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This paper inquires the ways in which paper folding constitutes a mathematical practice and may prompt a mathematical culture. To do this, we first present and investigate the common mathematical activities shared by this culture, i.e. we present mathematical paper folding as a material reasoning practice. We show that the patterns of mathematical activity observed in mathematical paper folding are, at least since the end of the nineteenth century, sufficiently stable to be considered as a practice. Moreover, we will argue that this practice is material. The permitted inferential actions when reasoning by folding are controlled by the physical realities of paper-like material, whilst claims to generality of some reasoning operations are supported by arguments from other mathematical idioms. The controlling structure provided by this material side of the practice is tight enough to allow for non-textual shared standards of argument and wide enough to provide sufficiently many problems for a practice to form. The upshot is that mathematical paper folding is a non-propositional and non-diagrammatic reasoning practice that adds to our understanding of the multi-faceted nature of the epistemic force of mathematical proof. We then draw on what we have learned from our contemplations about paper folding to highlight some lessons about what a study of mathematical cultures entails. [ABSTRACT FROM AUTHOR]

We consider discrete dynamical systems coming from the models of evolution of populations playing rock-paper-scissors game. Asymptotic behaviour of trajectories of these systems is described, occurrence of the Neimark-Sacker bifurcation and nonexistence of time averages are proved. [ABSTRACT FROM AUTHOR]

In Fritz John’s famous paper (1979), he discovered that for the wave equation □ u = ∣ u ∣ p , where 1 < p < 1 + 2 and □ denoting the d’Alembertian, there is no global solution for any nontrivial and compactly supported initial data. This paper is intended to simplify his proof by applying a Gronwall’s type inequality. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to provide examples of ‘non-traditional’ proof-related activities or theorems that can be explored through paper folding by university and high-school students. These theorems were encountered through playful acts of paper folding by the author. The author used these activities successfully with preservice teachers. The paper contains proof outlines for each theorem. [ABSTRACT FROM PUBLISHER]

In the paper the second order half-linear delay differential equation ( r ( t ) | u ′ ( t ) | α - 1 u ′ ( t ) ) ′ + p ( t ) | u ( τ ( t ) ) | α - 1 u ( τ ( t ) ) = 0 , α > 1 is studied under the condition ∫ ∞ r - 1 / α ( t ) d t < ∞ . The oscillation criterion proved by Sun and Meng (2006, Theorem 2.2) is carefully examined and improved by simplifying and sharpening the estimates used in the original proof and removing unnecessary assumptions. Among others, using elementary arguments we show that an arbitrary positive nondecreasing function ρ from this criterion can be safely and with no loss of generality replaced by a constant function. An extension to other related equations, such as neutral equations is also provided. [ABSTRACT FROM AUTHOR]

Purpose: The current paper proposes a prediction model for a cryptocurrency that encompasses three properties observed in the markets for cryptocurrencies—namely high volatility, illiquidity, and regime shifts. As far as the authors' knowledge extends, this paper is the first attempt to introduce a stochastic differential equation (SDE) for pricing cryptocurrencies while explicitly integrating the mentioned three significant stylized facts. Design/methodology/approach: Cryptocurrencies are increasingly utilized by investors and financial institutions worldwide as an alternative means of exchange. To the authors' best knowledge, there is no SDE in the literature that can be used for representing and evaluating the data-generating process for the price of a cryptocurrency. Findings: By using Ito calculus, the authors provide a solution for the suggested SDE along with mathematical proof. Numerical simulations are performed and compared to the real data, which seems to capture the dynamics of the price path of two main cryptocurrencies in the real markets. Originality/value: The stochastic differential model that is introduced and solved in this article is expected to be useful for the pricing of cryptocurrencies in situations of high volatility combined with structural changes and illiquidity. These attributes are apparent in the real markets for cryptocurrencies; therefore, accounting explicitly for these underlying characteristics is a necessary condition for accurate evaluation of cryptocurrencies. [ABSTRACT FROM AUTHOR]

Liseo and Loperfido [A note on reference priors for the scalar skew-normal distribution. J Statist Plann Inference. 2006;136(2):373–389] studied some peculiar features of default Bayes analysis of the scalar skew-normal model. In particular, they showed that, by considering the simplest model with a single unknown parameter λ of skewness, the reference – or Jeffreys’ – prior for this parameter is proper. They proved that tails of Jeffreys’ prior are of orderO(λ−3/2). But they made a mistake in their proof. In this note, we will modify their proof. [ABSTRACT FROM AUTHOR]

With the new revolution in data technology, many types of streaming data are automatically generated in our living environment. The vast amount of information carried by this streaming data demands decision support from new online learning algorithms. In this paper, we propose a new online learning algorithm to monitor changes in stream data/system status and provide automatic decision support using streaming data. The new online learning algorithm we propose consists of two components. Firstly, we adapt an existing Bayesian approach into an online learning algorithm for change detection, which serves as Component A. Then, we integrate this algorithm with a modified online learning method from repeated games (Play against Random Past), which forms Component B. Theoretical support for Component B is provided with a mathematical proof at the end. We demonstrate the algorithm's performance in this paper through simulations using both artificial data from a random process and data from the 2009 H1N1 pandemic. [ABSTRACT FROM AUTHOR]

In this theoretical paper, an epistemic logical model for analysis of students' argumentation and proof processes is presented. The model is conceived as a methodological tool addressed to the researcher in mathematics education that aims to shed light on the relations between argumentation and proof, highlighting the continuities and discontinuities within and between them. It reconciles the epistemic logic approach, which takes into account the exploratory phases of a statement, linked to argumentative processes, and the deductive logic approach, which takes into account the phases linked to proof in a classical sense. The model is based on Vergnaud's concepts- and theorems-in-action, on Duval's distinction between the epistemic and logical value of verbalised propositions, and on elements of Oostra's intuitionistic existential graphs, a kind of graphical topological logic rooted in Peircean thought, adapted to mathematics education research by considering also shifts in the classical existential graphs. After exposing the theoretical grounding the model is based on, some examples taken from the literature are examined to exemplify how it works. [ABSTRACT FROM AUTHOR]

We consider the class S(λ, β, τ) of convergent for all x ≥ 0 Taylor-Dirichlet type series of the form F(x) = X+∞ n=0 bnexλn+τ(x)βn, bn ≥ 0 (n ≥ 0), where τ: [0,+∞) → (0,+∞) is a continuously differentiable non-decreasing function, λ = (λn) and β = (βn) are such that λn ≥ 0, βn ≥ 0 (n ≥ 0). In the paper we give a partial answer to a question formulated by Salo T.M., Skaskiv O.B., Trusevych O.M. on International conference "Complex Analysis and Related Topics" (Lviv, September 23-28, 2013) ([2]). We prove the following statement: For each increasing function h(x): [0,+∞) → (0,+∞), h′(x) ↗ +∞ (x → +∞), every sequence λ = (λn) such that X+∞ n=0 1 λn+1 - λn < +∞ and for any non-decreasing sequence β = (βn) such that βn+1 - βn ≤ λn+1 - λn (n ≥ 0) there exist a function τ (x) such that τ ′(x) ≥ 1 (x ≥ x0), a function F ∈ S(α, β, τ), a set E and a constant d > 0 such that h-meas E := R E dh(x) = +∞ and (∀x ∈ E): F(x) > (1 + d)μ(x, F), where μ(x, F) = max{|an|exλn+τ(x)βn: n ≥ 0} is the maximal term of the series. At the same time, we also pose some open questions and formulate one conjecture. [ABSTRACT FROM AUTHOR]

In this paper we apply social epistemology to mathematical proofs and their role in mathematical knowledge. The most famous modern collaborative mathematical proof effort is the Classification of Finite Simple Groups. The history and sociology of this proof have been well-documented by Alma Steingart (2012), who highlights a number of surprising and unusual features of this collaborative endeavour that set it apart from smaller-scale pieces of mathematics. These features raise a number of interesting philosophical issues, but have received very little attention. In this paper, we will consider the philosophical tensions that Steingart uncovers, and use them to argue that the best account of the epistemic status of the Classification Theorem will be essentially and ineliminably social. This forms part of the broader argument that in order to understand mathematical proofs, we must appreciate their social aspects. [ABSTRACT FROM AUTHOR]

Combining data on locations with career and educational histories of mathematicians, we study how distance and ties affect citation patterns. The ties considered include coauthorship, past colocation, and relationships mediated by advisers and the alma mater. With fixed effects capturing subject similarity and article quality, we find linkages are strongly associated with citation. Controlling for ties generally halves the negative impact of geographic barriers on citations. Ties matter more for less prominent and more recent papers and have retained their quantitative importance in recent years. The impact of distance, controlling for ties, has fallen and is statistically insignificant after 2004. [ABSTRACT FROM AUTHOR]

O. Einstein (2008) proved Bollobás-type theorems on intersecting families of ordered sets of finite sets and subspaces. Unfortunately, we report that the proof of a theorem on ordered sets of subspaces had a mistake. We prove two weaker variants. [ABSTRACT FROM AUTHOR]

In this paper we provide new refinements of Aczel-type inequality and give some applications. Furthermore, we show that two of three theorems in the work by J. Tian and M.-H. Ha (J. Math. Inequal. 12 (1) (2018), 175-189) are incorrect whereas the proof of the other is technically wrong. We establish an improvement of the correctly stated theorem with a simple proof and give counterexamples to the wrong ones. [ABSTRACT FROM AUTHOR]

The article announces a call for papers for a special issue of the journal "Educational Studies in Mathematics" that will focus on research-based interventions in mathematics classrooms that are aimed at improving students' abilities to create mathematical proofs.

The k-deck of a graph is the multiset of its subgraphs induced by k vertices which is denoted by Dk(G). A graph or graph property is l-reconstructible if it is determined by the deck of subgraphs obtained by deleting l vertices. Manvel proved that from the (n−l)-deck of a graph and the numbers of vertices with degree i for all i, n−l ≤ i ≤ n−1, the degree list of the graph is determined. In this paper, we extend this result and prove that if G is a graph with n vertices, then from the (n−l)-deck of G and the numbers of vertices with degree i for all i, n−l ≤ i ≤ n−3, where l ≥ 4 and n ≥ l + 6, the degree list of the graph is determined. [ABSTRACT FROM AUTHOR]

The aim of the paper is to re-visit the 1990 Khamsi-Kozlowski-Reich Fixed Point Theorem, which initiated a flourishing field of fixed point theory in modular function spaces. Our result generalises this theorem as well as other classical fixed point theorems, including celebrated 1965 result of Kirk. As the common setting for our investigation, we choose the modulated LTI-spaces defined as modular spaces equipped with a sequential convergence structure, which allows also to use convergence types not associated with any topology (like convergence almost everywhere). [ABSTRACT FROM AUTHOR]

The paper is devoted to two problems pivotal in the generation theory. The first concerns verifying whether a normalized holomorphic function is an infinitesimal generator. It is also important to trace the influence of properties of a generator on the dynamic behavior of the generated semigroup. Recently, these problems have been studied using the so-called filtrations of generators. In this paper we study three filtrations. The study of the first began earlier, the second is a new one, while the third consists of functions studied previously with no connection to complex dynamics. For all filtration classes studied in the paper, dynamic properties of the corresponding semigroups are established and estimates for the Fekete-Szegö functional are given. [ABSTRACT FROM AUTHOR]

Background: To address the limitations of commonly used cross-validation methods, the linear regression method (LR) was proposed to estimate population accuracy of predictions based on the implicit assumption that the fitted model is correct. This method also provides two statistics to determine the adequacy of the fitted model. The validity and behavior of the LR method have been provided and studied for linear predictions but not for nonlinear predictions. The objectives of this study were to 1) provide a mathematical proof for the validity of the LR method when predictions are based on conditional means, regardless of whether the predictions are linear or non-linear 2) investigate the ability of the LR method to detect whether the fitted model is adequate or inadequate, and 3) provide guidelines on how to appropriately partition the data into training and validation such that the LR method can identify an inadequate model. Results: We present a mathematical proof for the validity of the LR method to estimate population accuracy and to determine whether the fitted model is adequate or inadequate when the predictor is the conditional mean, which may be a non-linear function of the phenotype. Using three partitioning scenarios of simulated data, we show that the one of the LR statistics can detect an inadequate model only when the data are partitioned such that the values of relevant predictor variables differ between the training and validation sets. In contrast, we observed that the other LR statistic was able to detect an inadequate model for all three scenarios. Conclusion: The LR method has been proposed to address some limitations of the traditional approach of cross-validation in genetic evaluation. In this paper, we showed that the LR method is valid when the model is adequate and the conditional mean is the predictor, even when it is a non-linear function of the phenotype. We found one of the two LR statistics is superior because it was able to detect an inadequate model for all three partitioning scenarios (i.e., between animals, by age within animals, and between animals and by age) that were studied. [ABSTRACT FROM AUTHOR]

Accelerometers, devices that measure body movements, have become valuable tools for studying the fragmentation of rest-activity patterns, a core circadian rhythm dimension, using metrics such as inter-daily stability (IS), intradaily variability (IV), transition probability (TP), and self-similarity parameter (named α ). However, their use remains mainly empirical. Therefore, we investigated the mathematical properties and interpretability of rest-activity fragmentation metrics by providing mathematical proofs for the ranges of IS and IV, proposing maximum likelihood and Bayesian estimators for TP, introducing the activity balance index (ABI) metric, a transformation of α , and describing distributions of these metrics in real-life setting. Analysis of accelerometer data from 2,859 individuals (age=60-83 years, 21.1% women) from the Whitehall II cohort (UK) shows modest correlations between the metrics, except for ABI and α . Sociodemographic (age, sex, education, employment status) and clinical (body mass index (BMI), and number of morbidities) factors were associated with these metrics, with differences observed according to metrics. For example, a difference of 5 units in BMI was associated with all metrics (differences ranging between -0.261 (95% CI -0.302, -0.220) to 0.228 (0.18, 0.268) for standardised TP rest to activity during the awake period and TP activity to rest during the awake period, respectively). These results reinforce the value of these rest-activity fragmentation metrics in epidemiological and clinical studies to examine their role for health. This paper expands on a set of methods that have previously demonstrated empirical value, improves the theoretical foundation for these methods, and evaluates their empirical use in a large dataset. [ABSTRACT FROM AUTHOR]

This paper proposes a new framework for post-synthesis obfuscation of digital circuits using a mixture of cells combined with an Anti-SAT block. Furthermore, a novel integrated framework has been established wherein obfuscation, along with Anti-SAT and validation of the benchmarks, progress through MATLAB®, Python, Cadence RTL Encounter® and Cadence LEC® to implement the proposed methodology. Area, delay, leakage power and total power are adopted as elements of the evaluation matrix. These parameters are compared between the original circuit, the circuit after obfuscation, the circuit after integration with Anti-SAT and the circuit after implementation of the proposed method of multiple-cell obfuscation with Anti-SAT. The probability of breaking a circuit is taken as the security criterion. It is mathematically proven that as the number of types of obfuscated cells used increases, then the probability of breaking the circuit decreases. The results obtained accord with the mathematical proof. The framework minimizes the delay by inserting obfuscation cells (OCs) in the non-critical paths, strengthens the security by using several types of OCs and allows the user to select a design based on justified area, leakage power and total power. However, against a Boolean SATisfiability (SAT) attack, obfuscation with multiple cells is not a sufficient defense. An Anti-SAT block performs better than obfuscation but has its own limitations. Thus, use of an Anti-SAT block in combination with multiple-cell obfuscation is proposed and implemented, giving better results against an efficient SAT attack. The number of iterations, as well as runtime to obtain the correct keys, increase significantly for the Anti-SAT block combined with multiple-cell obfuscation compared to the Anti-SAT or obfuscation block alone. [ABSTRACT FROM AUTHOR]

The advent of Homotopy Type Theory/Univalent Foundations (HoTT/UF) in recent years has given a new impetus to the ever ongoing philosophical discussion about foundations of mathematics. [Ternullo] addresses a specific currently much debated issue in between philosophy of set theory and set theory itself, the concept of multiverses, while [Welch] is a rather advanced account of using reflection principles for elucidating questions of independence in set theory. The papers devoted to set theory [Dzamonja/Kant], [Fontanella], [Ternullo], [Welch] should be accessible for any philosopher of mathematics, although they require a good understanding of classical set theory including models, independence results, and descriptive set theory for a deeper appreciation. However, structural set theory is arguably closer to mathematical practice outside of set theory, where the intricate |$\in$| -structure of material set theory is irrelevant and objects need only be defined up to isomorphism.". [Extracted from the article]

Memristors have several chaotic dynamic models and have been used successfully in various fields, including secure communication systems, information storage, and artificial neural networks. The memristor-based four-dimensional chaotic (FDMC) systems generate unpredictable and intricate time domain signals. Parameter fluctuations in the FDMC system may give birth to chaos, making it difficult to suppress. Stabilizing chaos in the FDMC system improves the circuit's performance. This paper synthesizes a novel time-efficient chatter-free nonlinear robust adaptive control (NLRAC) technique that stabilizes chaos in the FDMC system affected by time-varying unknown bounded exogenous disturbances and model uncertainties. The proposed NLRAC strategy decimates the time-varying unknown bounded exogenous disturbances and model uncertainties effects; it establishes a faster, smoother state-variable trajectories convergence to the zero vicinity. The theoretical analysis and mathematical proofs are based on the Lyapunov stability technique. Computer simulation results show that the proposed NLRAC technique effectively brings the FDMC system's state-variable trajectories to zero with reduced fluctuations for control input signals and state-variable trajectories. This feedback controller's attribute enhances closed-loop stability performance, improves precision, and reduces risk overshoot. The paper includes comparative computer simulation results to endorse the proposed controller performance. [ABSTRACT FROM AUTHOR]

In this paper, we study the nonlinear stability of a shear layer profile for Navier-- Stokes equations near a boundary. More precisely, we investigate the effect of cubic interactions on the growth of the linear instability. In the case of the exponential profile, we obtain that the nonlinearity tames the linear instability. We thus conjecture that small perturbations grow until they reach a magnitude O(v1/4) only, forming small rolls in the critical layer near the boundary. The mathematical proof of this conjecture is open. [ABSTRACT FROM AUTHOR]

This paper showcases the effectiveness of the discrete generalized Burr-Hatke distribution in analyzing insurance claims data, specifically focusing on scenarios with over-dispersed and zero-inflated claims. Key contributions include presenting foundational statistical theories with mathematical proofs to enrich the paper's mathematical and statistical aspects. Through the application of this discrete distribution, the study conducted a thorough risk analysis across five diverse sets of insurance claims data, evaluating critical risk indicators at specified quantiles. These indicators provided detailed insights into potential losses across different risk levels, supporting effective risk management strategies. The research emphasizes the importance of selecting appropriate probability distributions when analyzing zero-inflated data, as commonly observed in insurance claims. The discrete distribution accommodated these unique data characteristics and facilitated a robust analysis of risk metrics, enhancing the accuracy of potential loss assessments and reducing associated uncertainties. Furthermore, the study highlights the practical relevance of the discrete distribution in addressing specific challenges inherent to insurance claims data. By leveraging this distribution, insurers and risk analysts can improve their risk modeling capabilities, leading to more informed decision-making and enhanced financial exposure management. [ABSTRACT FROM AUTHOR]

Purpose: The active tuned mass damper (ATMD) system is underactuated where the number of actuators is less than the number of system's DOF. It is well known that the regulation of an underactuated system is challenging. Few reports can be found in the literature relating to the design of ATMD systems from the view point of their underactuated dynamics. In this paper, we firstly clarify the underactuated behavior of an ATMD system whose ill-conditioned controllability matrix largely limits the flexibility of controller design. Controllers are proposed to circumvent the underactuated dynamics to ensure vibration attenuation during the earthquake and rapid state convergence in the ATMD when the earthquake vanishes. Methods: This paper proposes a redefined output function to destroy the underactuated dynamics so that a stabilizing controller can be designed to regulate the new output. A low pass filter is suggested in the control loop to give boundedness of primal and auxiliary systems simultaneously supported by BIBO stability. Results: The simulation results show that the proposed ATMD design can give much better performance in both time and frequency responses compared with the traditional TMD design under the 1940 El Centro earthquake and 1985 Mexico City earthquake. Conclusion: The proposed output redefinition design removes the underactuated dynamics in the ATMD system effectively, and the low-pass filter inclusion preserves system passivity so that convergence of the states can be obtained. These have been justified by mathematical proofs and simulation verifications. [ABSTRACT FROM AUTHOR]

In most wireless networks, users/peers are connected to backbone networks using access links. The access link capacity is divided into upload and download capacity, and these capacities are used for upload and download of data, respectively. A peer can modify this division of link capacity between upload and download. A peer will allocate its entire link capacity for download to maximize utility. But incentive mechanism forces them to allocate some portion of capacity for upload. This paper investigates how to optimally divide link capacity so that peers receive maximum utility in a P2P network. We model this scenario as a game and determine capacity partitioning of peers during the Nash equilibrium NE. We also prove the social optimality of NE. As this portioning maximizes individual as well as social benefit, so NE is an optimal state of capacity partition. Using simulation, we verify that NE is an optimal state. BitTorrent network and a distributed algorithm for dividing access capacities are simulated, and partition providing maximum utility is compared with NE. This work provides a generalized expression and the mathematical proof for capacity partitioning at which peers receive maximum utility. [ABSTRACT FROM AUTHOR]

The optimal selection of pinning nodes for multi-agent flocking is a challenging NP-hard problem. Current pinning node selection strategies mainly rely on centrality measures of complex networks, which lack rigorous mathematical proof for effective flocking control. This paper proposes a pinning node selection strategy based on matrix eigenvalue theory. First, the effect of the pinning node on the eigenvalue of the Laplacian matrix is analyzed. Then, a synchronization index representing the topology uniformity of the multi-agent system is proposed to exert maximum influence on the system synchronizability. A practical optimal pinning node selection method based on the synchronization index is proposed and analyzed using the eigenvalue perturbation method. Finally, simulations demonstrate that the convergence rate of the system obtained using the optimal synchronizability pinning node selection method is better than that achieved with the maximum degree centrality node selection strategy. [ABSTRACT FROM AUTHOR]

This paper presents a new fuzzy non‐singular fast terminal decoupled sliding mode control based on the finite‐time extended state observer for trajectory tracking control of flexible joint robot manipulators under chaos and uncertainty. First, a suitable chaotic reference is used in a synchronization mechanism to create chaos in the dynamics. Then, the essential modifications made to the conventional decoupled sliding mode control are introduced. A new coupling variable and a non‐singular fast terminal sliding surface are proposed in combination with an extended state observer. A complete convergence analysis is also performed for all the sliding surfaces. Furthermore, two fuzzy systems are designed in such a way that not only they can eliminate the chattering phenomenon but can also improve the convergence time without increasing the control input in a voltage‐based manner. The mathematical proof guarantees the finite‐time asymptotic stability of the closed‐loop control system under the proposed scheme for a class of under‐actuated systems in the presence of structured and un‐structured uncertainties. Finally, some simulations and experimental implementations as hardware‐in‐the‐loop are performed to evaluate the proposed method's performance. The proposed controller is an effective method, as confirmed by the numerical simulation results and experimental validation. [ABSTRACT FROM AUTHOR]

In this paper, an optimal extended state observer (ESO) is designed for nonlinear integral chain system. For the first time, in this paper, proportional–integral (PI) extremum seeking control (ESC; PIESC) approach is applied to design a perturbation observer by minimizing the estimation error. The PI type structure of the ESC provides fast transient response for the closed-loop system to attain optimum equilibrium point. A mathematical proof is presented to show that the average error dynamic asymptotically converges to zero while the estimation error is minimized. The simulation results verify that the estimation error achieved by the proposed PIESC-based ESO (PIESC-ESO) is found significantly less than that achieved by the classic ESO and PSO-based ESO. Moreover, if slow changes occur within the system or external to it, the PIESC-ESO is still able to modify the observer gains such that the estimation error is minimized again. Therefore, this may help PIESC-ESO to be used in practical applications. [ABSTRACT FROM AUTHOR]

This paper is concerned with the propagation phenomena for a nonlocal dispersal Lotka–Volterra competition model with shifting habitats. It is assumed that the growth rate of each species is nondecreasing along the x-axis, positive near ∞ and nonpositive near - ∞ , and shifting rightward with a speed c > 0 . In the case where both species coexist near ∞ , we established three types of forced waves connecting the origin, respectively to the coexistence state with any forced speed c; to itself with forced speed c > c ∗ (∞) ; and to a semi-trivial steady state with forced speed c > c ¯ (∞) , where c ∗ (∞) and c ¯ (∞) are two positive numbers. In the case where one species is competitively stronger near ∞ , we also obtain the existence and nonexistence of forced waves connecting the origin to the semi-trivial steady state. Our results show the existence of multiple types of forced waves with the same forced speed. The mathematical proofs involve integral equations and Schauder's fixed point theorem, and heavily rely on the construction of various upper-lower solutions, which adds new techniques to deal with the "shifting environments" problem. [ABSTRACT FROM AUTHOR]

In Colombeau (Z Angew Math Phys 66:2575–2599, 2015), Colombeau (Z Angew Math Phys 71:112, 2020) we constructed weak asymptotic solutions to some systems of fluid dynamics on the n-D torus T n . In Colombeau (Z Angew Math Phys 71:112, 2020), we passed to the limit by compactness to obtain Radon measures on T n that satisfy the equations in a natural sense as sum of distributions. The construction was done on physical ground and, though we obtained full rigorous mathematical proofs, we had to introduce arbitrary ingredients such as an approximation of the initial condition or a function to obtain bounds on the variables of density and velocity, which are used in the demonstrations. In this paper, from a more precise construction and under the assumptions of finiteness of velocity and absence of void region in the flow, we prove that the Radon measure limits are independent on all arbitrary ingredients used in their construction: they depend only on the initial condition ρ 0 ∈ L 1 (T n) , u 0 ∈ L ∞ (T n) n . We describe them as particular vanishing viscosity limits. [ABSTRACT FROM AUTHOR]

In 1977, the first computer-assisted proof of a mathematical theorem was presented by K. Appel and W. Haken. The proof was met with a lot of criticism from both mathematicians and philosophers. In this paper, I present some examples of computer-assisted proofs, including Appel and Haken's work. Then, I analyze the most famous arguments against the equal acceptance of computer-based and human-based proofs in mathematics and examine the philosophical assumptions behind the presented criticism. In the conclusion, I talk about whether the philosophical assumptions are justified as they are, or one needs to take a specific philosophical position to accept them. [ABSTRACT FROM AUTHOR]

In this paper, we study the concept of 2⊗-auto Engel groups. Among other results, we prove that for any group G, if every element of G⊗Aut(G) is 2⊗-Engel group, then ⟨ (g ⊗ α),(g ⊗ α) g′⊗α′ ⟩ is a nilpotent subgroup of class at most 2 in G ⊗ Aut(G), for all g, g′ ∈ G and α, α′ ∈ Aut(G). [ABSTRACT FROM AUTHOR]

This paper describes an empirical study examining action research papers submitted by a group (n = 81) of out-of-field mathematics teachers as part of a professional development programme in Ireland. Utilising document analysis, the papers were qualitatively analysed for evidence of teachers' beliefs and practices with respect to the teaching and learning of mathematics. In particular, demonstration of direct transmission and constructivist beliefs and practices was explored. The authors draw on Valsiner's zone theory in conceptualising the 'teacher-as-learner' in this out-of-field context and it provided a means for analysing teachers' development in terms of their beliefs and practices. Findings indicate a prevalence of direct transmission or traditional teaching practices prior to the out-of-field teachers' action research, with some inconsistency with professed constructivist beliefs. There was evidence of a majority shift towards constructivist beliefs and practices post action research for a myriad of reasons, including increased pedagogical confidence in mathematics and the successful experience of constructivist approaches. Findings also highlight the role of reflective self-study in facilitating the creation of productive tensions, as well as the importance of the teachers' interpretation of their zone of free movement in resolving complex issues and enabling out-of-field teachers' development of constructivist beliefs and practices. [ABSTRACT FROM AUTHOR]

The encryption process and the decryption process of the new image encryption approaches mainly include pixel scrambling, exclusive NOR, and diffusion operations. The aspect that differentiates the proposed algorithm from the thousands of image encryption algorithms in the literature is that it is designed within the framework of the provable security design principle. In the last paper, entitled "Fractional-Order Analysis of Modified Chua's Circuit System with the Smooth Degree of 3 and Its Microcontroller-Based Implementation with Analog Circuit Design", Junxia Wang et al. present a fractional-order system from a modified Chua's circuit system with the smooth degree of 3 proposed by Fu et al. [Extracted from the article]

Weather-related or other types of faults can negatively impact the continuity of the electricity delivery to the consumers and, thus, lead to poorer reliability scores. 90% of customer outage-minutes are due to events which affect local distribution systems. The radial configuration is simple and has a low cost design; however, when a fault occurs it often causes power outages beyond the location of the fault itself. Therefore, it is beneficial for utilities to automatically isolate the fault and, thus, restore power to as many consumers as possible. The benefits of fault isolation and service restoration include, but are not limited to: enhanced reliability and resiliency, reduced loss of revenue to the utility, and enhanced comfort and satisfaction for the utility’s consumers. This paper provides an algorithm to quickly isolate a fault and automatically restore power to as many nodes of the distribution system as possible by opening and closing switches already placed within the system. The expert model and centralized algorithm that are proposed in this paper guarantee an optimal solution and a short run-time regardless of the size of the distribution system. A mathematical proof of the optimal nature of the solution is provided. In addition, the model is validated in practice, solving two different-sized systems. [ABSTRACT FROM AUTHOR]

In this paper, the concepts of equiprime ideals and equiprime semi-modules in Boolean like semirings (now onwards denote RB ) are introduced and illustrated with examples. It is established that if P is the equiprime RB –Ideal of the RB –module Ń then [P : Ń] is equiprime ideal of RB . Also it is approved that if Ń is an equiprime RB–module then (0: Ń) is an RB equiprime ideal of RB . Also, an important characterization of the equiprime ideal of RB in the semimodule structure over RB is put forward. Using this result, it can be proved that RB is equiprime if there exists a faithful equiprime RB –module Ń. Finally, it is proved that if Ń is an equiprime RB –module and ... is an invariant subgroup of RB such that ... is not contained in (0: Ń), then Ń is an equiprime... –module [ABSTRACT FROM AUTHOR]

Let A,B,C ∈ Cn×n be positive semidefinite matrices and let |A|, |B|, |C| be determinants of A, B, C ∈ Cn×n respectively. In this paper, the authors prove two determinantal inequalities |A + B + C| + |C| ≥ |A + C| + |B + C| + (2n - 2) |AB|1/2 + 3(3n-1 - 2n + 1) |ABC|1/3 and |A + B + C| + |A| + |B| + |C| ≥ |A + B| + |A + C| + |B + C| + 3(3n-1 - 2n + 1)|ABC|1/3. These two inequalities refine known ones. [ABSTRACT FROM AUTHOR]

In our earlier work, we suggested an axiomatic framework for deducing the fundamental entities which constitute the building block of the elementary particles in physics. The basic concept of this theory, named apriorics, is the ontological structure (OS)—an undirected simple graph satisfying specified conditions. The vertices of this graph represent the fundamental entities (FEs), its edges are binary compounds of the FEs (which are the fundamental bosons and fermions), and the structures constituting more than two connected vertices are composite particles. The objective of this paper is to focus the attention on several mathematical theorems and ideas associated with such graphs of order n, including their enumeration, showing what is the information content of apriorics. [ABSTRACT FROM AUTHOR]

In the last decades, the importance of argumentation as a human competence in general and its close relationship with mathematical comprehension in particular has been highlighted. Thus, in this paper we focus our interest on analyzing the argumentation skills shown by three-year-old children in a STEAM classroom experience. For this purpose, a qualitative study was carried out. For data collection, a STEAM task has been designed in which mathematics and science are worked together from the study of the physical properties of water. Specifically, what color and taste of water has by means of two experiments and, subsequently, photos and video-recordings have been collected during the implementation carried out in a classroom with 20 children of three-year-old. All data collect have subsequently been transcribed and categorized. The results indicate that the children, during the experiments done, mainly use arguments based on mathematical language, numerical figures, or make use of words or spelling, although with considerable imprecision in the type of language used. In addition, the experiments carried out have allowed a high percentage of children to change their previous ideas and have a more comprehensive approach to the concepts introduced, highlighting the importance of working on argumentation in early childhood education through contexts, in this case STEAM, which encourage reasoning and proof. [ABSTRACT FROM AUTHOR]

One of the important problem in finite groups theory is group characterization by specific property. Properties, such as element order, the set of element with the same order, etc. In this paper, we prove that Ree group ²G2(q), where q±√3q+1 is a prime number can be uniquely determined by its order and one conjugacy class size. [ABSTRACT FROM AUTHOR]

The notions of a local automorphism for Lie algebras are defined as similar to the associative case. Every automorphism of a Lie algebra L is a local automorphism. For a given Lie algebra L, the main problem concerning these notions is to prove that they automatically become an automorphism or to give examples of local automorphisms of L, which are not automorphisms. In this paper, we study local automorphisms on quasi-filiform Leibniz algebras. It is proved that quasi-filiform Leibniz algebras of type I, as a rule, admit local automorphisms which are not automorphisms. [ABSTRACT FROM AUTHOR]

In this paper, we investigate f-Kenmotsu 3-dimensional manifolds admitting generalized η-Ricci solitons with respect to the Schouten-van Kampen connection. We provide an example of generalized η-Ricci solitons with respect to the Schouten-van Kampen connection on an f-Kenmotsu 3-dimensional manifold to prove our results. [ABSTRACT FROM AUTHOR]

Vehicle platooning is seen as an enabler to improve driving safety, traffic congestion and stress‐free driving. A platoon is a highly non‐linear system and fuzzy control is able to deal with those systems. This paper presents a pioneering solution to the complex problem of vehicle platooning. Our adaptive controller seamlessly integrates fuzzy logic with conventional PID control technology, thereby ensuring safe and efficient travel for vehicles within a platoon. The key objectives of this controller encompass the maintenance of a consistent inter‐vehicle time gap and the consideration of time‐varying parameter uncertainties and communication delays between vehicles. It provides mathematical proof of the stability of the platoon formation using the Routh–Hurwitz theorem and a set of equations that describe the platoon's overall performance. Simulation results show that the proposed controller is effective in controlling vehicle platoons, even in the presence of random communication delays, resulting in improved driving safety and reduced traffic congestion. [ABSTRACT FROM AUTHOR]

With the increasing presence of blockchain-based distributed applications in various aspects of daily life, there has been a growing focus on the privacy protection of blockchain ledgers and the corresponding regulatory technologies. However, current mainstream solutions primarily concentrate on the verifiable encryption of blockchain transaction addresses and contents, neglecting the regulatory requirements for private transactions. Moreover, the few monitorable solutions suffer from issues such as excessive centralization and a single-minded approach to regulatory content. To address these deficiencies, this paper proposes a blockchain privacy-preserving scheme that supports multi-level regulation through the utilization of zero-knowledge proofs (zk-SNARKs) and attribute-based encryption (ABE). Firstly, by leveraging zk-SNARKs, this scheme achieves blockchain privacy-preserving within an account model, enabling the concealment of user transaction addresses and values. Secondly, by employing attribute-based encryption, a multi-level regulatory model is developed alongside the privacy protection measures, allowing for selective disclosure of transaction content. Finally, we analyze the security of the proposed scheme and compare it with other schemes, discussing its advantages in terms of privacy, security, and regulatory capabilities, we also provide a preliminary evaluation of the scheme's efficiency through experiments. In conclusion, the scheme demonstrates strong privacy by relying on mathematical proofs through zk-SNARKs to ensure security while comprehensively safeguarding content. It also achieves multi-level regulation on the foundation of privacy protection, with comprehensive regulatory coverage and decentralized regulatory authority. [ABSTRACT FROM AUTHOR]