Guidelines for the validation of machine learning predictions of species interactions

1. The prediction of species interactions is gaining momentum as a way to circumvent limitations in data volume. Yet, ecological networks are challenging to predict because they are typically small and sparse. Dealing with extreme class imbalance is a challenge for most binary classifiers, and there are currently no guidelines as to how predictive models can be trained for this specific problem.2. Using simple mathematical arguments and numerical experiments in which a variety of classifiers (for supervised learning) are trained on simulated networks, we develop a series of guidelines related to the choice of measures to use for model selection, and the degree of unbiasing to apply to the training dataset.3. Neither classifier accuracy nor the ROC-AUC are informative measures for the performance of interaction prediction. PR-AUC is a fairer assessment of performance. In some cases, even standard measures can lead to selecting a more biased classifier because the effect of connectance is strong. The amount of correction to apply to the training dataset depends on network connectance, on the measure to be optimized, and only weakly on the classifier.4. These results reveal that training machines to predict networks is a challenging task, and that in virtually all cases, the composition of the training set needs to be experimented on before performing the actual training. We discuss these consequences in the context of the low volume of data.

Market equilibria and money

By the first welfare theorem, competitive market equilibria belong to the core and hence are Pareto optimal. Letting money be a commodity, this paper turns these two inclusions around. More precisely, by generalizing the second welfare theorem we show that the said solutions may coincide as a common fixed point for one and the same system.Mathematical arguments invoke conjugation, convolution, and generalized gradients. Convexity is merely needed via subdifferentiablity of aggregate “cost”, and at one point only.Economic arguments hinge on idealized market mechanisms. Construed as algorithms, each stops, and a steady state prevails if and only if price-taking markets clear and value added is nil.

Supporting Students in Critiquing Math Arguments

Using cases from early childhood, elementary, and secondary classrooms, we showcase the work that teachers do to support students in building a collective argument and critiquing an individual’s argument. We identify four areas of work central to teaching students to build and critique mathematical arguments.

Language Constraints in Constructing Arguments for Mathematical Proofs

A learning trajectory for constructing mathematical proof has been developed. The trajectory is to provide the students with a step-by-step procedure in constructing arguments for proving mathematical statements. However, in proving activities, the students were found to encounter difficulties in completing a deductive axiomatic argument constituting an accepted mathematical proof. An investigation has been conducted to explore the problems the students experienced in constructing proofs. It was found that they faced language constraints in constructing mathematical arguments. They encountered challenges in how to correctly express the mathematical statements in their constructed proofs.

Modeling of the physical phenomenon of heat generation by using partial differential equations

The equations of mathematical physics are a natural environment for modeling physical phenomena, an example of the above is evidenced by the heat equation in relation to its use in a variety of applications; directly related to the equations of mathematical physics are the solution methods that are used to construct the predictive models. This paper describes step by step the analytical method of separation of variables to perform a complete description of the heat conduction phenomenon in the presence of a heat generation source. The investigation by using mathematical arguments allowed to calculate the temperature function as the addition of a Fourier series and a function which represents the steady state; by performing a computational simulation, it was possible to demonstrate the accuracy of the results achieved.

Preschool children’s mathematical arguments in play-based activities

The present study examines the structure and mathematical content of children’s mathematical arguments as part of communication in play-based activities. It shows how Nordin and Boistrup’s (The Journal of Mathematical Behavior 51:15–27, 2018) framework for identifying and reconstructing mathematical arguments, which includes Toulmin’s model of argumentation, the notion of anchoring (Lithner, Educational Studies in Mathematics 67:255–276, 2008) and a multimodal approach, can be used to identify and explore preschool children’s mathematical arguments. Two different types of argument that occurred during play-based activities were identified: partial arguments and full arguments. The findings reveal the extensive use of multimodal interactions in all parts of the children’s mathematical arguments. Moreover, the findings point to the crucial role of adults as dialogue collaborators in the argumentation that emerges in the play-based activities.

Promoting Precision in a Virtual Environment

The application of exact terminology benefits students when forming and supporting mathematical arguments virtually.

Dissimilarity of species interaction networks: quantifying the effect of turnover and rewiring

Despite having established its usefulness in the last ten years, the decomposition of ecological networks in components allowing to measure their β-diversity retains some methodological ambiguities. Notably, how to quantify the relative effect of mechanisms tied to interaction rewiring vs. species turnover has been interpreted differently by different authors. In this contribution, I present mathematical arguments and numerical experiments that should (i) establish that the decomposition of networks as it is currently done is indeed fit for purpose, and (ii) provide guidelines to interpret the values of the components tied to turnover and rewiring.

On the Option Pricing by the Binomial Model

In this note we study the binomial model applied to European, American and Bermudan type of derivatives. Our aim is to give the necessary and sufficient conditions under which we can define a fair value via replicating portfolios for any derivative using simple mathematical arguments and without using no arbitrage techniques. Giving suitable definitions we are able to define rigorously the fair value of any derivative without using concepts from probability theory or stochastic analysis therefore is suitable for students or young researchers. It will be clear in our analysis that if $e^{r \delta} \notin [d,u]$ then we can not define a fair value by any means for any derivative while if $d \leq e^{r \delta} \leq u$ we can. Therefore the definition of the fair value of a derivative is not so closely related with the absence of arbitrage. In the usual probabilistic point of view we assume that $d < e^{r \delta} < u$ in order to define the fair value but it is not clear what we can (or we can not) do in the cases where $e^{r \delta} \leq d$ or $e^{r \delta} \geq u$.

What affordances do open-ended real-life tasks offer for sharing student agency in collaborative problem-solving?

The important role of student agency in collaborative problem-solving has been acknowledged in previous mathematics education research. However, what remains unknown are the processes of agency in open-ended tasks that draw on real-life contexts and demand argumentation beyond “mathematical”. In this study, we analyse a video recording of two student groups (each consisting of four students) taking part in collaborative problem-solving. We draw on the framework for collaborative construction of mathematical arguments and its interplay with student agency by Mueller et al. (2012). This original framework is supplemented by (i) testing and revising it in the context of open-ended real-life tasks, with (ii) student groups rather than pairs working on the tasks, and by (iii) offering a strengthened methodological pathway for analysing student agency in such a context. Based on our findings, we suggest that the framework suits this new context with some extensions. First, we note that differences in student agency were not only identified in terms of the discourse students drew on, but in how students were able to shift between various discourses, such as between “mathematical” and “non-mathematical” discourses. We identify a novel discourse reflecting student agency, invalidation discourse, which refers to denying other students’ agency by framing their contribution as invalid. Finally, we discuss the need to reframe “mathematical” arguments—and indeed student agency—while the task at hand is open-ended and concerns real-life contexts.