technique. The first model is a flat frame on separate foundations, and the second is two slab foundations located next to each other. When performing calculations, real (first model) and actual (second model) characteristics of the underlying soils and materials from which the foundations and buildings. Also, to substantiate the convergence criterion of the iteration process, the method of mathematical induction was applied. Originality. For the first time, it was possible to substantiate the conditions for the convergence of the iteration process when determining the SSS of the BFBS system for several buildings located on separate and (or) slab foundations. Practical value. The convergence criteria for the iteration process in determining the VAT of the BFBS system were obtained. This allowed us to solve the problem of determining the mutual influence of several neighboring foundations and structures located on them on each other's SSS, provided that the foundation settlements are determined using the DBN method, and the forces and deformations in the foundations and buildings structures are determined using the theory of elasticity.

ABSTRACT This article investigates the distributed fusion (DF) filtering issue for multi‐sensor multi‐rate systems (MSMRSs) with packet disorders (PDs) under hybrid cyber attacks (HCAs). The PDs caused by random transmission delay are characterized by a random variable that has the known probability distribution, and two groups of Bernoulli random variables are adopted to depict the occurrence of HCAs. To facilitate the design of the filtering scheme, a virtual measurement method is introduced to transform the original multi‐rate systems into single‐rate systems. The purpose of this paper is to design a new local filtering scheme handling PDs, HCAs as well as stochastic nonlinearities simultaneously, and minimize the upper bound (UB) on each local filtering error covariance (LFEC) by selecting an appropriate gain matrix. By using the mathematical induction method, a sufficient condition is obtained to ensure that each LFEC is uniformly bounded. In addition, local estimates are fused based on the inverse covariance intersection (ICI) fusion method. Finally, it is shown that the proposed fusion filtering algorithm is effective through a simulation experiment.

Abstract Mathematical Induction (MI) plays a central role in mathematics, both for its theoretical and foundational implications and for the variety of its applications as a proving or defining scheme. Over the last decades, research in Mathematics Education has investigated MI from a didactic perspective, producing numerous studies with different foci of analysis. This paper presents a review of this body of research. Relevant studies have been firstly searched on the online database SCOPUS and then snowballing was applied to include other not yet found papers. This process resulted in a corpus of 30 studies, whose analysis is presented in this paper. Three main directions of discussion are identified: logical-epistemological aspects involved in MI, students’ difficulties with MI, and effective teaching experiences for the learning of MI. Overall, this review underscores the multifaceted complexity of MI as an educational issue, spanning logical-epistemological and didactic dimensions. Hopefully the presented discussion can provide a useful starting point for educators and researchers looking to delve into this theme.

This article presents the proof of a number of theorems using the convenient method of mathematical induction.

This study investigates the approximation of the natural logarithm function ln(x) for integer values x=2,3,…,100 using two approaches: an infinite geometric series and a Riemann integral–based method developed in this research. The study first proves the recursive formula of ln(x) through the Riemann integral using mathematical induction, establishing a theoretical foundation for the method. Numerical evaluations are then carried out, with the Riemann integral computed both analytically and numerically using partition values n = 10,100,1000,10000,100000,1000000. The two approaches are compared in terms of accuracy and computational efficiency, with accuracy measured using the Mean Absolute Percentage Error (MAPE). The results show that while the geometric series provides higher accuracy, the proposed Riemann integral method demonstrates superior execution speed and serves as a fundamental basis for developing more advanced numerical integration techniques.

The aim of the present work is to investigate the efficiency of spectral element method for the numerical solution of the two-dimensional distributed-order fractional cable equation on regular and irregular domains. To this end, we first employ Gauss-Lobatto-Legendre quadrature to approximate the distributed-order integrals. The equation is then transformed into a multi-term fractional equation. Afterward, we use the finite difference method in the time direction and obtain a semi-discrete scheme of order O ( τ 2 ) . We prove that the semi-discrete method is unconditionally stable using the mathematical induction. Then we make the fully discrete scheme using the spectral element method in spatial directions and obtain an error bound for the proposed method. Finally, to show the versatility and applicability of the method, we implement it on both regular and irregular domains. The results of the proposed method are compared with other well-known methods in the literature, tested on both regular and irregular convex domains. Furthermore, we demonstrate the method's efficiency and high accuracy on non-convex domains in which has not been addressed in prior works.

Real analysis is one of the main branches of mathematics that serves as a fundamental foundation for the development of science and technology. This study emphasizes the understanding of basic concepts such as real numbers, limits, continuity, the principle of mathematical induction, as well as deductive and axiomatic approaches as a strong framework for mathematical proofs. This research employs the Systematic Literature Review (SLR) method by examining articles and journals related to both the theoretical aspects and the implementation of real analysis in solving mathematical problems. The findings indicate that the application of Polya’s method, visualization through demonstration, and the use of modern technology such as augmented reality (AR) can enhance the effectiveness of real analysis learning. These approaches help students connect abstract concepts with real applications, thereby strengthening logical, systematic, critical, and rigorous thinking skills. However, several learning difficulties are also identified, including the complexity of the material, weak learning habits, social influences, and inappropriate teaching strategies. Therefore, more applicative, interactive, and contextual learning strategies are needed to support the achievement of learning objectives. The conclusion of this review highlights that the integration of theoretical understanding, problem-solving methods, and innovative technology is a strategic step to improve students’ thinking quality in real analysis courses. Thus, real analysis is not only positioned as a theoretical subject but also as a medium for developing problem-solving abilities and higher-order thinking skills that are relevant to both academic needs and real-world applications.

This study aims to describe the ability of mathematical proof using mathematical induction method of students who have visual, auditorial and kinesthetic learning styles. The subjects of this research are 3 students who are representatives of each learning style, namely 1 student who has a visual learning style, 1 student who has an auditory learning style and 1 student who has a kinesthetic learning style. This research is a descriptive research that describes students' mathematical proof ability using mathematical induction method adapted to visual, auditory and kinesthetic learning styles. The results showed that students with visual and auditorial learning styles were able to perform mathematical proof using mathematical induction in the basic steps and induction steps but only up to the correct assumption for n = k while students with kinesthetic learning styles could not apply the principle of mathematical induction correctly, in th basic steps students substituted n = 1 and n = 2 into the statement and ther were error in arithmetic operations.