This paper presents a mathematical explanation of one of the Multi-Attribute Decision-Making (MADM) methods—the Ranking based on Distance and Range (RADAR) method—along with its modified variant, RADAR II. Through mathematical proofs, the influence of each step of the method on the final ranking of alternatives is analyzed. The methods are tested on three numerical examples with varying criterion weights. The robustness of the methods, as well as their fundamental characteristics, is demonstrated. A comparative analysis reveals that although both methods prioritize alternatives based on their stability across all criteria—particularly the most important ones—the RADAR II method is somewhat more rigorous and stringent, whereas the original RADAR method is more flexible and yields more objective results.

Numeracy constitutes a critical professional competency for pre-service mathematics teachers (PMTs), particularly in relation to applying mathematical ideas meaningfully within interdisciplinary contexts. Guided by contemporary frameworks of numeracy and spatial reasoning, this study investigates how PMTs demonstrate conceptual understanding, mathematical reasoning, contextual application, and representational flexibility when engaging in art-integrated STEAM geometry tasks. The study employed a qualitative-dominant convergent parallel mixed-methods case study design, supported by supplementary descriptive survey data. Participants were 28 PMTs enrolled in a private teacher education program in Central Java, Indonesia. STEAM-based problem-solving artifacts served as the primary data source, while a post-task perception survey provided complementary descriptive insights to corroborate and extend the qualitative findings. From the full cohort, four representative cases were selected to illustrate the range of reasoning approaches observed, based on variation in conceptual clarity, representational strategies, and the completeness of mathematical explanations. Thematic analysis identified four interrelated themes: mathematical reasoning and conceptual understanding; spatial visualization and representation; creative approaches to problem solving; and balancing artistic intuition with mathematical precision. The findings indicate that PMTs were able to connect geometric reasoning with visual and artistic design; however, they also encountered challenges related to maintaining representational accuracy and coordinating creative exploration with analytical rigor. This study contributes to the literature on STEAM pedagogy by elucidating how art-integrated geometry tasks can elicit multiple dimensions of numeracy, including spatial reasoning and representational flexibility. It further offers implications for teacher education, underscoring the need for targeted instructional scaffolding to support PMTs in progressing from open-ended creative engagement toward mathematically rigorous reasoning when designing interdisciplinary learning experiences.

The asymmetry between spite and altruism.

Clemmesen’s hook refers to a commonly observed slowdown and rebound in breast cancer incidence around the age at menopause. It suggests a shift in the underlying carcinogenic dynamics, but the mechanistic basis remains poorly understood. Building on our previously developed Extended Multistage Clonal Expansion Tumor (MSCE-T) model, we perform a theoretical analysis to determine the conditions under which Clemmesen’s hook would occur. Our results show that Clemmesen’s hook can be quantitatively explained by time-specific changes in the proliferative and apoptotic balance of early-stage mutated cell populations, corresponding to the decline in progesterone levels and progesterone-driven proliferation due to reduced menstrual cycles preceding menopause, and changing dominant carcinogenic impact from alternative growth pathways post-menopause (e.g., adipose-derived growth signals). In contrast, variation in last-stage clonal dynamics cannot effectively reproduce the observed non-monotonic incidence pattern. Analytical results further demonstrate that midlife incidence dynamics corresponding to the hook are governed primarily by intrinsic proliferative processes rather than detection effects. Overall, this study provides a mechanistic and mathematical explanation for Clemmesen’s hook and establishes a quantitative framework linking hormonal transitions during menopause to age-specific breast cancer incidence curve.

A patient with a medical condition can have medical tests or symptoms scored that generate numerical results before a treatment, during a treatment or after a treatment, usually over several days, to determine if any benefits have occurred. The changes in the numerical measurements or scores over time can be readily plotted using computer software to show an equation for the line of best fit for either linear or log equations, together with the coefficient of determination (R2). Despite the ease of generating this type of graphical representations caution is required in interpreting the R2 value with reference to medical case reports. To understand why this is so, at a basic level, four scenarios using hypothetical patient scores were used to generate scatter plots showing the equation for the line of best fit and R2 values with comparison to the average and standard deviation (SD) values. The graphical examples are used to supplement the more complex mathematical and statistical explanations and choice for effect measures that are available. It was found R2 values for log equations for the line of best fit did not follow a trend with increasing treatment days. For linear equations, higher R2 value may not necessarily correspond to a lower standard deviation (SD) value for the averaged scores. The R2 value can be influenced by the day on which the scores were recorded, despite the equivalence of the average scores and SD values. R2 values may not indicate the strength of a treatment benefit or the magnitude of scatter between data sets. Score averaging can increase R2 values, while average values remain the same but with the SD value decreasing. The graphical examples shown provide an explanation why line graphs may be the simplest and best option for reporting, particularly non-linear numerical data, in case reports.

Student-generated explanations are increasingly recognised as a key strategy for promoting active learning in undergraduate mathematics and statistics education. This systematic literature review examines three explanation methods - Self-Explanation (SE), Peer Explanation (PE), and Explanation to Fictitious Others (EFO) - each emerging from different research programmes. Following the PRISMA protocol, we analysed 45 studies published between 2014 and 2024. Our synthesis reveals these methods' characteristics, effects on learning, and boundary conditions: SE is most effective when supported by prompts and training, enhancing conceptual and procedural knowledge, though its benefits decline with material complexity and low prior knowledge. PE, based on collaborative peer interaction, shows potential motivational gains but lacks controlled studies isolating causal effects. EFO, in which students explain to a fictitious audience, appears promising, particularly when delivered through video, likely due to social presence and teaching expectancy effects. In the affective domain, some preliminary evidence suggests that both PE and EFO can enhance learner motivation, with indications that PE may improve confidence among female students in specific contexts. Future research should clarify the cognitive, metacognitive, and motivational mechanisms underlying each method, while employing rigorous designs to allow direct comparisons across SE, PE, and EFO.

Abstract Optical interferometry is an observational technique that provides the highest spatial resolutions available in the optical. By interfering light from separate telescopes, and measuring the properties of the resulting interference pattern, it is possible to retrieve information about the night sky at spatial resolutions equal to the separation of the telescopes, overcoming the diffraction limit of a single telescope. In long baseline amplitude optical interferometry, the beams of light from the telescopes are transported to a central location and physically interfered. The interference is achieved via an instrument known as a beam combiner. In this review, I discuss the functionality of a beam combiner. I begin with a mathematical explanation of how interference fringes are produced and what information these interference fringes contain. This is followed by a discussion of how interference fringes are generated and measured in practise for the most common beam combination schemes, for both pupil plane and image plane combination and how these schemes can be realised in bulk optics or integrated optics. I also provide a detailed summary of the various design considerations that can affect the functionality of a beam combiner. Finally, I discuss current and future work in long baseline amplitude optical interferometry.

Abstract Recently, a class of algorithms combining classical fixed‐point iterations with repeated random sparsification of approximate solution vectors has been successfully applied to eigenproblems with matrices as large as . So far, a complete mathematical explanation for this success has proven elusive. The family of methods has not yet been extended to the important case of linear system solves. In this paper, we propose a new scheme based on repeated random sparsification that is capable of solving sparse linear systems in arbitrarily high dimensions. We provide a complete mathematical analysis of this new algorithm. Our analysis establishes a faster‐than‐Monte Carlo convergence rate and justifies use of the scheme even when the solution is too large to store as a dense vector.

What mathematical explanation need not be

ABSTRACTInterference mitigation remains a persistent challenge in satellite communication (SATCOM). Especially, a cost‐effective and efficient approach is demanded to full‐fill the 6G vision of ubiquitous coverage with low‐cost satellites. To this aim, we advocate applying sidelobe cancellation (SLC) in combination with analog phased arrays for SATCOM uplink interference mitigation. In this paper, we conduct theoretical performance analysis of such SLC system by proposing an approximate signal‐to‐interference‐plus‐noise (SINR) model. Specifically, we provide a mathematical explanation for the relationship between SINR and auxiliary array gain, which is a pivotal inquiry in system design but remains inadequately addressed. Based on these novel findings, we also propose an approach to optimize the system performance via online control of the auxiliary array gain. The proposed models and methods are rigorously validated through extensive simulations.

In this paper, we advance the network theory of aging and mortality by developing a causal mathematical model for the mortality rate. First, we show that in large networks, where health deficits accumulate at nodes representing health indicators, the modelling of network evolution with Poisson processes is universal and can be derived from fundamental principles. Second, with the help of two simplifying approximations, which we refer to as mean-field assumption and homogeneity assumption, we provide an analytical derivation of Gompertz law under generic and biologically relevant conditions. Third, we identify for which network parameters Gompertz law is accurate, express the parameters in Gompertz law as a function of the network parameters, and illustrate our computations with simulations and analytic approximations. Our paper is the first to offer a full mathematical explanation of Gompertz law and its limitations based on network theory.

Modern public-key cryptography relies on the hardness of mathematical problems such as integer factorization and discrete logarithms. However, the development of quantum computing poses an imminent threat to these assumptions. Shors algorithm, in particular, can factor large semiprimes exponentially faster than classical algorithms, compromising systems like RSA, DSA, and ECC. This paper explores the mathematical foundations of pre-quantum cryptography, discusses the limitations of classical security models when confronted with quantum capabilities, and then pays attention to post-quantum cryptography (PQC), a field dedicated to developing cryptographic schemes resilient against both classical and quantum attacks. Among the proposed families, this paper focuses specifically on hash functionbased cryptography for its simplicity and minimal reliance on algebraic structure. This study focuses in particular on SPHINCS+, a stateless hash-based digital signature scheme currently under consideration by NIST. Through detailed mathematical explanation and a visual example, we analyze its construction using Winternitz One-Time Signatures and Merkle trees. The results highlight SPHINCS+ as a robust candidate for post-quantum security due to its reliance on well-understood hash primitives and its resistance to known quantum algorithms such as Grovers. Finally, this paper discusses ongoing challenges such as performance trade-offs, standardization, and real-world deployment. This research underscores the urgency of adopting quantum-resistant cryptographic systems before large-scale quantum computers become a reality.

Background: Mathematical communication is the clear expression of ideas using math language and symbols, either in speech or writing. Without effective writing skill, students may not write steps of problem-solving clearly, and this leads to confusion and ineffective learning. This negatively affects performance in academic work since mathematics is based on logical progression and effective communication can lead to misunderstood problems and incorrect solutions.Aims: This research aims to study the mathematical communication ability in writing, focusing on their ability to use mathematical language and symbols to express concepts.Methods: Quantitative survey was conducted by defining research questions, selecting 40 Grade 10 students from Sarakhampittayakhom School, designing a mathematical communication ability test in writing, distributing the test, collecting post-instruction data, analyzing the data using descriptive statistics, and presenting the research results.Result: Students score an average of 6 out of 16 points (37.50%), with a standard deviation of 2.80. The test focuses on their ability to use mathematical language, symbols, and explanations in problem-solving. While 17.5% have good writing and reasoning ability, 82.5% found it difficult due to limited understanding, low confidence, and insufficient practice, resulting in unclear communication and disorganized thought processesConclusion: Students' mathematical writing ability remains below 50%, with notable score disparities. Developing this skill is essential for effective learning. Teachers enhance mathematical writing through problem-solving explanations, structured writing practices, and precise terminology. Identifying the causes of these disparities supports more effective instruction, while ongoing research explores contributing factors and teaching strategies for skill development.

The study of neuronal electrical activity and spatial organization is essential for uncovering the mechanisms that regulate neuronal electrophysiology and function. Mathematical models have been utilized to analyze the structural properties of neuronal networks, predict connectivity patterns, and examine how morphological changes impact neural network function. In this study, we aimed to explore the role of the actin cytoskeleton in neuronal signaling via primary cilia and to elucidate the role of the actin network in conjunction with neuronal electrical activity in shaping spatial neuronal formation and organization, as demonstrated by relevant mathematical models. Our proposed model is based on the polygamma function, a mathematical application of ramification, and a geometrical definition of the actin cytoskeleton via complex numbers, ring polynomials, homogeneous polynomials, characteristic polynomials, gradients, the Dirac delta function, the vector Laplacian, the Goldman equation, and the Lie bracket of vector fields. We were able to reflect the effects of neuronal electrical activity, as modeled by the Van der Pol equation in combination with the actin cytoskeleton, on neuronal morphology in a 2D model. In the next step, we converted the 2D model into a 3D model of neuronal electrical activity, known as a core-shell model, in which our generated membrane potential is compatible with the neuronal membrane potential (in millivolts, mV). The generated neurons can grow and develop like an organoid brain based on the developed mathematical equations. Furthermore, we mathematically introduced the signal transduction of primary cilia in neurons. Additionally, we proposed a geometrical model of the neuronal branching pattern, which we described as ramification, that could serve as an alternative mathematical explanation for the branching pattern emanating from the neuronal soma. In conclusion, we highlighted the relationship between the actin cytoskeleton and the signaling processes of primary cilia. We also developed a 3D model that integrates the geometric organization unique to neurons, which contains soma and branches, such that the mathematical model represents the interaction between the actin cytoskeleton and neuronal electrical activity in generating action potentials. Next, we could generalize the model into a cluster of neurons, similar to an organoid brain model. This mathematical framework offers promising applications in artificial intelligence and advancements in neural networks.

The geometrical modelling of weft knitted structures is useful to predict fabric properties and for creating physical models. This technology allows fabric simulation to reduce the intensity of the designer and improve work efficiency. This article explores the use of parametric and mathematical techniques to capture the complexities of knitted structures. Finite element analysis (FEA) and mathematical explanations of loop formations are among the topics discussed. This study attempts to offer a thorough resource for scholars, designers, and practitioners navigating the challenging field of weft knitted textile geometrical modelling by combining ideas from various techniques. Mathematical models are based on mathematical equations, these models are easier to understand, but they are unable to forecast how cloth will behave in real time. Since parametric models provide a 3D geometrical model based on the actual values of the substrate, they provide a more realistic representation of the fabric simulation than mathematical models. By breaking down difficulties into smaller components, the Finite Element Analysis (FEA) is used to address physics and engineering problems. Things with asymmetrical structures can easily simulated by using FEA. This study provides a platform for future research and innovation in textile engineering and design by synthesizing the present level of geometrical modelling in the context of plain weft knitted structures through a thorough analysis of the existing literature

Hyperparameter optimization (HPO), which is also called hyperparameter tuning, is a vital component of developing machine learning models. These parameters, which regulate the behavior of the machine learning algorithm and cannot be directly learned from the given training data, can significantly affect the performance of the model. In the context of relevance vector machine hyperparameter optimization, we have used zero-mean Gaussian weight priors to derive iterative equations through evidence function maximization. For a general Gaussian weight prior and Bayesian linear regression, we similarly derive iterative reestimation equations for hyperparameters through evidence function maximization. Subsequently, after using relative entropy and Bayesian optimization, the aforementioned non-closed-form reestimation equations can be partitioned into E and M steps, providing a clear mathematical and statistical explanation for the iterative reestimation equations of hyperparameters. The experimental result shows the effectiveness of the EM algorithm of hyperparameter optimization, and the algorithm also has the merit of fast convergence, except that the covariance of the posterior distribution is a singular matrix, which affects the increase in the likelihood.

Pitanje iz naslova ovoga članka postavio je Daniele Molinini prije gotovo deset godina, uz tvrdnju da do tada na njega nije bio jasno artikuliran odgovor. Povećana učestalost pojma autentično matematičko objašnjenje može se pratiti unatrag do rada Alana Bakera, objavljenog prije otprilike dvadeset godina. Nažalost i danas, dva desetljeća kasnije, teško možemo tvrditi da je došlo do značajnijeg napretka u razumijevanju pojmovnih nijansi toga izraza. Bakerov pojačani argument neizostavnosti, za razliku od Quine–Putnamova argumenta neizostavnosti, temelji se na važnosti matematičkog objašnjenja u znanosti, s posebnim naglaskom na ono što se naziva pravim matematičkim objašnjenjem. Ovaj se pojam navodi kako u tekstovima autora koji zagovaraju platonistička stajališta, tako i onih koji zastupaju nominalističke pozicije. Razmotrit ćemo tumačenja toga izraza kod trojice autora, nastojeći uočiti zajedničke elemente, s nadom da će naša analiza makar skromno pridonijeti kristalizaciji njegova značenja, u mjeri u kojoj je to moguće u skladu s intuicijom.

In this article, the authors have discussed the Alice and Bob experimental set-up for the quantum teleportation, giving a detailed step-by step mathematical explanation of each unit. Quantum teleportation of a qubit requires the creation of an entangled quantum state between the sender’s (Alice) and the receiver’s (Bob) shared quantum state/s. At the end of this experiment, actual measurements of the Alice’s share of quantum information are to be taken and shared with Bob via a classical communication channel. The teleportation is expected to open up a whole new regime of quantum communication, data encryption to be one of its applications.

Abstract It was recently discovered that higher-order dynamics are intrinsically variational, in the sense that higher-derivative versions of the classical equations of motion can always be derived from a minimum-action principle similar to Hamilton's principle, even when the physical system is nonconservative. This discovery has already led to several applications, including a new and more efficient algorithm for computing a nonproportionally damped system's resonant frequencies, based on the fourth-order system dynamics. The purpose of this paper is to investigate the source of this improved efficiency in greater detail. We find that the improved efficiency of the new resonant frequency algorithm is due almost entirely to savings in computing the eigenvalues of the system's stiffness matrix Ω˜. This result is surprising in light of the ostensible complexity of this matrix. Nevertheless, the savings are shown to be statistically significant, with attained significance levels below machine precision. Although a rigorous mathematical explanation remains elusive, empirical results presented here lead us to conjecture that the reason may have to do with the unique block structure of the stiffness matrix, which it inherits from the mathematically Hamiltonian structure of the fourth-order formulation. The present authors believe there may be additional applications of higher-order dynamics waiting to be discovered, and a few potential ideas to explore are given in the conclusion.

The use of artificial intelligence (AI) and machine learning (ML) in healthcare has seen significant growth in recent years. In this study, we explored the potential of deep learning techniques for dental teeth detection and the identification of teeth as “normal”, “implant”, “root”, “erupting”, and “missing” using X-ray images. Traditionally, dentists rely on visual-tactile methods to diagnose oral conditions. However, these methods have limitations, such as inefficiency in time spent on diagnosis, the high cost of diagnosis, and subjectivity in the diagnosis. To address these limitations, we developed an automated algorithm that recognizes teeth structures using computer vision technology and ML methods. Our algorithm was trained on a dataset of 340 adult teeth X-ray radiographs and optimized through a series of experiments to determine the best training hyper-parameters. The development of an AI-based clinical decision support system for dental diagnosis can increase efficiency and accuracy in clinical decision-making. Our study contributes to the field of dentistry by exploring the potential of AI and ML techniques for teeth recognition and detection. Additionally, we provided mathematical explanations of our observations to aid in the interpretation of our results. Our optimized algorithm achieved a good precision of 78.2%. Overall, our study successfully demonstrated the potential of AI and ML in dental healthcare, specifically in teeth detection and implant identification using X-ray images.