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To build understanding of, and the ability to use, first-year science and engineering calculus, students were introduced to differential equations and asked to carry out research on differential equation modeling projects. Describes a model for the research paper; appendices provide a copy of the student handout and five examples of student research topics. (MDH)

There is a need for further studies on students' learning of Differential Equations (DEs), especially in advanced undergraduate and graduate courses. Research on the mathematical education of engineers shows a conflict between students' demands for practical, contextualized pedagogies and the need for abstract reasoning and appropriate use of mathematical results. Few papers focus on engineering students' interpretation of theorems and their use as tools in argumentation and problem-solving. This paper takes a sociocultural stance on learning and employs dialogical inquiry – a methodology rooted in Bakhtinian theory, newly developed for collaborative inquiry and qualitative data analysis – to investigate the meanings that senior engineering students made while working on a task designed to evaluate their understanding of Existence and Uniqueness Theorems (EUTs) of solutions of DEs. We identified two important epistemological disconnections that explain the difficulties that some of our students faced in making meaning of solutions of DEs and the EUT. • There is a need for more studies on the learning of Differential Equations (DEs). • Few research papers focus on students' meaning-making of theoretical results in DEs. • Dialogical inquiry methodology was used to analyze students' meaning-making processes. • Two epistemological disconnections were found that explain students' difficulties. • The use of warrants in students' dialogue was important in their meaning-making. [ABSTRACT FROM AUTHOR]

The study of the dynamics of the size of a population via mathematical modelling is a problem of interest and widely studied. Traditionally, continuous deterministic methods based on differential equations have been used to deal with this problem. However, discrete versions of some models are also available and sometimes more adequate. In this paper, we randomize the Pielou logistic equation in order to include the inherent uncertainty in modelling. Taking advantage of the method of transformation of random variables, we provide a full probabilistic description to the randomized Pielou logistic model via the computation of the probability density functions of the solution stochastic process, the steady state, and the time until a certain level of population is reached. The theoretical results are illustrated by means of two examples: The first one consists of a numerical experiment and the second one shows an application to study the diffusion of a technology using real data. [ABSTRACT FROM AUTHOR]

Abstract: Two players of Rock-Paper-Scissors are modeled as adaptive agents which use a reinforcement learning algorithm and exhibit chaotic behavior in terms of trajectories of probability in mixed strategies space. This paper demonstrates that an external super-agent can exploit the behavior of the other players to predict favorable moments to play against one of the other players the symbol suggested by a sub-optimal strategy. This third agent does not affect the learning process of the other two players, whose only goal is to beat each other. The choice of the best moment to play is based on a threshold associated with the Local Lyapunov Exponent or the Entropy, each computed by using the time series of symbols played by one of the other players. A method for automatically adapting such a threshold is presented and evaluated. The results show that these techniques can be used effectively by a super-agent in a game involving adaptive agents that exhibit collective chaotic behavior. [Copyright &y& Elsevier]

Discretizing a differential equation may change the qualitative behaviour drastically, even if the stepsize is small. We illustrate this by looking at the discretization of a piecewise continuous differential equation that models a population of agents playing the Rock-Paper-Scissors game. The globally asymptotically stable equilibrium of the differential equation turns, after discretization, into a repeller surrounded by an annulus shaped attracting region. In this region, more and more periodic orbits emerge as the discretization step approaches zero. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study a new equivalent form of the weak maximum principle for a large class of differential operators on Riemannian manifolds. This new form has been inspired by the work of Berestycki, Hamel and Rossi for trace operators, and allows us to shed new light on it and to introduce a new sufficient bounded Khas'minskii type condition for its validity. We show its effectiveness by applying it to obtain some uniqueness results in a geometric setting. [ABSTRACT FROM AUTHOR]

Abstract Our reply on the above comment [1] shows that the new mapping method which proposed by Zeng and Yong [2] is absolutely right. Consequently, the solutions of the nonlinear evolution equations obtained in Zayed and Al-Nowehy [3,4] are true. Unfortunately, the author of this comment [1] failed in using this method and wrote a wrong comment. [ABSTRACT FROM AUTHOR]

A two-phase method for estimation of the scattering and absorption coefficients and the asymmetry factor (σs, σa and g) in the radiative transfer problem is presented. The first phase parameterizes σs and σa through g via a simplified model and performs-at a relatively low cost-a scalar optimization over g. It is shown that this gives such a good starting point that the second phase can be accurately performed by a simple Gauss-Newton method. It is also shown that as a part of the first phase can be used on its own when only σs and σa are wanted, and it is noted that this gives higher accuracy than the commonly used Kubelka-Munk method when using standardized paper industry reflectance factor measurements. The parameter estimation problem is shown to be non-trivial and ill-conditioned, and its character is analysed. It is discussed that as standard optimization methods are so sensitive to the choice of starting point for this problem that it is hard to find a starting point that gives convergence at all. The new two-phase method is illustrated by application to relevant paper industry problems, and efficiency and sensitivity measures are given. [ABSTRACT FROM AUTHOR]

We assessed 6122 environmental modelling papers published since 2005 to determine whether the number of citations each paper had received by September 2014 could be predicted with no knowledge of the paper's quality. A random forest was applied, using a range of easily quantified or classified variables as predictors. The 511 papers published in two key journals in 2008 were further analysed to consider additional variables. Papers with no differential equations received more citations. The topic of the paper, number of authors and publication venue were also significant. Ten other factors, some of which have been found significant in other studies, were also considered, but most added little to the predictive power of the models. Collectively, all factors predicted 16–29% of the variation in citation counts, with the remaining variance (the majority) presumably attributable to important subjective factors such as paper quality, clarity and timeliness. [ABSTRACT FROM AUTHOR]

Parametric vibration and dynamic stability of the printing paper web with variable speed were studied. The speed of the moving paper web was regarded as the sum of the constant mean speed and the sine pulse speed. The dynamical model and the transverse vibration differential equation were established. The period coefficient state equation was obtained by discretizing the space variable of the differential equation with the differential quadrature method and imported state vector. The instability region and the stability region of the paper web were determined by the implicit Runge-Kutta method. The relationships of the instability region versus the tension ratio, length-width ratio and the amplitude of the sine pulse speed were analyzed. The results provide a theoretical basis for optimizing the structure of the printing machine and improving the working stability of the paper web. [ABSTRACT FROM AUTHOR]

This document is an editorial for a special issue of the journal Mathematics titled "Differential Equations of Mathematical Physics and Related Problems of Mechanics." The special issue covers a range of topics related to differential equations in mathematical physics and mechanics, including wave equations, spectral theory, scattering, and inverse problems. The editorial provides a summary of the published papers in the special issue, highlighting their contributions to the field. The document emphasizes the importance of the special issue in covering both applied and fundamental aspects of mathematics, physics, and their applications in various fields. The author expresses gratitude to the authors, reviewers, assistants, associate editors, and editors for their contributions to the special issue. The report does not provide specific details about the content of the papers or the nature of the special issue. [Extracted from the article]

The paper proposes applying an in-house mathematical model of a liquid flat-plate solar collector to calculate the collector time constant. The described model, proposed for the first time in an earlier study, is a one-dimensional distributed parameter model enabling simulations of the collector operation under arbitrarily variable boundary conditions. The model is based on the solution of energy balance equations for all collector components. The formulated differential equations are solved iteratively using an implicit difference scheme. To obtain a stable numerical solution, it is necessary to use appropriate steps of time and spatial division. These were found by comparing the results obtained from the model with the results of the analytical solution available in the literature for the transient state, which constitutes the novelty of the present study. The accuracy of the results obtained from the model was verified experimentally by comparing the measured and calculated history of the fluid temperature at the outlet of the collector. The calculation of the collector time constant is proposed in the paper as an example of the model's practical application. The results of the time constant calculation were compared with the values obtained experimentally on the test stand. This is another novelty of the presented research. The analysed collector instantaneous efficiency was then calculated for selected outdoor conditions. The presented mathematical model can also be used to verify the correctness of the collector operation. By comparing, on an ongoing basis, the measured and calculated values of the fluid temperature at the collector outlet, conclusions can be drawn about the process of solar glass fouling or glycol gelling. The simplicity of the model and the low computational demands enable such comparisons in an online mode. [ABSTRACT FROM AUTHOR]

Although the knowledge concerning dewatering and consolidation of a paper web in a press machine has increased tremendously over the past few decades, there is still a need to improve the actual understanding, especially if high-temperature conditions and compression rates are involved, as in the case of impulse drying. In an effort to further understand the transport phenomena involved in impulse drying (i.e., conductive and convective heat and mass transfer with phase change), a heat- and mass-transfer model was developed and the results from the simulation runs were compared with experimental data collected in a fast-response laboratory platen press. Insight into the structural changes that occur when paper is compressed in the paper machine were also gained by the modeling and simulation of its viscoelastoplastic behavior. To characterize the nonlinear rheologic behavior of the material, a cellular solids theory approach was used. This approach resulted in an unsteady-state problem and the resulting set of ordinary differential equations (ODEs) was solved by means of the DLSODES subroutine, which uses an integrator based in the Gear method. The subroutine was integrated in a C-based program that contained the constitutive equations of the heat- and mass-transfer model, together with the differential equations describing the mechanical behavior of the material and all the relations required to determine the transport and physical properties at any point in the paper web. The global model was used to predict the evolution of the dependent variables (such as temperature, gas and vapor pressure, water volume fraction, moisture ratio, web thickness and density profiles) throughout a typical impulse drying event. The model qualitatively predicts the heat and mass transfer in the porous medium, but it would benefit most from more-refined boundary conditions. On the other hand, the viscoelastoplastic mechanical model seems to represent, in an acceptable way, the material''s behavior under compression. [ABSTRACT FROM AUTHOR]

Abstract  In this work, important aspects of bacteriocin producing bacteria and their interplay are elucidated. Various attempts to model the resistant, producer and sensitive Escherichia coli strains in the so-called rock–scissors–paper (RSP) game had been made in the literature. The question arose whether there is a continuous model with a cyclic structure and admitting an oscillatory dynamics as observed in various experiments. The May–Leonard system admits a Hopf bifurcation, which is, however, degenerate and hence inadequate. The traditional differential equation model of the RSP-game cannot be applied either to the bacteriocin system because it involves positive interaction terms. In this paper, a plausible competitive Lotka–Volterra system model of the RSP game is presented and the dynamics generated by that model is analyzed. For the first time, a continuous, spatially homogeneous model that describes the competitive interaction between bacteriocin-producing, resistant and sensitive bacteria is established. The interaction terms have negative coefficients. In some experiments, for example, in mice cultures, migration seemed to be essential for the reinfection in the RSP cycle. Often statistical and spatial effects such as migration and mutation are regarded to be essential for periodicity. Our model gives rise to oscillatory dynamics in the RSP game without such effects. Here, a normal form description of the limit cycle and conditions for its stability are derived. The toxicity of the bacteriocin is used as a bifurcation parameter. Exact parameter ranges are obtained for which a stable (robust) limit cycle and a stable heteroclinic cycle exist in the three-species game. These parameters are in good accordance with the observed relations for the E. coli strains. The roles of growth rate and growth yield of the three strains are discussed. Numerical calculations show that the sensitive, which might be regarded as the weakest, can have the longest sojourn times. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we will establish some new oscillation criteria for the second-order retarded differential equation of the formThe results obtained essentially improve and extend those of Džurina and Stavroulakis [Oscillation criteria for second-order delay differential equations, Appl. Math. Comput., 140 (2003) 445–453]. An open problem is proposed at the end of this paper. [Copyright &y& Elsevier]

A continuous time model for multiagent systems governed by reinforcement learning with scale-free memory is developed. The agents are assumed to act independently of one another in optimizing their choice of possible actions via trial-and-error search. To gain awareness about the action value the agents accumulate in their memory the rewards obtained from taking a specific action at each moment of time. The contribution of the rewards in the past to the agent current perception of action value is described by an integral operator with a power-law kernel. Finally a fractional differential equation governing the system dynamics is obtained. The agents are considered to interact with one another implicitly via the reward of one agent depending on the choice of the other agents. The pairwise interaction model is adopted to describe this effect. As a specific example of systems with non-transitive interactions, a two agent and three agent systems of the rock-paper-scissors type are analyzed in detail, including the stability analysis and numerical simulation. Scale-free memory is demonstrated to cause complex dynamics of the systems at hand. In particular, it is shown that there can be simultaneously two modes of the system instability undergoing subcritical and supercritical bifurcation, with the latter one exhibiting anomalous oscillations with the amplitude and period growing with time. Besides, the instability onset via this supercritical mode may be regarded as “altruism self-organization”. For the three agent system the instability dynamics is found to be rather irregular and can be composed of alternate fragments of oscillations different in their properties. [ABSTRACT FROM AUTHOR]

Abstract: This paper considers certain neutral differential equation with the distributed delayWithout imposing a nonnegative restriction on coefficients of the equation, we establish a delay-dependent asymptotic stability condition for the equation by introducing a new Lyapunov function. The condition presented here is different from that of Park [J.H. Park, LMI optimization approach to asymptotic stability of certain neutral delay differential equation with time-varying coefficients, Appl. Math. Comput. 160 (2005) 355–361] in the sense that it is dependent on all delays of the equation. Numerical examples illustrate effectiveness and sharpness of our theoretical result. [Copyright &y& Elsevier]

The Savage-Hutter (SH) equations of granular avalanche flows are a hyperbolic system of equations determining the distribution of depth and depth-averaged velocity components tangential to the sliding bed. We review the equations and point out the geometrical complexities to which these equations have been generalized. Because of the hyperbolicity of the equations, successful numerical modelling is challenging, particularly when large gradients of the physical variables occur, e.g. for a moving front or possibly formed shock waves in avalanche flows if velocities change from supercritical to subcritical e.g. during the deposition. Numerical schemes solving these free surface flows must be able to cope with smooth as well as non-smooth solutions. In this paper several numerical methods are applied to solve the SH equations and compared, including traditional difference schemes, e.g. central and upstream difference schemes, as well as high-resolution NOC (Non-Oscillatory Central Differencing) schemes, in which several second-order TVD (Total Variation Diminishing) limiters and a third-order ENO (Essentially Non-Oscillatory) cell reconstruction scheme are used. Results show that the high-resolution schemes, particularly the NOC scheme with the Minmod TVD limiter or the van Leer limiter, provide excellent performances. In the SH theory the material response is expressed by only two phenomenological parameters – the internal and the bed friction angles. Parameter investigations show that avalanche flows are much more sensitive against variations of the bed friction angle than that of the internal angle of friction. Effects due to a pressure dependence of the bed friction angle and lateral variations of the basal topography are therefore also numerically examined. [ABSTRACT FROM AUTHOR]

Reliability analysis is crucial for understanding the performance and failure characteristics of lifetime systems. This paper presents a comprehensive study on the reliability analysis of lifetime systems using the Weibull distribution. The Weibull distribution, known for its flexibility in modeling failure times, provides a versatile framework for capturing diverse failure behaviors. A useful model for redundancy systems is proposed in this paper. The model consists of (n+1) components, where n components serve as spare parts for the main component. The failure rate of the working component is time-dependent, denoted as λ(t), while the failure rates of the non-working components are assumed to be zero. Whenever a component fails, one of the spare parts immediately takes over its role. The failed components in this model are considered non-repairable. To analyze this model, we establish the differential equations that describe the system states. By solving these equations, we calculate important parameters such as system reliability and mean time to failure (MTTF) in real-time scenarios. These parameters provide valuable insights into the performance and behavior of the system under study. By employing the Weibull distribution and the proposed model, this paper contributes to enhancing the understanding of reliability analysis in lifetime systems and enables the estimation of important reliability parameters for practical applications. [ABSTRACT FROM AUTHOR]

In this note, we take care of a gap in Remark 2.1 of Mishra and Laha ('On approximately star-shaped functions and approximate vector variational inequalities', Mishra and Laha in J. Optim. Theory Appl. 156:278-293, ) and modify the corresponding results. [ABSTRACT FROM AUTHOR]

A complex dynamical network (CDN) can be considered as the composition system with the nodes subsystem (NS) and the links subsystem (LS), and both subsystems are coupled with each other. In this paper, two vector differential equations (VDE) are used to describe the dynamical behaviours of NS and LS, respectively, in which the dynamical behaviour of NS is considered as the VDE with the second derivative term (SDT). This paper mainly focuses on the dynamics of LS, which is represented as VDE with the intuitive topologic feature of outgoing links, and investigates the design of the tracking controller for NS and the auxiliary tracking objectives (ATO) for LS. Firstly, the dynamical models of NS and LS in CDN are proposed, and the corresponding assumptions are given. Secondly, based on Lyapunov stability theory, the controller of NS and the ATO of LS are designed so that the state of NS can asymptotically track the given reference signal. Finally, the effectiveness of the proposed control strategy in this paper is verified by the numerical simulation example with N two-links robots. Abbreviations: ATO: auxiliary tracking objectives; CDN: complex dynamical network; LS: links subsystem; MDE: matrix differential equation; NS: nodes subsystem; SDT:second derivative term; VDE: vector differential equation; [ABSTRACT FROM AUTHOR]

In this paper, we continue to consider the problem of recovering the port-Hamiltonian structure for an arbitrary system of differential equations. We complement our previous study on this topic by explaining the choice of machine learning algorithms and discussing some details of their application. We also consider the possibility provided by this approach for a potentially new definition of canonical forms and classification of systems of differential equations. [ABSTRACT FROM AUTHOR]

In order to effectively prevent and combat online public opinion crises triggered by major emergencies, this paper explores the dissemination mechanism of uncertain information on online social platforms. According to the decisionmaking behavior of netizens after receiving uncertain information, they are divided into eight categories. Considering that there will be a portion of netizens who clarify uncertain information after receiving it, this paper proposes a SEFTFbTbMR model of uncertain information clarification behavior. The propagation dynamics equations of the model are given based on the theory of differential equations, the basic regeneration number R0 of the model is calculated, and the existence and stability of the equilibrium point of the model are analyzed. The theoretical analysis of the model is validated using numerical simulation software, and sensitivity analysis is performed on the parameters related to R0. In order to reduce the influence caused by uncertain information, the optimal control strategy of the model is proposed using the Hamiltonian function. It is found that the dissemination of uncertain information among netizens can be suppressed by strengthening the regulation of social platforms, improving netizens' awareness of identifying the authenticity of information, and encouraging netizens to participate in the clarification of uncertain information. The results of this work can provide a theoretical basis for future research on the uncertain information dissemination mechanism triggered by major emergencies. In addition, the results can also provide methodological support for the relevant government departments to reduce the adverse effects caused by uncertain information in the future. [ABSTRACT FROM AUTHOR]

The objective work of this paper is to transform the nonlinear space-time fraction Whitham–Broer–Kaup equation into ordinary differential equation by using the conformal fractional derivative, and find the exact solutions through the complete polynomial discriminant system. At the same time, we build the appropriate solution for the identified parameters to show the existence of the solution. In addition, we provide the 3D and 2D graphics to show that the solutions are real and effective. [ABSTRACT FROM AUTHOR]

Abstract: In this technical note, we give a short extension application for nonlinear sum of ratios problem (P) considered in [Y.J. Wang, K.C. Zhang, Global optimization of nonlinear sum of ratios problem, Applied Mathematics and Computation 158 (2004) 319–330]. Actually our result is slightly more general, since we do not specify additional positive coefficient for each ratio. In this note, we use different equivalent problem as done in Wang and Zhang (2004). Our method introduce p variables less than other method (Wang and Zhang, 2004), and our approach need not additional special program to obtain the upper and lower bound of numerator and denominator for each ratio in the objective function. [Copyright &y& Elsevier]

Kolmogorov-Arnold-Moser (or KAM) theory was developed for conservative dynamical systems that are nearly integrable. Integrable systems in their phase space usually contain lots of invariant tori, and KAM theory establishes persistence results for such tori, which carry quasi-periodic motions. We sketch this theory, which begins with Kolmogorov's pioneering work. [ABSTRACT FROM AUTHOR]

We prove oscillation and nonoscillation theorems for the second order linear differential equation (E) y″+q(t)y=0, where q(t)⩾0 and locally integrable on [t*,∞). These results are extensions of earlier results of Huang [J. Math. Anal. Appl. 210 (1997) 712–723]. Furthermore, we show that the oscillation criterion established for Eq. (E) can be extended to the delayed differential equation y″+q(t)y(σ(t))=0, where σ(t)⩽t and limt→∞σ(t)=∞. [Copyright &y& Elsevier]