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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]

Communication and interaction of individuals in the real world cannot be represented by one-layer networks, and in our paper we use multiplex networks capturing the complexity of their interactions. We define players' utilities taking into account a multilayer structure of the network by specifying different ways of accounting players' costs. Using two approaches, we obtain stability conditions for special multilplex structures including all-layer complete, all-layer star with unique or different centers, and all-layer unique networks. [ABSTRACT FROM AUTHOR]

The constructive mathematics developed by Bishop in Foundations of Constructive Analysis succeeded in gaining the attention of mathematicians, but discussions of its underlying philosophy are still rare in the literature. Commentators seem to conclude, from Bishop's rejection of choice sequences and his severe criticism of Brouwerian intuitionism, that he is not an intuitionist–broadly understood as someone who maintains that mathematics is a mental creation, mathematics is meaningful and eludes formalization, mathematical objects are mind-dependent constructions given in intuition, and mathematical truths are experienceable. This paper develops and defends an intuitionistic interpretation of Bishop's philosophical views. [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 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

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

Especially nice models of intuitionistic set theories are realizability models |$V({\mathcal A})$|⁠ , where |$\mathcal A$| is an applicative structure or partial combinatory algebra. This paper is concerned with the preservation of various choice principles in |$V({\mathcal A})$| if assumed in the underlying universe |$V$|⁠ , adopting Constructive Zermelo–Fraenkel as background theory for all of these investigations. Examples of choice principles are the axiom schemes of countable choice, dependent choice, relativized dependent choice and the presentation axiom. It is shown that any of these axioms holds in |$V(\mathcal{A})$| for every applicative structure |$\mathcal A$| if it holds in the background universe. 1 It is also shown that a weak form of the countable axiom of choice, |$\textbf{AC}^{\boldsymbol{\omega , \omega }}$|⁠ , is rendered true in any |$V(\mathcal{A})$| regardless of whether it holds in the background universe. The paper extends work by McCarty (1984, Realizability and Recursive Mathematics, PhD Thesis) and Rathjen (2006, Realizability for constructive Zermelo–Fraenkel set theory. In Logic Colloquium 03 , pp. 282–314). [ABSTRACT FROM AUTHOR]

Google Scholar OpenURL Placeholder Text WorldCat 11 Gödel, Kurt [ 1934 ]: ' On undecidable propositions of formal mathematical systems ', in [Davis, 1965], pp. 39 - 74. Google Scholar OpenURL Placeholder Text WorldCat 9 Gödel, Kurt [ 1931a ]: ' Diskussion zur Grundlegung der Mathematik ', Erkenntnis 2, 147 - 151. Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC 6 Gödel, Kurt [ 1930 ]: ' Die Vollstandigkeit der Axiome des logischen Funktionenkalküls ', Monatshefte für Mathematik und Physik 37, 349 - 360. Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC 19 Wang Hao [ 1996 ]; A Logical Journey, from Gödel to Philosophy. [Extracted from the article]

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]

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 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]

This paper investigates the convergence of the randomized Kaczmarz algorithm for the problem of phase retrieval of complex-valued objects. Although this algorithm has been studied for the real-valued case in [ 28 ], its generalization to the complex-valued case is nontrivial and has been left as a conjecture. This paper applies a different approach by establishing the connection between the convergence of the algorithm and the convexity of an objective function. Based on the connection, it demonstrates that when the sensing vectors are sampled uniformly from a unit sphere in |${\mathcal{C}}^n$| and the number of sensing vectors |$m$| satisfies |$m>O(n\log n)$| as |$n, m\rightarrow \infty $|⁠ , then this algorithm with a good initialization achieves linear convergence to the solution with high probability. The method can be applied to other statistical models of sensing vectors as well. A similar convergence result is established for the unitary model, where the sensing vectors are from the columns of random orthogonal matrices. 2000 Math Subject Classification: 68W20, 68W27, 92D25. [ABSTRACT FROM AUTHOR]

In higher education, student learning support centres are examples of walk-in services with nonstationary demand. For many centres, the major expenditure is tutor wages; thus, optimizing tutor numbers and ensuring value for money in this area are key. In University College Dublin, the mathematics support centre (MSC) has developed a software system, which electronically records the time each student enters the queue, their start time with a tutor and time spent with a tutor. In this paper, we show how data analysis of 25,702 student visits and tutor timetable data, spanning 6 years, is used to identify busy and quiet periods. Prediction modelling is then used to estimate the waiting time for future MSC visitors. Subsequently, we discuss how this is used for staffing optimization, i.e. to ensure there is sufficient coverage for busy times and no resource wastage during quieter periods. The analysis described resulted in the MSC reducing the number of queue abandonments and releasing funds from overstaffed hours to increase opening hours. The methods used are easily adapted for any busy walk-in service, and the code and data referenced are freely available: https://github.com/ehoward1/Math-Support-Centre-. [ABSTRACT FROM AUTHOR]

This paper outlines an account of numbers based on the numerical equivalence schema (NES), which consists of all sentences of the form ' # x. F x = n if and only if ∃ n x   F x ', where # is the number-of operator and ∃ n is defined in standard Russellian fashion. In the first part of the paper, I point out some analogies between the NES and the T-schema for truth. In light of these analogies, I formulate a minimalist account of numbers, based on the NES, which strongly parallels the minimalist (deflationary) account of truth. One may be tempted to develop the minimalist account in a fictionalist direction, according to which arithmetic is useful but untrue, if taken at face value. In the second part, I argue that this suggestion is not as attractive as it may first appear. The NES suffers from a similar problem to the T-schema: it is deductively weak and does not enable the derivation of any non-trivial generalizations. In the third part of the paper, I explore some strategies to deal with the generalization problem, again drawing inspiration from the literature on truth. In closing this paper, I briefly compare the minimalist to some other accounts of numbers. [ABSTRACT FROM AUTHOR]

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]

Ensuring that patients with opioid use disorder (OUD) have access to optimal medication therapies is a critical challenge in substance use epidemiology. Rudolph et al. (Am J Epidemiol. 2023;XXX(X):XXXX-XXXX) demonstrated that sophisticated data-adaptive statistical techniques can be used to learn optimal, individualized treatment rules that can aid providers in choosing a medication treatment modality for a particular patient with OUD. This important work also highlights the effects of the mathematization of epidemiologic research. Here, we define mathematization and demonstrate how it operates in the context of effectiveness research on medications for OUD using the paper by Rudolph et al. as a springboard. In particular, we address the normative dimension of mathematization and how it tends to resolve a fundamental tension in epidemiologic practice between technical sophistication and public health considerations in favor of more technical solutions. The process of mathematization is a fundamental part of epidemiology; we argue not for eliminating it but for balancing mathematization and technical demands equally with practical and community-centric public health needs. [ABSTRACT FROM AUTHOR]

Recently we have found a couple of errors in our paper entitled An extension problem and trace Hardy inequality for the sub-Laplacian on |$H$| -type groups , Int. Math. Res. Not. IMRN (2020), no. 14, 4238–4294. They concern Propositions 3.12–3.13, and Theorem 1.5, Corollary 1.6 and Remark 4.10. The purpose of this corrigendum is to point out the errors and supply necessary modifications where it is applicable. [ABSTRACT FROM AUTHOR]

Recently we have found a couple of errors in our paper entitled An extension problem and trace Hardy inequality for the sub-Laplacian on |$H$| -type groups , Int. Math. Res. Not. IMRN (2020), no. 14, 4238–4294. They concern Propositions 3.12–3.13, and Theorem 1.5, Corollary 1.6 and Remark 4.10. The purpose of this corrigendum is to point out the errors and supply necessary modifications where it is applicable. [ABSTRACT FROM AUTHOR]

Recent debate about the error theory has taken a 'formal turn'. On the one hand, there are those who argue that the error theory should be rejected because of its difficulties in providing a convincing formal account of the logic and semantics of moral claims. On the other hand, there are those who claim that such formal objections fail, maintaining that arguments against the error theory must be of a substantive rather than a formal kind. In this paper, we argue that formal objections to the error theory cannot be eschewed but must be met head-on. [ABSTRACT FROM AUTHOR]

The core of Beverton and Holt's seminal work of the 1950s had been anticipated by Baranov 30 years earlier, but his contributions, published in Russian, remained poorly known to Anglophone scientists until after 1945. By that time, Russell and Graham had presented parallel ideas, though without Baranov's mathematics. Limited subsequent acknowledgement of the Russian's contributions has left an impression that each of his advances was achieved independently in England, hence that they contributed little to the Beverton–Holt model. I here construct a timeline of events linking and separating the Russian and English studies. From it, I argue that both Russell's presentation of the underlying concepts and Graham's decision to pursue a mathematical realization of his "Modern Theory" may have been influenced by translations of Baranov's papers. Moreover, when Holt developed the "simple" Beverton–Holt model during 1946–1947, he certainly drew on Baranov's exponential model of mortality, though perhaps only via Ricker. [ABSTRACT FROM AUTHOR]

Some mathematical proofs explain why the theorems they prove hold. This paper identifies several challenges for any counterfactual account of explanation in mathematics (that is, any account according to which an explanatory proof reveals how the explanandum would have been different, had facts in the explanans been different). The paper presumes that countermathematicals can be nontrivial. It argues that nevertheless, a counterfactual account portrays explanatory power as too easy to achieve, does not capture explanatory asymmetry, and fails to specify why certain proofs are explanatory and others are not. Greater informativeness about counterfactual dependence can even yield less explanatory power. [ABSTRACT FROM AUTHOR]

Gerhard Jäger and Wilfried Sieg. Feferman on Foundations: Logic, Mathematics, Philosophy. [Extracted from the article]

The article discusses the proceedings of the 17th Brazilian Logic Conference 2014 held on the premises of the National Laboratory for Scientific Computing (LNCC) in Petrópolis, Rio de Janeiro from April 7-11, 2014. Information about the keynote speakers and the theme for the topics which included Algebraic Logic, Foundations of Mathematics, and Set Theory and Theory of Computation are presented. An introduction in which the editor discusses various topics within the issue is also presented.

The containment companion of a logic |$\vdash $| consists of the consequence relation |$\vdash ^{r}$| which satisfies all the inferences of |$\vdash $|⁠ , where the variables of the conclusion are contained into those of the set of premises, in case this is not inconsistent. Following the algebraic analysis started in Bonzio and Pra Baldi (2021, Studia Logica , 109, 969–994), this paper characterizes the algebraic counterpart of a finitary containment logic |$\vdash ^{r}$| and investigates the structure of the Leibniz and Suszko reduced models. The analysis is carried within the framework of abstract algebraic logic. Mathematics Subject Classification : Primary: 03G27. Secondary: 03G25 [ABSTRACT FROM AUTHOR]

Reliability evaluation of interconnection networks is of significant importance to the design and maintenance of interconnection networks. The component connectivity is an important parameter for the reliability evaluation of interconnection networks and is a generalization of the traditional connectivity. The |$g$| -component connectivity |$c\kappa _g (G)$| of a non-complete connected graph |$G$| is the minimum number of vertices whose deletion results in a graph with at least |$g$| components. Determining the |$g$| -component connectivity is still an unsolved problem in many interconnection networks. Let |$Q_{n,k}$| (⁠|$1\leq k\leq n-1$|⁠) denote the |$(n, k)$| -enhanced hypercube. In this paper, let |$n\geq 7$| and |$1\leq k \leq n-5$|⁠ , we determine |$c\kappa _{g}(Q_{n,k}) = g(n + 1) - \frac{1}{2}g(g + 1) + 1$| for |$2 \leq g \leq n$|⁠. The previous result in Zhao and Yang (2019, Conditional connectivity of folded hypercubes. Discret. Appl. Math. , 257 , 388–392) is extended. [ABSTRACT FROM AUTHOR]

The paper introduces a new characterization of strict positive definiteness for kernels on the 2-sphere without assuming the kernel to be radially (isotropic) or axially symmetric. The results use the series expansion of the kernel in spherical harmonics. Then additional sufficient conditions are proven for kernels with a block structure of expansion coefficients. These generalize the result derived by Chen et al. (2003, A necessary and sufficient condition for strictly positive definite functions on spheres. Proc. Amer. Math. Soc. , 131 , 2733–2740) for radial kernels to nonradial kernels. [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]

Forecasting non-stationary time series, especially when the data generating processes contains a random walk component, is a difficult and sometimes impossible task. In this paper we suggest an intuitive, computationally fast and expedient way of forecasting time series of the above type using distance-based nearest neighbours (NN). We exploit to advantage the path and scale dependence present in a random walk model and so we provide a number of theoretical results (a) on the distances used for selecting the NN, (b) on a number of new forecasting models that use these distances and (c) on the properties of the resulting forecasts. We illustrate the efficacy of our method via a comprehensive empirical application on time series of exchange rates and commodities, where we present the resulting performance enhancements and discuss the importance of such results in a decision-making context, linking our forecasting approach with management mathematics and predictive analytics problems. [ABSTRACT FROM AUTHOR]

This paper investigates the impact of publication pressure on the ethics and the scientific integrity in the domain of mathematics and of the arts. Both research fields are specific in their methodology, being that they do not start from a classical hypothesis and researchers in these areas are not knowing what the outcome will be. The research design is open, and creativity is a main part of the research investigation. Both research fields do not rely on empirical cases nor on data collection or data handling. This could be a reason why mathematics and arts seem to be less subject to scientific misconduct. After presenting the (inter)national regulations on ethics and integrity, we will investigate a philosophical analysis in which we consider possible influences from publication pressure that became widespread over all disciplines. We will clarify if and how mathematics and the arts are sensitive to scientific research misconduct or questionable research practices. [ABSTRACT FROM AUTHOR]

The aim of the paper is to answer some arguments raised against mathematical structuralism developed by Michael Resnik. These arguments stress the abstractness of mathematical objects, especially their causal inertness, and conclude that mathematical objects, the structures posited by Resnik included, are inaccessible to human cognition. In the paper I introduce a distinction between abstract and ideal objects and argue that mathematical objects are primarily ideal. I reconstruct some aspects of the instrumental practice of mathematics, such as symbolic manipulations or ruler-and-compass constructions, and argue that instrumental practice can secure epistemic access to ideal objects of mathematics. [ABSTRACT FROM AUTHOR]

We wish to point out errors in the paper "Abelian Arithmetic Chern–Simons Theory and Arithmetic Linking Numbers", International Mathematics Research Notices, Vol. 2017, No. 00, pp. 1–29. The main error concerns the symmetry of the "ramified case" of the height pairing, which relies on the vanishing of the Bockstein map in Proposition 3.5. The surjectivity claimed in the 1st line of the proof of Proposition 3.5 is incorrect. The specific results that are affected are Proposition 3.5; Lemmas 3.6, 3.7, 3.8, and 3.9; and Corollary 3.11. The definition of the (S,n)-height pairing following Lemma 3.9 is also invalid, since the symmetry of the pairing was required for it to be well defined. The results of Section 3 before Proposition 3.5 as well as those of the other Sections are unaffected. Proposition 3.10 is correct, but the proof is unclear and has some sign errors. So we include here a correction. As in the paper, let I be an ideal such that In is principal in OF,S⁠. Write In = (ƒ−1)⁠. Then the Kummer cocycles kn(ƒ) will be in Z1(U, Z/nZ)⁠. For any a ∈ F⁠ , denote by aS its image in ∏v∈S Fv⁠. Thus, we get an element   [ƒ]S,n := [(kn(&#402), kn2(ƒS), 0)] ∈ Z1(U, Z/nZ × SZ/n2Z), which is well defined in cohomology independently of the choice of roots used to define the Kummer cocycles. (We have also trivialized both μn2 and μn⁠.) [ABSTRACT FROM AUTHOR]

The article focuses on Robert Ferber and the creation of the Association for Consumer Research's Robert Ferber Award. It states that Ferber was born in New York City in 1922 and had a college education in mathematics and economics. It mentions he was hired as an assistant professor at the University of Illinois, publishing 17 books and a number of articles and monographs. It comments on research he did about consumer behavior and in marketing research. It states that in the early 1960s he was the editor of the "Journal of Marketing Research" where he encouraged the publication of papers concerning consumer research and behavioral articles.

Discusses the multivariate extreme value theory that emerged in several studies in the 1950s during the popularity of asymptomatic distributions in probability. Examination on the mathematical abstraction of the formulation of the theory; Discovery of the deficiencies of this appraoch during the 1990s; Information on proposed correction to the deficiencies of the approach.

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]

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]

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]

This paper aims at introducing the concept of time-dependent polarization tensors (TDPTs) for the wave equation associated to a diametrically small acoustic inclusion, with constitutive parameters different from those of the background and size smaller than the operating wavelength. Firstly, the solution to the Helmholtz equation is considered, and a rigourous systematic derivation of a complete asymptotic expansion of the scattered field due to the presence of the inclusion is presented. Then, by applying the Fourier transform, the corresponding time-domain expansion is readily obtained after truncating the high frequencies. The new concept of TDPTs is shown to be promising for performing imaging. In particular, the optimization approach proposed by Ammari et al. (Ammari, H. Kang, H. Kim, & E. Lee, J.-Y. (2012)The generalized polarization tensors for resolved imaging. Part II: shape and electromagnetic parameters reconstruction of an electromagnetic inclusion from multistatic measurements. Math. Comp. , 81, 839–860.) is extended to TDPTs. Numerical simulations are presented, showing that the TDPTs reconstructed from noisy measurements allow to image fine shape details of the inclusion. [ABSTRACT FROM AUTHOR]

This article presents three hitherto unpublished letters by Frank Plumpton Ramsey on the foundations of mathematics with commentary. One of the letters was sent to Abraham Fraenkel and the other two letters to Heinrich Behmann. The transcription of the letters is preceded by an account that details the extent of Ramsey's known contacts with mathematical logicians on the Continent. [ABSTRACT FROM AUTHOR]

A multirepeat in a string is a substring (factor) that appears a predefined number of times. A multirepeat is maximal if it cannot be extended either to the right or to the left and produce a multirepeat. In this paper, we present algorithms for two different versions of the problem of finding maximal multirepeats in a set of strings. In the case of arbitrary gaps, we propose an algorithm with O(σN2n + α) time complexity. When the gap is bounded in a small range c, we propose an algorithm with O((c2 + σ2)mN2n log(Nn) + α) time complexity. Here, N is the number of strings, n the mean length of each string, m the multiplicity of the multirepeat and α the number of reported occurrences. Our results extend previous work by considering sets of strings as well as by generalizing pairs to multirepeats. [ABSTRACT FROM AUTHOR]