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Introduction Advancements in information technology have facilitated information exchange practices within the Military Health System (MHS), enabling "systems of systems" approaches that broaden and coordinate the set of capabilities available to enhance patient outcomes. This is applicable for MHS modeling and simulation (M&S) applications as well. Learning from successful approaches applied in current interoperability solutions used in the military helps to ensure interoperability practices yield trusted compositions of simulations. Materials and Methods The use of formal methods provides the rigor necessary to unambiguously communicate these approaches across the MHS community. Here, 3 formal methods are proposed to ensure the harmonization of models and alignment of M&S data needed for simulation interoperability for MHS applications. Results To clarify considerations relevant for establishing simulation interoperability, the proposed formal methods are examined within a notional example of an injury sustained because of blast exposure. The first method applies the principles of semiotics, addressing the coding of information via syntax and semantics, to understand how to align and transform data across simulations within a composition. The second method applies the concepts of well-specified co-simulations, and the use of different techniques, tools, and algorithms to address the composition and synchronization of M&S components. The third method applies the mathematical branch of model theory to codify expert knowledge about concepts, assumptions, and constraints to ensure conceptual alignment within the simulation composition. Conclusions Biomedical research must contend with complexity inherent to computational human body modeling, enlisting expert knowledge from multiple domains supporting the development of cross-disciplinary research tools that resolve research foci and associated differences in underlying theories, methods, and applied tools. This is closely related to the broader context of digital engineering for military systems engineering. [ABSTRACT FROM AUTHOR]

Let |$f:[0,1]^{d}\to{\mathbb{R}}$| be a completely monotone integrand as defined by Aistleitner and Dick (2015, Acta Arithmetica, 167, 143–171) and let points |$\boldsymbol{x}_{0},\dots ,\boldsymbol{x}_{n-1}\in [0,1]^{d}$| have a non-negative local discrepancy (NNLD) everywhere in |$[0,1]^{d}$|⁠. We show how to use these properties to get a non-asymptotic and computable upper bound for the integral of |$f$| over |$[0,1]^{d}$|⁠. An analogous non-positive local discrepancy property provides a computable lower bound. It has been known since Gabai (1967, Illinois J. Math. 11, 1–12) that the two-dimensional Hammersley points in any base |$b\geqslant 2$| have NNLD. Using the probabilistic notion of associated random variables, we generalize Gabai's finding to digital nets in any base |$b\geqslant 2$| and any dimension |$d\geqslant 1$| when the generator matrices are permutation matrices. We show that permutation matrices cannot attain the best values of the digital net quality parameter when |$d\geqslant 3$|⁠. As a consequence the computable absolutely sure bounds we provide come with less accurate estimates than the usual digital net estimates do in high dimensions. We are also able to construct high-dimensional rank one lattice rules that are NNLD. We show that those lattices do not have good discrepancy properties: any lattice rule with the NNLD property in dimension |$d\geqslant 2$| either fails to be projection regular or has all its points on the main diagonal. Complete monotonicity is a very strict requirement that for some integrands can be mitigated via a control variate. [ABSTRACT FROM AUTHOR]

Background Estimates of the prevalence of antimicrobial resistance (AMR) underpin effective antimicrobial stewardship, infection prevention and control, and optimal deployment of antimicrobial agents. Typically, the prevalence of AMR is determined from real-world antimicrobial susceptibility data that are time delimited, sparse, and often biased, potentially resulting in harmful and wasteful decision-making. Frequentist methods are resource intensive because they rely on large datasets. Objectives To determine whether a Bayesian approach could present a more reliable and more resource-efficient way to estimate population prevalence of AMR than traditional frequentist methods. Methods Retrospectively collected, open-source, real-world pseudonymized healthcare data were used to develop a Bayesian approach for estimating the prevalence of AMR by combination with prior AMR information from a contextualized review of literature. Iterative random sampling and cross-validation were used to assess the predictive accuracy and potential resource efficiency of the Bayesian approach compared with a standard frequentist approach. Results Bayesian estimation of AMR prevalence made fewer extreme estimation errors than a frequentist estimation approach [ n  = 74 (6.4%) versus n  = 136 (11.8%)] and required fewer observed antimicrobial susceptibility results per pathogen on average [mean = 28.8 (SD = 22.1) versus mean = 34.4 (SD = 30.1)] to avoid any extreme estimation errors in 50 iterations of the cross-validation. The Bayesian approach was maximally effective and efficient for drug–pathogen combinations where the actual prevalence of resistance was not close to 0% or 100%. Conclusions Bayesian estimation of the prevalence of AMR could provide a simple, resource-efficient approach to better inform population infection management where uncertainty about AMR prevalence is high. [ABSTRACT FROM AUTHOR]

This paper addresses the long-standing question of the predicativity of the Mahlo universe. A solution, called the extended predicative Mahlo universe, has been proposed by Kahle and Setzer in the context of explicit mathematics. It makes use of the collection of untyped terms (denoting partial functions) which are directly available in explicit mathematics but not in Martin-Löf type theory. In this paper, we overcome the obstacle of not having direct access to untyped terms in Martin-Löf type theory by formalizing explicit mathematics with an extended predicative Mahlo universe in Martin-Löf type theory with certain indexed inductive-recursive definitions. In this way, we can relate the predicativity question to the fundamental semantics of Martin-Löf type theory in terms of computation to canonical form. As a result, we get the first extended predicative definition of a Mahlo universe in Martin-Löf type theory. To this end, we first define an external variant of Kahle and Setzer's internal extended predicative universe in explicit mathematics. This is then formalized in Martin-Löf type theory, where it becomes an internal extended predicative Mahlo universe. Although we make use of indexed inductive-recursive definitions that go beyond the type theory |$\mathbf {IIRD}$| of indexed inductive-recursive definitions defined in previous work by the authors, we argue that they are constructive and predicative in Martin-Löf's sense. The model construction has been type-checked in the proof assistant Agda. [ABSTRACT FROM AUTHOR]

A zero-one matrix |$M$| is said to contain another zero-one matrix |$A$| if we can delete some rows and columns of |$M$| and replace some |$1$| -entries with |$0$| -entries such that the resulting matrix is |$A$|⁠. The extremal number of |$A$|⁠ , denoted |$\operatorname{ex}(n,A)$|⁠ , is the maximum number of |$1$| -entries that an |$n\times n$| zero-one matrix can have without containing |$A$|⁠. The systematic study of this function for various patterns |$A$| goes back to the work of Füredi and Hajnal from 1992, and the field has many connections to other areas of mathematics and theoretical computer science. The problem has been particularly extensively studied for so-called acyclic matrices, but very little is known about the general case (i.e. the case where |$A$| is not necessarily acyclic). We prove the first asymptotically tight general result by showing that if |$A$| has at most |$t$| |$1$| -entries in every row, then |$\operatorname{ex}(n,A)\leq n^{2-1/t+o(1)}$|⁠. This verifies a conjecture of Methuku and Tomon. Our result also provides the first tight general bound for the extremal number of vertex-ordered graphs with interval chromatic number |$2$|⁠ , generalizing a celebrated result of Füredi and Alon, Krivelevich, and Sudakov about the (unordered) extremal number of bipartite graphs with maximum degree |$t$| in one of the vertex classes. [ABSTRACT FROM AUTHOR]

In the present paper, we consider a specific class of nonautonomous wave equations on a smooth, bounded domain and their discretization in space by isoparametric finite elements and in time by the implicit Euler method. Building upon the work of Baker and Dougalis (1980, On the |${L}^{\infty }$| -convergence of Galerkin approximations for second-order hyperbolic equations. Math. Comp. , 34 , 401–424), we prove optimal error bounds in the |$W^{1,\infty } \times L^\infty $| -norm for the semidiscretization in space and the full discretization. The key tool is the gain of integrability coming from the inverse of the discretized differential operator. For this, we have to pay with (discrete) time derivatives on the error in the |$H^{1} \times L^2$| -norm, which are reduced to estimates of the differentiated initial errors. To confirm our theoretical findings, we also present numerical experiments. [ABSTRACT FROM AUTHOR]

We present new high order approximations schemes for the Cox–Ingersoll–Ross (CIR) process that are obtained by using a recent technique developed by Alfonsi and Bally (2021, A generic construction for high order approximation schemes of semigroups using random grids. Numer. Math. , 148 , 743–793) for the approximation of semigroups. The idea consists in using a suitable combination of discretization schemes calculated on different random grids to increase the order of convergence. This technique coupled with the second order scheme proposed by Alfonsi (2010, High order discretization schemes for the CIR process: application to affine term structure and Heston models. Math. Comp. , 79 , 209–237) for the CIR leads to weak approximations of order |$2k$|⁠ , for all |$k\in{{\mathbb{N}}}^{\ast }$|⁠. Despite the singularity of the square-root volatility coefficient, we show rigorously this order of convergence under some restrictions on the volatility parameters. We illustrate numerically the convergence of these approximations for the CIR process and for the Heston stochastic volatility model and show the computational time gain they give. [ABSTRACT FROM AUTHOR]

Practical approaches to the notions of subsets and extension sets are compared, coming from broadly set-theoretic and category-theoretic traditions of mathematics. I argue that the set-theoretic approach is the most practical for 'looking down' or 'in' at subsets and the category-theoretic approach is the most practical for 'looking up' or 'out' at extensions, and suggest some guiding principles for using these approaches without recourse to either category theory or axiomatic set theory. [ABSTRACT FROM AUTHOR]

The COVID-19 pandemic has highlighted the need to use infection testing databases to rapidly estimate effectiveness of prior infection in preventing reinfection (⁠|$P{E}_S$|⁠) by novel severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) variants. Mathematical modeling was used to demonstrate a theoretical foundation for applicability of the test-negative, case–control study design to derive |$P{E}_S$|⁠. Apart from the very early phase of an epidemic, the difference between the test-negative estimate for |$P{E}_S$| and true value of |$P{E}_S$| was minimal and became negligible as the epidemic progressed. The test-negative design provided robust estimation of |$P{E}_S$| and its waning. Assuming that only 25% of prior infections are documented, misclassification of prior infection status underestimated |$P{E}_S$|⁠ , but the underestimate was considerable only when > 50% of the population was ever infected. Misclassification of latent infection, misclassification of current active infection, and scale-up of vaccination all resulted in negligible bias in estimated |$P{E}_S$|⁠. The test-negative design was applied to national-level testing data in Qatar to estimate |$P{E}_S$| for SARS-CoV-2. |$P{E}_S$| against SARS-CoV-2 Alpha and Beta variants was estimated at 97.0% (95% CI, 93.6-98.6) and 85.5% (95% CI, 82.4-88.1), respectively. These estimates were validated using a cohort study design. The test-negative design offers a feasible, robust method to estimate protection from prior infection in preventing reinfection. [ABSTRACT FROM AUTHOR]

We correct an error in Lemma 4.1 [pages 10420–2] in our article listed in the title. This concerns the cases |$s=n$| and |$\gamma \leqslant 2$| below, but it does not affect the main results of the article, that is, Theorems 1.1 and 1.2, as only the case |$s

Health literacy is an important skill for people receiving care. Those with limited literacy face disparities in their care and health outcomes when strategies for addressing literacy are not used when delivering health information. In this article, we introduce the importance of considering health literacy, defining it and related concepts including numeracy, graph literacy, and digital literacy, and discuss open questions about measuring health literacy in clinical care. Finally, we present best practices, including assuming 'universal precautions', carefully considering wording, leveraging visualizations, recognizing cultural differences in interpretation, providing guidance on pilot testing, and considering digital literacy when developing electronic materials. [ABSTRACT FROM AUTHOR]

Faithfully flat descent of pseudo-coherent complexes in rigid geometry was proved by Mathew. In this paper, we generalize the result of Mathew to solid quasi-coherent complexes on rigid analytic varieties, which have been introduced by Clausen and Scholze by means of condensed mathematics. [ABSTRACT FROM AUTHOR]

Background Few studies have examined the associations between pain trajectories and cognitive function in older adults. This study explored the associations between pain trajectories and different cognitive domains in older adults from a network perspective. Methods Data on pain trajectories were derived from the Health and Retirement Study between 2010 and 2020 using latent class growth analyses. Measurements of key cognition domains, including memory, attention, calculation, orientation and language, were included. Linear regression and network analysis were performed to evaluate the associations between different pain trajectories and cognition. Results A total of 9,551 older adults were included in this study and three trajectories of pain were identified. After controlling for the covariates, persistent severe pain trajectory was associated with poorer overall cognition, memory and calculation ability when compared to mild or non-persistent pain trajectory. In the pain and cognition network model, memory (expected influence (EI) = 0.62), language (EI = 0.58) and calculation (EI = 0.41) were the most central domains. Conclusions Pain trajectories appeared stable over time among older adults in this study. Severity of persistent pain was an important risk factor for poor cognition, especially in relation to memory and calculation domains. Interventions targeting memory, language and calculation domains might be useful in addressing cognitive decline in older adults with persistent pain. [ABSTRACT FROM AUTHOR]

The text discusses two books on the philosophy of mathematics education. The first book, "Connecting Humans to Equations," explores the human dimensions of mathematics and covers topics such as the ontological and epistemological aspects of mathematics. The second book, "Critical Mathematics Education," presents the Critical Mathematics Education (CME) paradigm, which aims to empower students to think critically about mathematics and its societal implications. Both books offer valuable insights into the philosophy of mathematics education and are recommended for philosophers and mathematics teachers. [Extracted from the article]

Just as mathematics helps us to represent and reason about the natural world, in its internal applications one branch of mathematics helps us to represent and reason about the subject matter of another. Recognition of the close analogy between internal and external applications of mathematics can help resolve two persistent philosophical puzzles concerning its applicability: a platonist puzzle arising from the abstractness of mathematical objects; and an empiricist puzzle arising from mathematical propositions' lack of empirical factual content. In order to see how this is the case, we will examine what it is to apply mathematics internally and describe examples. [ABSTRACT FROM AUTHOR]

The article discusses effect estimators on average treatment effects (ATE) that depend on key assumptions for validity. Topics covered include the alternative of instrumental variables (IV) and their conditions of validity and targeted effects, the estimates of the local ATE (LATE), and the derivation of the IV estimator. Also noted are the uses of IV methods without confounding assumptions, and on quantifying effects other than the complier average causal effect (CACE).

We consider the problem of Ω bounds for the partial sums of a modified character, i.e. , a completely multiplicative function f such that |$f(p)=\chi(p)$| for all but a finite number of primes p , where χ is a primitive Dirichlet character. We prove that in some special circumstances, |$\sum_{n\leq x}f(n)=\Omega((\log x)^{|S|})$|⁠ , where S is the set of primes p , where |$f(p)\neq \chi(p)$|⁠. This gives credence to a corrected version of a conjecture of Klurman et al. Trans. Amer. Math. Soc. 374 (11), 2021, 7967–7990. We also compute the Riesz mean of order k for large k of a modified character and show that the Diophantine properties of the irrational numbers of the form |$\log p / \log q$|⁠ , for primes p and q , give information on these averages. [ABSTRACT FROM AUTHOR]

The discrete-time Wright–Fisher (DTWF) model and its diffusion limit are central to population genetics. These models can describe the forward-in-time evolution of allele frequencies in a population resulting from genetic drift, mutation, and selection. Computing likelihoods under the diffusion process is feasible, but the diffusion approximation breaks down for large samples or in the presence of strong selection. Existing methods for computing likelihoods under the DTWF model do not scale to current exome sequencing sample sizes in the hundreds of thousands. Here, we present a scalable algorithm that approximates the DTWF model with provably bounded error. Our approach relies on two key observations about the DTWF model. The first is that transition probabilities under the model are approximately sparse. The second is that transition distributions for similar starting allele frequencies are extremely close as distributions. Together, these observations enable approximate matrix–vector multiplication in linear (as opposed to the usual quadratic) time. We prove similar properties for Hypergeometric distributions, enabling fast computation of likelihoods for subsamples of the population. We show theoretically and in practice that this approximation is highly accurate and can scale to population sizes in the tens of millions, paving the way for rigorous biobank-scale inference. Finally, we use our results to estimate the impact of larger samples on estimating selection coefficients for loss-of-function variants. We find that increasing sample sizes beyond existing large exome sequencing cohorts will provide essentially no additional information except for genes with the most extreme fitness effects. [ABSTRACT FROM AUTHOR]

Objective To compare the diagnostic performance of 10 mathematical formulae for identifying thalassemia trait in blood donors. Methods Compete blood counts were conducted on peripheral blood specimens using the UniCel DxH 800 hematology analyzer. Receiver operating characteristic curves were used to evaluate the diagnostic performance of each mathematical formula. Results In the 66 donors with thalassemia and 288 subjects with no thalassemia analyzed, donors with thalassemia trait had lower values for mean corpuscular volume and mean corpuscular hemoglobin than subjects without thalassemia donors (77 fL vs 86 fL [ P  < .001]; 25 pg vs 28 pg [ P  < .001]). The formula developed by Shine and Lal in 1977 showed the highest area under the curve value, namely, 0.9. At the cutoff value of <1812, this formula had maximum specificity of 82.35% and sensitivity of 89.58%. Conclusions Our data indicate that the Shine and Lal formula has remarkable diagnostic performance in identifying donors with underlying thalassemia trait. [ABSTRACT FROM AUTHOR]

Phage therapy is the application of bacterial viruses to control and, ideally, to eliminate problematic bacteria from patients. Usually employed are so-called strictly lytic phages, which upon adsorption of a bacterium should give rise to both bacterial death and bacterial lysis. This killing occurs with single-hit kinetics, resulting in relatively simple ways to mathematically model organismal-level, phage-bacterium interactions. Reviewed here are processes of phage therapy as viewed from these simpler mathematical perspectives, starting with phage dosing, continuing through phage adsorption of bacteria, and then considering the potential for phage numbers to be enhanced through in situ phage population growth. Overall, I suggest that a basic working knowledge of the underlying "simple maths" of phage therapy can be helpful toward making dosing decisions and predicting certain outcomes. This especially is during controlled in vitro experimentation but is relevant to thinking about in vivo applications as well. [ABSTRACT FROM AUTHOR]

Let |$G=\textrm {GL}_{2n}({\mathbb {R}})$| or |$G=\textrm {GL}_n({\mathbb {H}})$| and |$H=\textrm {GL}_n({\mathbb {C}})$| regarded as a subgroup of |$G$|⁠. Here, |${\mathbb {H}}$| is the quaternion division algebra over |${\mathbb {R}}$|⁠. For a character |$\chi $| on |${\mathbb {C}}^\times $|⁠ , we say that an irreducible smooth admissible moderate growth representation |$\pi $| of |$G$| is |$\chi _H$| -distinguished if |$\operatorname {Hom}_H(\pi , \chi \circ \det _H)\neq 0$|⁠. We compute the root number of a |$\chi _H$| -distinguished representation |$\pi $| twisted by the representation induced from |$\chi $|⁠. This proves an Archimedean analogue of the conjecture by Prasad and Takloo-Bighash (J. Reine Angew. Math. 2011). The proof is based on the analysis of the contribution of |$H$| -orbits in a flag manifold of |$G$| to the Schwartz homology of principal series representations. A large part of the argument is developed for general real reductive groups of inner type. In particular, we prove that the Schwartz homology |$H_\ast (H, \pi \otimes \chi)$| is finite-dimensional and hence it is Hausdorff for a reductive symmetric pair |$(G, H)$| and a finite-dimensional representation |$\chi $| of |$H$|⁠. [ABSTRACT FROM AUTHOR]

I examine the classical idea of 'algorithm' as a sequential, step-by-step, deterministic procedure (i.e. , the idea of 'algorithm' that was already in use by the 1930s), with respect to three themes, its relation to the notion of an 'effective procedure', its different roles and uses in logic, computer science, and mathematics (focused on numerical analysis), and its different formal definitions proposed by practitioners in these areas. I argue that 'algorithm' has been conceptualized and used in contrasting ways in the above areas, and discuss challenges and prospects for adopting a final foundational theory of (classical) 'algorithms'. [ABSTRACT FROM AUTHOR]

Using ideas proposed in Aboutness and developed in 'If-thenism', Stephen Yablo has tried to improve on classical if-thenism in mathematics, a view initially put forward by Bertrand Russell in his Principles of Mathematics. Yablo's stated goal is to provide a reading of a sentence like 'The number of planets is eight' with a sort of content on which it fails to imply 'Numbers exist'. After presenting Yablo's framework, our paper raises a problem with his view that has gone virtually unnoticed in the literature. If we are right, then Yablo's version of if-thenism cannot succeed. [ABSTRACT FROM AUTHOR]

Who gets to contribute to knowledge production of an epistemic community? Scholarship has focussed on unjustified forms of exclusion. Here I study justified forms of exclusion by investigating the phenomenon of so-called 'cranks' in mathematics. I argue that workload-management concerns justify the exclusion of these outsiders from mathematical knowledge-making practices. My discussion reveals three insights. There are reasons other than incorrect mathematical argument that justify exclusions from mathematical practices. There are instances in which mathematicians are justified in rejecting even correct mathematical arguments. Finally, the way mathematicians spot mathematical crankery does not support the pejorative connotations of the 'crank' terminology. [ABSTRACT FROM AUTHOR]

Estimating the fraction of dementia cases in a population attributable to a risk factor or combination of risk factors (the population attributable fraction (PAF)) informs the design and choice of dementia risk-reduction activities. It is directly relevant to dementia prevention policy and practice. Current methods employed widely in the dementia literature to combine PAFs for multiple dementia risk factors assume a multiplicative relationship between factors and rely on subjective criteria to develop weightings for risk factors. In this paper we present an alternative approach to calculating the PAF based on sums of individual risk. It incorporates individual risk factor interrelationships and enables a range of assumptions about the way in which multiple risk factors will combine to affect dementia risk. Applying this method to global data demonstrates that the previous estimate of 40% is potentially too conservative an estimate of modifiable dementia risk and would necessitate subadditive interaction between risk factors. We calculate a plausible conservative estimate of 55.7% (95% confidence interval: 55.2, 56.1) based on additive risk factor interaction. [ABSTRACT FROM AUTHOR]

Mathematics modelling is a vital competency for students of all ages. In this study, we aim to fill the research gap about valid and reliable tools for assessing and grading mathematical modeling problems, particularly those reflecting multiple steps of the modelling cycle. We present in this paper the design of a reliable and valid assessment tool aimed at gauging the level of mathematical modelling associated with real-world modeling problems in a scientific-engineering context. The study defines and bases the central modelling processes on the proficiency levels identified in PISA Mathematics. A two-dimensional rubric was developed, reflecting the combined assessment of the type and level of a modelling process. We identified criteria that enable a clear comparison and differentiation among the different levels across each of the modelling processes. These criteria allow for concrete theoretical definitions for the various modelling processes, introducing a well-defined mathematical modelling framework from a didactical viewpoint, which can potentially contribute to promoting modelling competencies or the understanding of modelling by teachers and students. Theoretical, methodological and practical implications are discussed. [ABSTRACT FROM AUTHOR]

Mathematics Learning Support (MLS) has been widely available in Higher Education Institutions in the United States (US) for decades. However, until recently, there has been little research that considered the extent of provision. In this paper, we present the results of a survey of MLS with responses from 268 institutions across the US, which indicate that such support is well established across responding institutions. To place these results in context, we compare them to other recent international surveys of provision. We discuss what these findings mean and the collaborative opportunities that are available for US and international practitioners and researchers in this area. [ABSTRACT FROM AUTHOR]

This study explores the types of mathematical connections established in the classroom in the teaching of functions. An extended model for mathematical connections (different representations (DR), procedural (PC), if-then (I-T), part-whole connections (PWC), feature/property (F/P), analogies, and instruction-oriented connections (IOC)) is used as the analytical framework. The context for the study is classroom observations of two secondary mathematics teachers teaching functions to Grade 9 students. The strength of teachers' mathematical knowledge for teaching (MKT) the concept of function is different: one has stronger MKT, while the other has weaker MKT. A total of 485 connections are identified in a sample of 24 ninth-grade lessons observed (12 lessons per teacher). The teacher with stronger MKT produces far more connections (f  = 317) than the teacher with weaker MKT (f  = 168), and she mostly establishes I-T, DR, PWC and F/P type of connections. The teacher with weaker MKT frequently makes procedural types of connections. This 'connections gap' may reflect differences in the teachers' MKT and in their beliefs about the teaching and learning of mathematics. The study also documents some of the important internal connections within functions based on the observed lessons, and an additional IOC has emerged from the data. [ABSTRACT FROM AUTHOR]

This paper reports a study of university lecturers' and students' experiences of teaching and learning mathematics following the abrupt requirement to switch to online teaching in 2020. A goal of the study is to share experiences that could be useful to improve the teaching and learning of mathematics in online settings. The qualitative research described is a phenomenological study and draws on interviews with ten mathematics lecturers and six undergraduate students who were enrolled in at least one university mathematical course during the lockdown in 2020. The interview data were analysed using a thematic approach. This paper reports findings regarding perceptions of lecturers and students about the challenges and benefits of online teaching and learning of mathematics, how the transition to online education has influenced assessment and sharing useful approaches for teaching and learning mathematics in online settings. [ABSTRACT FROM AUTHOR]

The main difficulty in studying numerical methods for stochastic evolution equations (SEEs) lies in the treatment of the time discretization (Printems, 2001, ESAIM Math. Model. Numer. Anal. 35 , 1055–1078). Although fruitful results on numerical approximations have been developed for SEEs, as far as we know, none of them include that of stochastic incompressible Euler equations. To bridge this gap, this paper proposes and analyzes a splitting semi-implicit Euler method in temporal direction for stochastic incompressible Euler equations on torus |$\mathbb {T}^2$| driven by additive noises. By a Galerkin approximation and the fixed-point technique, we establish the unique solvability of the proposed method. Based on the regularity estimates of both exact and numerical solutions, we measure the error in |$L^2(\mathbb {T}^2)$| and show that the pathwise convergence order is nearly |$\frac {1}{2}$| and the convergence order in probability is almost |$1$|⁠. [ABSTRACT FROM AUTHOR]

Regression calibration is a popular approach for correcting biases in estimated regression parameters when exposure variables are measured with error. This approach involves building a calibration equation to estimate the value of the unknown true exposure given the error-prone measurement and other covariates. The estimated, or calibrated, exposure is then substituted for the unknown true exposure in the health outcome regression model. When used properly, regression calibration can greatly reduce the bias induced by exposure measurement error. Here, we first provide an overview of the statistical framework for regression calibration, specifically discussing how a special type of error, called Berkson error, arises in the estimated exposure. We then present practical issues to consider when applying regression calibration, including: 1) how to develop the calibration equation and which covariates to include; 2) valid ways to calculate standard errors of estimated regression coefficients; and 3) problems arising if one of the covariates in the calibration model is a mediator of the relationship between the exposure and outcome. Throughout, we provide illustrative examples using data from the Hispanic Community Health Study/Study of Latinos (United States, 2008–2011) and simulations. We conclude with recommendations for how to perform regression calibration. [ABSTRACT FROM AUTHOR]

Recent enthusiasm for grounding often begins by observing that inquiry in metaphysics (and other areas) features a distinctive species of noncausal explanation. Having labeled this species "grounding explanation," it's a short step to the conclusion that we need a philosophical theory of grounding itself: an allegedly fundamental relation of metaphysical dependency between facts, such that a "grounding explanation" of some fact succeeds by providing information about what "grounds" that fact. This short step is hasty. For another live option is to accept that grounding explanation is a legitimate form of explanation, but to give it a thoroughly epistemic treatment, one that does not see it as involving any sort of special metaphysical relationship or structure at all. This paper sketches such a treatment, drawing inspiration from reflections on explanatory structure in mathematics. [ABSTRACT FROM AUTHOR]

An old and well-known objection to non-classical logics is that they are too weak; in particular, they cannot prove a number of important mathematical results. A promising strategy to deal with this objection consists in proving so-called recapture results. Roughly, these results show that classical logic can be used in mathematics and other unproblematic contexts. However, the strategy faces some potential problems. First, typical recapture results are formulated in a purely logical language, and do not generalize nicely to languages containing the kind of vocabulary that usually motivates non-classical theories—for example, a language containing a naive truth predicate. Second, proofs of recapture results typically employ classical principles that are not valid in the targeted non-classical system; hence non-classical theorists do not seem entitled to those results. In this paper we analyse these problems and provide solutions on behalf of non-classical theorists. To address the first problem, we provide a novel kind of recapture result, which generalizes nicely to a truth-theoretic language. As for the second problem, we argue that it relies on an ambiguity, and that once the ambiguity is removed there are no reasons to think that non-classical logicians are not entitled to their recapture results. [ABSTRACT FROM AUTHOR]

Epidemiologic researchers generalizing or transporting effect estimates from a study to a target population must account for effect-measure modifiers (EMMs) on the scale of interest. However, little attention is paid to how the EMMs required may vary depending on the mathematical nuances of each effect measure. We defined 2 types of EMMs: a marginal EMM, where the effect on the scale of interest differs across levels of a variable, and a conditional EMM, where the effect differs conditional on other variables associated with the outcome. These types define 3 classes of variables: class 1 (conditional EMM), class 2 (marginal but not conditional EMM), and class 3 (neither marginal nor conditional EMM). Class 1 variables are necessary to achieve a valid estimate of the RD in a target population, while an RR requires class 1 and class 2 and an OR requires classes 1, 2, and 3 (i.e. all variables associated with the outcome). This does not mean that fewer variables are required for an externally valid RD (because variables may not modify effects on all scales), but it does suggest that researchers should consider the scale of the effect measure when identifying an EMM necessary for an externally valid treatment effect estimate. [ABSTRACT FROM AUTHOR]

An error analysis of a mixed discontinuous Galerkin (DG) method with lifting operators as numerical fluxes for the time-harmonic Maxwell equations with minimal smoothness requirements is presented. The key difficulty in the error analysis for the DG method is that due to the low regularity the tangential trace of the exact solution is not well defined on the faces of the computational mesh. This difficulty is addressed by adopting the face-to-cell lifting introduced by Ern & Guermond (2021, Quasi-optimal nonconforming approximation of elliptic PDEs with contrasted coefficients and |$H^{1+r}$|⁠ , |$r>0$|⁠ , regularity. Found. Comput. Math. , 1–36). To obtain optimal local interpolation estimates, we introduce Scott–Zhang-type interpolations that are well defined for |$H(\textrm {curl})$| and |$H(\textrm {div})$| functions with minimal regularity requirements. As a by-product of penalizing the lifting of the tangential jumps, an explicit and easily computable stabilization parameter is given. [ABSTRACT FROM AUTHOR]

In this paper, we offer a descriptive theory of analogical reasoning in mathematics, stating general conditions under which an analogy may provide genuine inductive support to a mathematical conjecture (over and above fulfilling the merely heuristic role of 'suggesting' a conjecture in the psychological sense). The proposed conditions generalize the criteria of Hesse in her influential work on analogical reasoning in the empirical sciences. By reference to several case studies, we argue that the account proposed in this paper does a better job in vindicating the use of analogical inference in mathematics than the prominent alternative defended by Bartha. [ABSTRACT FROM AUTHOR]

In this paper, I develop a "safety result" for applied mathematics. I show that whenever a theory in natural science entails some non-mathematical conclusion via an application of mathematics, there is a counterpart theory that carries no commitment to mathematical objects, entails the same conclusion, and the claims of which are true if the claims of the original theory are "correct": roughly, true given the assumption that mathematical objects exist. The framework used for proving the safety result has some advantages over existing nominalistic accounts of applied mathematics. It also provides a nominalistic account of pure mathematics. [ABSTRACT FROM AUTHOR]

We revisit construction of the Atiyah–Hitchin manifold in the generalized Legendre transform approach. This was originally studied by Ivanov and Roček [I. T. Ivanov, M. Roček, Commun. Math. Phys. 182, 291 (1996)] and subsequently investigated more by Ionas [R. A. Ionas, arXiv:0712.3598] , in the latter of which the explicit forms of the Kähler potential and the Kähler metric are calculated. There is a difference between the former and the latter. In the generalized Legendre transform approach, a Kähler potential is constructed from the contour integration of one function with holomorphic coordinates. The choice of the contour in the latter is different from that in the former; this difference may yield a discrepancy in the Kähler potential and eventually in the Kähler metric. We show that the former only gives the real Kähler potential, which is consistent with its definition, while the latter yields the complex one. We derive the Kähler potential and the metric for the Atiyah–Hitchin manifold in terms of holomorphic coordinates for the contour considered by Ivanov and Roček for the first time. [ABSTRACT FROM AUTHOR]

New graduate students in biology programs may lack the quantitative skills necessary for their research and professional careers. The acquisition of these skills may be impeded by teaching and mentoring experiences that decrease rather than increase students' beliefs in their ability to learn and apply quantitative approaches. In this opinion piece, we argue that revising instructional experiences to ensure that both student confidence and quantitative skills are enhanced may improve both educational outcomes and professional success. A few studies suggest that explicitly addressing productive failure in an instructional setting and ensuring effective mentoring may be the most effective routes to simultaneously increasing both quantitative self-efficacy and quantitative skills. However, there is little work that specifically addresses graduate student needs, and more research is required to reach evidence-backed conclusions. [ABSTRACT FROM AUTHOR]

We consider the problem of designing a prospective randomized trial in which the outcome data will be self-reported and will involve sensitive topics. Our interest is in how a researcher can adequately power her study when some respondents misreport the binary outcome of interest. To correct the power calculations, we first obtain expressions for the bias and variance induced by misreporting. We model the problem by assuming each individual in our study is a member of one "reporting class": a true-reporter, false-reporter, never-reporter, or always-reporter. We show that the joint distribution of reporting classes and "response classes" (characterizing individuals' response to the treatment) will exactly define the error terms for our causal estimate. We propose a novel procedure for determining adequate sample sizes under the worst-case power corresponding to a given level of misreporting. Our problem is motivated by prior experience implementing a randomized controlled trial of a sexual-violence prevention program among adolescent girls in Kenya. [ABSTRACT FROM AUTHOR]

Extra-mathematical explanations explain natural phenomena primarily by appeal to mathematical facts. Philosophers disagree about whether there are extra-mathematical explanations, the correct account of them if they exist, and their implications (for example, for the philosophy of scientific explanation and for the metaphysics of mathematics) (Baker 2005 , 2009 ; Bangu 2008 ; Colyvan 1998 ; Craver and Povich 2017 ; Lange 2013 , 2016 , 2018 ; Mancosu 2008 ; Povich 2019 , 2020 ; Steiner 1978). In this discussion, I present three desiderata for any account of extra-mathematical explanation and argue that Baron's (2020) U-Counterfactual theory fails to meet each of them. I conclude with some reasons for pessimism that a successful account will be forthcoming. [ABSTRACT FROM AUTHOR]

In this study, the properties of coevolution were analysed and presented as a promising avenue to investigate the interaction between institutions and scientific organizations, which has implications for the development of scientific fields and public scientific policies. These properties were used to analyse the interaction between the Institute for Pure and Applied Mathematics (IMPA) and national and international institutions from 1951 to 2019. The findings reveal that if an area of knowledge is weak in a country, it can develop through different institutional actions by interested parties such as researchers. The institutional agency of IMPA researchers has functioned as an element that feeds coevolutionary dynamics. This dynamic has caused changes in the IMPA (micro-level), scientific community (meso-level), and governmental institutions and civil society (macro-level). This study's main contribution is the cross-fertilization of coevolution with the agency approach of institutional theory. [ABSTRACT FROM AUTHOR]

We tested the relationship between gender-stereotypical beliefs (GSBs) of female and male classroom peers and female achievement in mathematics. Complete-class student survey data merged with family register data from 1,047 Danish 6th-grade students, nested in 50 classrooms within 28 schools, were analysed using multilevel linear regressions including controls for parental education and income, the percentage of females in the classroom, prior math test scores, and the parental education, income, and prior test scores of female and male peers. We found that the GSBs of female peers were negatively associated with girls' math achievement. Similar variables were not related to the math achievement of boys. [ABSTRACT FROM AUTHOR]

In this note we correct an oversight in [ A geometric model for the module category of a gentle algebra , Int. Math. Res. Not. IMRN 2021, no. 15, 11357–11392] regarding morphisms between indecomposable modules of a gentle algebra. The oversight only occurs when bands are involved, which can be easily corrected by considering infinite periodic strings associated to bands and the universal cover of the surface associated to the gentle algebra. [ABSTRACT FROM AUTHOR]

I use my reading of Plato to develop what I call as-ifism, the view that, in mathematics, we treat our hypotheses as if they were first principles and we do this with the purpose of solving mathematical problems. I then extend this view to modern mathematics showing that when we shift our focus from the method of philosophy to the method of mathematics, we see that an as-if methodological interpretation of mathematical structuralism can be used to provide an account of the practice and the applicability of mathematics while avoiding the conflation of metaphysical considerations with mathematical ones. [ABSTRACT FROM AUTHOR]

The mathematician G.F.C. Griss is known for his program of negationless intuitionistic mathematics. Although Griss's rejection of negation is regarded as characteristic of his philosophy, this is a consequence of an executability requirement that mental constructions presuppose agents' executing corresponding mental activity. Restoring Griss's executability requirement to a central role permits a more subtle characterization of the rejection of negation, according to which D. Nelson's strong constructible negation is compatible with Griss's principles. This exposes a 'holographic' theory of negation in negationless mathematics, in which a full theory of negation is 'flattened' in a putatively negationless setting. [ABSTRACT FROM AUTHOR]

The essay also addresses the implications of Fenstad's understanding of the development of human knowledge for the training of future scientists. By contrast, in Part 2, Fenstad provides more detailed and technical discussions of the mathematical modeling of human language and cognition, and defends a cultural account of mathematics. Like the fourth essay, the primary purpose of this essay is to demonstrate the importance of mathematical modeling to the human sciences, ensuring that no one can take Fenstad's reflections on human knowledge to apply only to the natural and social sciences. While I could devote significantly more space to exploring Fenstad's overall project, instead, I will proceed by outlining the content of each essay in the collection[1] and providing a sense of how it relates to Fenstad's overall project. [Extracted from the article]

A key question of the 'maverick' tradition of the philosophy of mathematical practice is addressed, namely what is mathematical progress. The investigation is based on an article by Penelope Maddy devoted to this topic in which she considers only contributions 'of some mathematical importance' as progress. With the help of a case study from contemporary mathematics, more precisely from tropical geometry, a few issues with her proposal are identified. Taking these issues into consideration, an alternative account of 'mathematical importance', broadly within the framework of progress Maddy offers, is developed with a special focus on mathematicians' peer-review practice. [ABSTRACT FROM AUTHOR]