Mathematical Challenges Research Articles

A comparative analysis of public key cryptographic algorithms: RSA, ELGamal, and elliptic curve encryption

Cryptography stands as an indispensable and efficient facet within the expansive field of information security. It offers a reliable method for ensuring the confidentiality and integrity of data during the complex process of information transmission between a sender and a recipient. Beyond this, exclusive decryption privileges are meticulously reserved for the designated recipient. This individual holds the exclusive authority to decipher the transmitted information, which has been encrypted by the key holder beforehand. This scholarly investigation introduces and delves into three prevalent cryptographic algorithms: RSA, El-Gamal, and Elliptic Curve Cryptography. It offers a discerning examination and contrast of the underlying mathematical challenges associated with each sophisticated method. The discourse unfolds a detailed comparative analysis of these three pivotal algorithms, zeroing in on their crucial aspects such as key size length and operational running time. The in-depth exploration within this study aims to shed light on the intricate workings, strengths, and potential limitations of RSA, El-Gamal, and Elliptic Curve Cryptography. By unraveling these aspects, the study contributes to a richer understanding and more informed choices in the practical application of cryptographic algorithms, enhancing the overarching realm of information security in an increasingly digital and interconnected world.

Analysis of nonlinear multi-field coupling responses of piezoelectric semiconductor rods via machine learning

ABSTRACT Piezoelectric semiconductors (PSs) have widespread applications in semiconductor devices due to the coexistence of piezoelectricity and semiconducting properties. It is very important to conduct a theoretical analysis of PS structures. However, the present of nonlinearity in the partial differential equations (PDEs) that describe those multi-field coupling mechanical behaviors of PSs poses a significant mathematical challenge when studying these PS structures. In this paper, we present a novel approach based on machine learning for solving multi-field coupling problems in PS structures. A physics-informed neural networks (PINNs) is constructed for predicting the multi-field coupling behaviors of PS rods with extensional deformation. By utilizing the proposed PINNs, we evaluate the multi-field coupling responses of a ZnO rod under static and dynamic axial forces. Numerical results demonstrate that the proposed PINNs exhibit high accuracy in solving both static and dynamic problems associated with PS structures. It provides an effective approach to predicting the nonlinear multi-field coupling phenomena in PS structures.

MetaPINNs: Predicting soliton and rogue wave of nonlinear PDEs via the improved physics-informed neural networks based on meta-learned optimization

Efficiently solving partial differential equations (PDEs) is a long-standing challenge in mathematics and physics research. In recent years, the rapid development of artificial intelligence technology has brought deep learning-based methods to the forefront of research on numerical methods for partial differential equations. Among them, physics-informed neural networks (PINNs) are a new class of deep learning methods that show great potential in solving PDEs and predicting complex physical phenomena. In the field of nonlinear science, solitary waves and rogue waves have been important research topics. In this paper, we propose an improved PINN that enhances the physical constraints of the neural network model by adding gradient information constraints. In addition, we employ meta-learning optimization to speed up the training process. We apply the improved PINNs to the numerical simulation and prediction of solitary and rogue waves. We evaluate the accuracy of the prediction results by error analysis. The experimental results show that the improved PINNs can make more accurate predictions in less time than that of the original PINNs.

Pengaruh Konsep Gamifikasi Dalam Model Pembelajaran IMPROVE Terhadap Kemampuan Berpikir Kritis Matematis Siswa SMP

All students should develop their mathematical critical thinking skills to aid their learning and overcome various mathematical challenges. The purpose of this study is to find out how the idea of gamification of IMPROVE learning model affects students' mathematical critical thinking skills. This study used a quasi-experimental design to collect data by using a test in the form of description questions. Normality test and homogeneity test are the two data analysis procedures used. One-way ANOVA test was used in testing the hypothesis of this study. Based on the results of the study, the idea of gamification of IMPROVE learning model has an impact on students' mathematical critical thinking skills.

Using Digital Learning Media to Increase Student Creativity in Solving Mathematics Problems

This research explores the use of digital learning media as a means of increasing students' creativity in solving mathematics problems. Conventional teaching methods are often inadequate to develop students' creative abilities in solving mathematical problems. Through the application of digital learning media, such as educational games and interactive simulations, this research aims to create a learning environment that motivates students to think creatively and apply mathematical concepts in real contexts. This research methodology involves students' active participation in digital interactive activities, followed by an evaluation of their level of creativity in responding to mathematical challenges. It is hoped that the results of this research will provide an in-depth view of the potential of digital learning media as an effective tool for increasing student creativity in learning mathematics

Pedagogy of Mathematics

This paper delves into the realm of mathematics pedagogy, exploring various facets of teaching and learning mathematics. Through the analysis of the modified Fennema-Sherman Attitude Scales and an in-depth examination of best practices, this paper aims to shed light on the attitudes of students toward mathematics and provide recommendations for improving mathematics education. The paper emphasizes the importance of cultivating a growth mindset, promoting diversity and inclusion, and incorporating active learning strategies in mathematics instruction. It highlights the significance of continuous professional development for educators and support systems for students facing challenges in mathematics. As we conclude this internship, it is clear that mathematics education is not static but a dynamic field that demands collaboration and a commitment to excellence. This paper serves as a compass for future endeavors, guiding the way toward a mathematics pedagogy that empowers learners and transforms perceptions of mathematics.

Problem-Based Learning in Improving Problem-Solving Ability and Interest in Learning Mathematics: An Empirical Study

The main challenge in learning mathematics is improving students' problem-solving abilities and strengthening interest in learning. This research focuses on the Problem-Based Learning (PBL) model's influence on students' problem-solving abilities and interest in learning mathematics. This research investigates how PBL can influence the improvement of students' problem-solving abilities and interest in learning mathematics. The method used was an experimental approach with one group following PBL-based learning and a control group using conventional learning. Data was collected through problem-solving ability tests and learning interest questionnaires. The research results showed that PBL significantly improved students' problem-solving abilities and resulted in a positive increase in interest in learning mathematics. Conclusion: These findings effectively strengthen students' problem-solving abilities and awaken their interest in mathematics. The implication is the importance of implementing PBL in mathematics learning to create more competent and enthusiastic students to explore the world of mathematics.

Cloud-Battery management system based health-aware battery fast charging architecture using error-correction strategy for electric vehicles

The health-aware battery fast charging (HABFC) strategy for electric vehicles (EVs) is predominantly dependent on the input conditions. These conditions are set using a set of various type of sensors (temperature, current, voltage, etc.), which can seldom be erroneous. Also, one of the primary challenges associated with physical battery management system (BMS) is the lack of memory required to execute intricate state estimation methodologies. Therefore, this paper proposes the Cloud-BMS based health-aware battery fast charging (HABFC) architecture with error correction strategy for reducing the charging time and increasing the cycle life of the EV battery. A cloud BMS has been integrated in the HABFC architecture to reduce the dependency on erroneous sensors. To design an effective cloud BMS, this paper reviews the seven most critical battery state estimation techniques (state of charge (SoC), state of health (SoH), state of power (SoP), state of energy (SoE), state of temperature (SoT), state of function (SoF), and state of safety (SoS)). Based on comparison of mathematical computational methods, its accuracy and challenges, the best computational method for each state estimation technique has been determined. The cloud BMS makes all state estimations based on parameters estimated using the virtual battery. Further, this paper presents an error correction strategy that helps correct the error in physical BMS employing a master-slave configuration. The proposed cloud BMS based HABFC architecture is validated using a cell cycling setup. It can be concluded that the proposed architecture produces 21.24% more cycle life as compared to fast charging without HABFC and 11% more cycle life as compared to cell charged using HABFC without cloud BMS.

Post-Quantum Security: Opportunities and Challenges

Cryptography is very essential in our daily life, not only for confidentiality of information, but also for information integrity verification, non-repudiation, authentication, and other aspects. In modern society, cryptography is widely used; everything from personal life to national security is inseparable from it. With the emergence of quantum computing, traditional encryption methods are at risk of being cracked. People are beginning to explore methods for defending against quantum computer attacks. Among the methods currently developed, quantum key distribution is a technology that uses the principles of quantum mechanics to distribute keys. Post-quantum encryption algorithms are encryption methods that rely on mathematical challenges that quantum computers cannot solve quickly to ensure security. In this study, an integrated review of post-quantum encryption algorithms is conducted from the perspective of traditional cryptography. First, the concept and development background of post-quantum encryption are introduced. Then, the post-quantum encryption algorithm Kyber is studied. Finally, the achievements, difficulties and outstanding problems in this emerging field are summarized, and some predictions for the future are made.

Eccentric catastrophes & what to do with them

Analytic modeling of gravitational waves (GWs) from inspiraling eccentric binaries poses an interesting mathematical challenge. When constructing analytic waveforms in the frequency domain, one has to contend with the fact that the phase of the Fourier integral in non-monotonic, resulting in a breakdown of the standard stationary phase approximation (SPA). In this work, we study this breakdown within the context of catastrophe theory. We find that the SPA holds in the context of eccentric Keplerian orbits when the Fourier frequency satisfies , where are integer multiples of the apocenter/pericenter frequencies, respectively. For values outside of this interval, the phase undergoes a fold catastrophe, giving rise to an Airy function approximation of the Fourier integral. Using these two different approximations, we generate a matched asymptotic expansion that approximates generic Fourier integrals of Keplerian motion for bound orbits across all frequency values. This asymptotic expansion is purely analytic and closed-form. We discuss several applications of this investigation and the resulting approximation, specifically: (1) the development and improvement of effective fly-by waveforms for binary black holes, (2) the transition from burst emission in the high eccentricity limit to wave-like emission in the quasi-circular limit, which results in an analogy between eccentric GW bursts and Bose–Einstein condensates, and (3) the calculation of f-mode amplitudes in eccentric binary neutron stars and black hole-neutron star binaries in terms of complex Hansen coefficients. The techniques and approximations developed herein are generic, and will be useful for future studies of GWs from eccentric binaries within the context of post-Newtonian theory.

Run-and-tumble oscillator: Moment analysis of stationary distributions

When it comes to active particles, even an ideal gas model in a harmonic potential poses a mathematical challenge. An exception is a run-and-tumble particles (RTP) model in one dimension for which a stationary distribution is known exactly. The case of two dimensions is more complex, but the solution is possible. Incidentally, in both dimensions the stationary distributions correspond to a beta function. In three dimensions, a stationary distribution is not known but simulations indicate that it does not have a beta function form. The current work focuses on the three-dimensional RTP model in a harmonic trap. The main result of this study is the derivation of the recurrence relation for generating moments of a stationary distribution. These moments are then used to recover a stationary distribution using the Fourier–Lagrange expansion.

The application of blended learning in mathematics teacher education: Protocol for a systematic review.

In recent decades, especially in higher education, blended learning has become the most commonly used active teaching strategy. Because of the COVID-19 pandemic, blended learning, which combines face-to-face and online components, is believed to overcome the shortcomings of conventional teaching methods, particularly in face-to-face interactions. Based on PRISMA guidelines, this study follows the protocol for a systematic review of blended learning applications in mathematics teacher education. This systematic review study aims to comprehend the potential of blended learning for various mathematical topics, the common blended learning models, and the benefits and challenges this teaching approach presents for educational stakeholders. Searches will be performed in various electronic databases, including Scopus, ScienceDirect, Taylor & Francis Online, Mendeley, Google Scholar, and ERIC. Selected studies that satisfy the inclusion criteria will document the use of various blended learning models in a range of mathematical topics as well as the advantages and disadvantages of this method of instruction. The data extraction process will be carried out independently by various authors, and the results of the data synthesis will be reported per the chosen studies, methodological considerations, and key findings. This review will provide information about the application of blended learning and its benefits and challenges in mathematics teacher education to support educational stakeholders in mathematics teacher education.

REcaNet: Residual neural networks with initialized weights and attention mechanism for image propagating in multimode optical fiber restoration

Training a neural network to reconstruct images from time-series waveforms obtained from fiber optic probes not only yields high-quality, content-aware images but can also acquire different types of images from lower quality training images. Image reconstruction, as an inverse problem, involves using collected signals and system models to retrieve desired images, encountering mathematical challenges like distortion and degradation. In this paper, we introduce REcaNet, a multi-mode fiber image restoration model based on an enhanced residual convolutional neural network (CNN). The network employs a symmetrical architecture that downscales the image before upscaling it for restoration, and it reconstructs the high-level semantic feature map generated by the encoder to the original image resolution. Additionally, we incorporate weight initialization, attention mechanisms, and residual connections to enhance the final restored feature map with more low-dimensional features and promote fusion of features from distinct layers. The algorithm performs well on three datasets collected by multi-mode fibers, namely Minist, Clothes, and Omiglot. Among them, various indicators such as SSIM have been significantly improved.

The gradual removal of Hertz pressure from the surface of elastic half-space

Contact stress determination in non-stationary dynamic loading of elastic bodies is crucial for modelling structures at high speeds, but it presents mathematical challenges due to the time-dependent and often unknown contact area size and shape. The study aims to obtain an energy remainder estimation that forms waves during the contact interaction of elastic bodies, based on the exact solutions of non-stationary problems for an elastic half-space. For this purpose, the problem of the instantaneous loading half-space as an additional research problem was reconstructed using the Hankel transform concerning a radial coordinate and the Laplace transform concerning a time variable. The method of derivation of the displacements at an elastic half-space loaded (unloaded) gradually by Hertz's contact pressure has been proposed. Its availability made it possible to pass to the solution of the main problem – the problem of gradual loading of the half-space surface by Hertz pressure. The possibility of changing of the order of differentiation and integration operations in the obtained representation is substantiated based on the integrand properties. The cases when the speed of the indenter was constant when its motion was uniformly accelerated and when the motion corresponded to the law of the first quarter of the cosine period in the time were considered. It was concluded that the distribution of dynamic contact stresses is similar to the Hertz distribution. An estimation of the part of the energy spent on the formation of elastic waves was made for various laws of unloading. The practical significance of this study lies in its development of an effective method for calculating normal displacements on a loading area in dynamic contact interactions of elastic bodies, which can be valuable for modelling structures at high speeds

Parametrization of embedded-atom method potential for liquid lithium and lead-lithium eutectic alloy

Liquid lead-lithium eutectic remains as a promising candidate for various breeding-blanket designs in future nuclear-fusion technologies. The lack of a generalized theory of interatomic forces in the liquid state is reflected on the wide variety of proposed functional forms to describe interatomic interactions even in simple liquids. Computer simulations facilitate the study of liquid metal properties, due to mathematical and experimental challenges. A classical-MD EAM potential is parametrized using mechanical and non-mechanical (melting-point) properties to minimize the arbitrariness of functional forms, where the employed pair potential stems from the liquid-state theory to avoid the issue of the uniqueness of the potential. Enhanced performance is obtained for liquid density, energy, structure, diffusivity and shear viscosity of Li, and their temperature-dependencies. In a similar manner, reference experimental and ab initio MD data are used to parametrize a functional to describe Pb-Li pairwise interactions in liquid Pb-Li alloy, which is used with the derived EAM of liquid Li and a reference EAM of liquid Pb to investigate properties of liquid Pb-Li alloy. Enhanced transferability characteristics are obtained for low-in-lithium liquid Pb-Li melts, where Coulombic interactions are negligible. In specific, the exhibited behaviour of Li in liquid lead-lithium eutectic is consistent with findings from ab initio MD methods, and drastically different from predictions of previous C-MD studies which suggested a substantial segregation of Li atoms instead of dispersion. It is concluded that the functional form of the pair potential and its uniqueness influence both the pure liquid-metal properties and the validity of the potential transferability in multi-component systems, where a theoretical functional results in enhanced performance in pure and alloyed liquid systems.

Fractional Reasoning: Fostering Fundamental Knowledge

Fractional reasoning is a crucial aspect of mathematical understanding that plays a fundamental role in various mathematical concepts, real-world applications, and higher-level mathematical skills. The ability to comprehend and work with fractions is essential for students to develop a solid foundation in mathematics. However, fractional reasoning is often a challenging area for many students, requiring a deep understanding of concepts such as equivalence, ordering, operations, and connections to other mathematical domains. This study aimed to investigate to what extent primary school pupils develop fractional reasoning and the ability to solve related problems. The research involved a sample of eight primary school pupils from Perak (in Malaysia) participating in an interview. The findings revealed that the participants relied on representation methods of enactive and symbolic representations when working on fractions of an area, while they predominantly utilised symbolic representations when determining fractions for a set of objects. These results shed light on the students' fractional reasoning strategies, which are required in solving many other problems in the context of mathematical tasks. Based on the findings, it is recommended that educators employ instructional strategies such as representations that promote fractional reasoning, such as incorporating real-world contexts, to foster students' understanding and proficiency in addressing complex mathematical challenges. Keywords: Representation, Enactive, Symbolic, Fractions, Fractional Reasoning

Sharp interface analysis of a diffuse interface model for cell blebbing with linker dynamics

AbstractWe investigate the convergence of solutions of a recently proposed diffuse interface/phase field model for cell blebbing by means of matched asymptotic expansions. It is a biological phenomenon that increasingly attracts attention by both experimental and theoretical communities. Key to understanding the process of cell blebbing mechanically are proteins that link the cell cortex and the cell membrane. Another important model component is the bending energy of the cell membrane and cell cortex which accounts for differential equations up to sixth order. Both aspects pose interesting mathematical challenges that will be addressed in this work like showing non‐singularity formation for the pressure at boundary layers, deriving equations for asymptotic series coefficients of uncommonly high order, and dealing with a highly coupled system of equations.

A Hybrid Non-Polynomial Spline Method and Conformable Fractional Continuity Equation

This paper presents a groundbreaking numerical technique for solving nonlinear time fractional differential equations, combining the conformable continuity equation (CCE) with the Non-Polynomial Spline (NPS) interpolation to address complex mathematical challenges. By employing conformable descriptions of fractional derivatives within the CCE framework, our method ensures enhanced accuracy and robustness when dealing with fractional order equations. To validate our approach’s applicability and effectiveness, we conduct a comprehensive set of numerical examples and assess stability using the Fourier method. The proposed technique demonstrates unconditional stability within specific parameter ranges, ensuring reliable performance across diverse scenarios. The convergence order analysis reveals its efficiency in handling complex mathematical models. Graphical comparisons with analytical solutions substantiate the accuracy and efficacy of our approach, establishing it as a powerful tool for solving nonlinear time-fractional differential equations. We further demonstrate its broad applicability by testing it on the Burgers–Fisher equations and comparing it with existing approaches, highlighting its superiority in biology, ecology, physics, and other fields. Moreover, meticulous evaluations of accuracy and efficiency using (L2 and L∞) norm errors reinforce its robustness and suitability for real-world applications. In conclusion, this paper presents a novel numerical technique for nonlinear time fractional differential equations, with the CCE and NPS methods’ unique combination driving its effectiveness and broad applicability in computational mathematics, scientific research, and engineering endeavors.

Quasiperiodicity in the α-Fermi-Pasta-Ulam-Tsingou problem revisited: An approach using ideas from wave turbulence.

The Fermi-Pasta-Ulam-Tsingou (FPUT) problem addresses fundamental questions in statistical physics, and attempts to understand the origin of recurrences in the system have led to many great advances in nonlinear dynamics and mathematical physics. In this work, we revisit the problem and study quasiperiodic recurrences in the weakly nonlinear α-FPUT system in more detail. We aim to reconstruct the quasiperiodic behavior observed in the original paper from the canonical transformation used to remove the three-wave interactions, which is necessary before applying the wave turbulence formalism. We expect the construction to match the observed quasiperiodicity if we are in the weakly nonlinear regime. Surprisingly, in our work, we find that this is not always the case and in particular, the recurrences observed in the original paper cannot be constructed by our method. We attribute this disagreement to the presence of small denominators in the canonical transformation used to remove the three-wave interactions before arriving at the starting point of wave turbulence. We also show that these small denominators are present even in the weakly nonlinear regime, and they become more significant as the system size is increased. We also discuss our results in the context of the problem of equilibration in the α-FPUT system and point out some mathematical challenges when the wave turbulence formalism is applied to explain thermalization in the α-FPUT problem. We argue that certain aspects of the α-FPUT system such as thermalization in the thermodynamic limit and the cause of quasiperiodicity are not clear, and that they require further mathematical and numerical studies.

Heat transfer in the Jeffery-Hamel flow of a yield-stress fluid

Heat transfer in the Jeffery-Hamel flow of a yield-stress fluid that obeys the Bingham-Papanastasiou model is investigated in the present study. Among various rheological models proposed to formulate the flow of yield-stress fluids, the Bingham model is usually used to address the effect of yield-stress. A fluid that obeys the Bingham model does not show any shear-thinning or shear-thickening behaviors; this makes the interpretation of the results much easier. Addressing the unyielded regions (regions at which the yield-stress material behaves like a solid) in a yield-stress fluid flow is not a trivial task, and it could be impossible, especially in complex geometries. Unfortunately, the Bingham model itself cannot be used to formulate complex fluid flow problems due to the mathematical difficulties (i.e., addressing the unyielded regions); hence it is usually regularized to avoid these mathematical challenges. One of these regularized models is the Bingham-Papanastasiou model. Using an exponential term enables the Bingham-Papanastasiou model to address the unyielded regions. However, this exponential term results in highly non-linear equations, which are difficult to be handled mathematically. In the present study, the effect of the yield-stress (Bingham number) on the hydrodynamics and heat transfer in the Jeffery-Hamel flow is investigated by using the Bingham-Papanastasiou model. The momentum transfer in the Jeffery-Hamel flow of a Bingham-Papanastasiou fluid is governed by a highly non-linear ordinary differential equation (ODE) which is a boundary value problem. Assuming that the fluid's properties are independent of the temperature and pressure, this highly non-linear ODE is solved numerically. The heat transfer in the flow, with a constant wall temperature boundary condition, is investigated, assuming the flow to be an equilibrium viscous dissipation flow (the fluid's mixed mean temperature variation is negligible along the channel). The resultant linear energy equation is also solved numerically to obtain the temperature distribution across the channel. The results obtained in this work show that the yield-stress increases the wall shear stress, Nusselt number, and Stanton number. In fact, the effect of the yield-stress on the Nusselt number is comparable to the effects of Prandtl and Eckert numbers. It is also found that the yield-stress can prevent the flow separation in a divergent channel. Considering the advantages of using the yield-stress fluids as the working fluids in the Jeffery-Hamel flow, they can be considered as an alternative to nanoparticles which are usually used to enhance the fluid's thermal conductivity and also the Nusselt number.