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We generalize Deligne's equivalence between the categories of regular-singular connections on the formal punctured disk and on the punctured affine line to the case where the base is a strictly Henselian discrete valuation ring of equal characteristic 0. We also provide a weaker result when the base is higher dimensional. [ABSTRACT FROM AUTHOR]

Purpose: The aim of this study was to assess the concurrent validity of a contact mat against force plates to measure jump height in countermovement jump (CMJ) and squat jump (SJ) in professional soccer players. Methods: 23 male professional soccer players performed the CMJ and SJ, which were concurrently recorded using a portable contact mat (SmartJump) and a portable dual force plate system (ForceDecks). Equivalence testing between both systems (contact mat vs. force plate) and the two methods (impulse- momentum vs. flight-time and flight-time vs. flight-time) was performed compared to equivalence bounds of ±1.1 cm for the CMJ and ±1.6 cm for the SJ. Additionally, 95% Limits of Agreement (LoA) and intraclass correlation coefficients (ICC) were computed. Results: Mean differences for the impulse-momentum vs. flight-time comparison for CMJ [3.2 cm, 95% CI (2.3-4.1)] and SJ [2.7 cm, (1.8-3.6)] were non-equivalent between both systems. LoA were larger than the equivalence bunds for CMJ and SJ, while ICCs were good [CMJ, 0.89, (0.76-0.95)] and excellent [SJ, 0.91, (0.79-0.96)]. As for the flight-time vs. flight-time comparison, mean differences were non-equivalent for the CMJ [1.0 cm (0.8 to 1.2 cm)] and equivalent for the SJ [0.9 cm (0.7-1.1 cm)]. LoA were narrower than the equivalence bounds for CMJ and SJ, while ICCs were excellent [CMJ, 0.995, 95% CI (0.989-0.998); SJ, 0.997, 95% CI (0.993-0.997)]. Conclusion: Our findings indicate that the SmartJump contact mat cannot be used interchangeably with the ForceDecks force platform to measure jump height for the CMJ and SJ. [ABSTRACT FROM AUTHOR]

Introduction: Hand gestures and actions-with-objects (hereafter ‘actions’) are both forms of movement that can promote learning. However, the two have unique affordances, which means that they have the potential to promote learning in different ways. Here we compare how children learn, and importantly retain, information after performing gestures, actions, or a combination of the two during instruction about mathematical equivalence. We also ask whether individual differences in children’s understanding of mathematical equivalence (as assessed by spontaneous gesture before instruction) impacts the effects of gesture- and action-based instruction. Method: Across two studies, racially and ethnically diverse third and fourthgrade students (N=142) were given instruction about how to solve mathematical equivalence problems (eg., 2+9+4=__+4) as part of a pretest-training-posttest design. In Study 1, instruction involved teaching students to produce either actions or gestures. In Study 2, instruction involved teaching students to produce either actions followed by gestures or gestures followed by actions. Across both studies, speech and gesture produced during pretest explanations were coded and analyzed to measure individual differences in pretest understanding. Children completed written posttests immediately after instruction, as well as the following day, and four weeks later, to assess learning, generalization and retention. Results: In Study 1 we find that, regardless of individual differences in pre-test understanding of mathematical equivalence, children learn from both action and gesture, but gesture-based instruction promotes retention better than actionbased instruction. In Study 2 we find an influence of individual differences: children who produced relatively few types of problem-solving strategies (as assessed by their pre-test gestures and speech) perform better when they receive action training before gesture training than when they receive gesture training first. In contrast, children who expressed many types of strategies, and thus had a more complex understanding of mathematical equivalence prior to instruction, performed equally with both orders. Discussion: These results demonstrate that action training, followed by gesture, can be a useful stepping-stone in the initial stages of learning mathematical equivalence, and that gesture training can help learners retain what they learn. [ABSTRACT FROM AUTHOR]

A triangular tile latching system consists of a set Σ of equilateral triangular tiles with at least one latchable side and an attachment rule which permits two tiles to get latched along a latchable side. In this paper we determine the language generated by a triangular tile latching system in terms of planar graphs. [ABSTRACT FROM AUTHOR]

We study faithful actions with a dense orbit of abelian unipotent groups on quintic del Pezzo varieties over a field of characteristic zero. Such varieties are forms of linear sections of the Grassmannian of planes in a 5-dimensional vector space. We characterize which smooth forms admit these types of actions and show that in case of existence, the action is unique up to equivalence by automorphisms. We also give a similar classification for mildly singular quintic del Pezzo threefolds and surfaces. [ABSTRACT FROM AUTHOR]

This qualitative research study uses middle-grades students' numerical reasoning to model their symbolic representations of the relationship between two multiplicatively related unknowns on an algebra task. Students in sixth grade through ninth grade participated in clinical interviews that assessed their numerical reasoning using the Number Sequences framework, their interpretation of the equal sign, and their algebraic reasoning in the context of writing equations. Analysis used the mental structures that define their numerical reasoning to explain their algebraic capabilities, and changes in their reasoning about the equal sign in different contexts. Findings show that splitting, one mental structure related to the students' numerical reasoning, is related to students' symbolic representations of multiplicatively related unknowns. Furthermore, students who had not constructed a splitting operation tended to use numerical examples in place of reasoning about unknowns, and to reason operationally about the equal sign to compensate for this limitation on an algebra task. [ABSTRACT FROM AUTHOR]

In this meta syntheses paper, I explore gender as a social factor that interacts with both the cognitive and affective domains of education by explaining what it means for mathematics to be gendered and how we can use mathematics education to promote gender fairness as a form of social justice. I start by discussing gender as a dichotomy and how educational research transitioned from a physiological to a socio-cultural perspective. Guided by Ferguson’s and Bjerrun-Nielsen’s Four-Perspective Gender model, and Cohen and Martin’s Four Dimensional Model of Gender. I argue that mathematics education can be designed to promote gender equity only by considering these dimensions, and when there is a deliberate effort to examine how our exercise of education widens or narrows gender gap in terms of the four dimensions of gender equality: equality to access, learning experience, educational outcomes, and most importantly, external results. I discuss mathematics as an area perceived to embody a symbolic gender and what can be done through mathematics education to break that stereotype and contribute to the furthering equality to external results. I focus this discussion into the context of the Philippines and why it may be essential for educational reforms to zoom out to include other equally important dimensions of gender other than equality to access. [ABSTRACT FROM AUTHOR]

This paper discusses the large deviations for stochastic predator–prey model driven by multiplicative Lévy noise. Using Galerkin approximation, we initially prove the existence and uniqueness of solution. Due to the equivalence between Laplace principle and large deviation principle under a Polish space, the method of weak convergence has been followed in order to establish our results for this coupled system of equations. [ABSTRACT FROM AUTHOR]

Summary: Testing for equivalence, rather than testing for a difference, is an important component of some scientific studies. While the focus of the existing literature is on comparing two groups for equivalence, real‐world applications arise regularly that require testing across more than two groups. This paper reviews the existing approaches for testing across multiple groups and proposes a novel framework for multigroup equivalence testing under a Bayesian paradigm. This approach allows for a more scientifically meaningful definition of the equivalence margin and a more powerful test than the few existing alternatives. This approach also allows a new definition of equivalence based on future differences. [ABSTRACT FROM AUTHOR]

Here, we propose, prove mathematically and discuss maximum and minimum measures of maximum parsimony evolution across 12 discrete phylogenetic character types, classified across 4467 morphological and molecular datasets. Covered character types are: constant, binary symmetric, multistate unordered (non‐additive) symmetric, multistate linear ordered symmetric, multistate non‐linear ordered symmetric, binary irreversible, multistate irreversible, binary Dollo, multistate Dollo, multistate custom symmetric, binary custom asymmetric and multistate custom asymmetric characters. We summarize published solutions and provide and prove a range of new formulae for the algebraic calculation of minimum (m), maximum (g) and maximum possible (gmax) character cost for applicable character types. Algorithms for exhaustive calculation of m, g and gmax applicable to all classified character types (within computational limits on the numbers of taxa and states) are also provided. The general algorithmic solution for minimum steps (m) is identical to a minimum spanning tree on the state graph or minimum weight spanning arborescence on the state digraph. Algorithmic solutions for character g and gmax are based on matrix mathematics equivalent to optimization on the star tree, respectively for given state frequencies and all possible state frequencies meeting specified numbers of taxa and states. We show that maximizing possible cost (gmax) with given transition costs can be equivalent to maximizing, across all possible state frequency combinations, the lowest implied cost of state transitions if any one state is ancestral on the star tree, via the solution of systems of linear equations. The methods we present, implemented in the Claddis R package, extend to a comprehensive range, the fundamental character types for which homoplasy may be measured under parsimony using m, g and gmax, including extra cost (h), consistency index (ci), retention index (ri) or indices based thereon. [ABSTRACT FROM AUTHOR]

Let E be an equivalence on a set X and let T ∃ (X) denote the semigroup (under composition) of all f : X → X that reflect E. Fix an element θ ∈ T ∃ (X) and for f , g ∈ T ∃ (X) , define a new operation ∗ on T ∃ (X) by f ∗ g = f θ g where f θ g denotes the product of g , θ and f in the original sense. In this paper, we characterize Green's relations on the variant semigroups T ∃ (X , θ) of T ∃ (X) with sandwich operation θ . [ABSTRACT FROM AUTHOR]

We consider the problem of estimating the mean of a random vector based on N independent, identically distributed observations. We prove the existence of an estimator that has a nearoptimal error in all directions in which the variance of the one-dimensional marginal of the random vector is not too small: with probability 1-δ, the procedure returns μ N ​ which satisfies, for every direction u∈Sd-1, ... where σ²(u)=Var(X,u) and C is a constant. To achieve this, we require only slightly more than the existence of the covariance matrix, in the form of a certain moment-equivalence assumption. [ABSTRACT FROM AUTHOR]

We classify all 2‐blocks with abelian defect groups of rank 4 up to Morita equivalence. The classification holds for blocks over a suitable discrete valuation ring as well as for those over an algebraically closed field. An application is that Broué's abelian defect group conjecture holds for all blocks under consideration here. [ABSTRACT FROM AUTHOR]

Two curves are affinely equivalent if there exists an affine mapping transforming one of them onto the other. Thus, detecting affine equivalence comprises, as important particular cases, similarity, congruence and symmetry detection. In this paper we generalized previous results by the authors to provide an algorithm for computing the affine equivalences between two parametric curves of certain types, in any dimension. In more detail, the algorithm is valid for rational curves, and for parametric curves with nonrational but meromorphic components, it admits an also meromorphic, and in fact rational, inverse. Unlike other algorithms already known for rational curves, the algorithm completely avoids polynomial system solving, and instead uses bivariate factoring as a fundamental tool. The algorithm has been implemented in the computer algebra system Maple and can be freely downloaded and used. [ABSTRACT FROM AUTHOR]

Psychological researchers may be interested in demonstrating that sets of scores are equivalent, as opposed to different. If this is true, use of equivalence analyses (equivalence and non-inferiority testing) are appropriate. However, the use of such tests has been found to be inconsistent and incorrect in other research fields (Lange and Freitag 2005). This study aimed to review the use of equivalence analyses in the psychological literature to identify issues in the selection, application, and execution of these tests. To achieve this a systematic search through several databases was conducted to identify psychological research from 1999 to the 2020 that utilized equivalence analyses. Test selection, choice of equivalence margin, equivalence margin justification and motivation, and data assessment practices for 122 studies were examined. The findings indicate wide variability in the reporting of equivalence analyses. Results suggest there is a lack of agreement amongst researchers as to what constitutes a meaningless difference. Additionally, explications of this meaninglessness (i.e., justifications of equivalence margins) are often vague, inconsistent, or inappropriate. This scoping review indicates that the proficiency of use of these statistical approaches is low in psychology. Authors should be motivated to explicate all aspects of their selected equivalence analysis and demonstrate careful consideration has been afforded to the equivalence margin specification with a clear justification. Additionally, there is also a burden of responsibility on journals and reviewers to identify sub-par reporting habits and request refinement in the communication of statistical protocols in peer-reviewed research. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of weakly semiprime ideals and weakly n -systems in noncommutative rings. We establish the equivalence between an ideal P being a weakly semiprime ideal and R − P being a weakly n -system. We provide alternative definitions and explore the properties of weakly semiprime ideals. Additionally, we investigate scenarios where all ideals in a given ring are weakly semiprime and demonstrate that in Noetherian rings, where every ideal is weakly semiprime, the prime radical and the Jacobson radical coincide. [ABSTRACT FROM AUTHOR]

This study focuses on the research question: How do prospective elementary school teachers (FT) make decisions when considering the algebraic thinking of 9-year-old children? A specific type of noticing was used to describe the decisions of 21 FT when observing the strategies employed by 3 children to solve the equation 6+4=+5. The FT participated in a six-session course on the teaching and learning of algebra, based on video analysis as a means to approach teaching practice. Focusing on sessions 1 and 2, the written responses of the FT to two distinct questions were examined. The main results show that, although the FT's decisions frequently lacked evidential support, their reasoning aligned with specific aspects of algebraic thinking and research covered during the course. Finally, two positions adopted by the FT in decision making were identified: a) arithmetic, centred on the calculations that children should follow, and (b) relational, focused on the interplay of operations through the equals sign. The role of noticing and video analysis as tools to bring future primary school teachers closer to the practice of teaching algebra in primary education was discussed. [ABSTRACT FROM AUTHOR]

This paper initiates a teaching sequence that focuses on building up equivalent definitions to the standard ones for the limit concept in Real Analysis. It comprises two parts: The first provides a classroom assignment where students, guided by Analysis lecturers, are led to develop an alternative definition to the $ \varepsilon - \delta $ ϵ − δ one for limits of one-variable real functions that are based on the realisation of the interval of $ \delta $ δ 's. In the second one, students are directed by their instructor to restore equivalence between two distinct definitions of limit for a function mapping a subset of $ {R^2} $ R 2 into $ R $ R. This is done by adding a condition to one of the definitions, resulting in a third definition that is equivalent to the other one. [ABSTRACT FROM AUTHOR]

Running time analysis of evolutionary algorithms for continuous optimization is one research challenge in the field of evolutionary algorithms (EAs). However, the theoretical analysis results have rarely been applied to evolutionary algorithms for continuous optimization in practice, let alone their variants for evolution strategy. In this paper, we regarded the first hitting time of evolution strategy as the stopping time of the renewal process on the basis of the renewal process and in combination with Wald's inequality and stopping time theory. Afterwards, to demonstrate the application of the proposed model in the first hitting time analysis of (1 + 1) ES, we analyzed it with different mutation operators on the sphere function. First, we significantly improved the lower bound on the first hitting time of (1 + 1) ES with a uniform mutation operator, i.e., from Ω (n) to Ω (e c n) . Next, O (n 2 n) was the upper bound on the first hitting time of (1 + 1) ES with a Gaussian mutation operator from the initial distance R to half of the initial distance R /2. The numerical experimental results showed that the theoretical calculation was consistent with the actual running time, which provides a novel method for analyzing the first hitting time of EAs. [ABSTRACT FROM AUTHOR]

We undertake a study of conic bundle threefolds p: X → W over geometrically rational surfaces whose associated discriminant covers Δ → Δ ⊂ W are smooth and geometrically irreducible. We first show that the structure of the Galois module CH² Xk of rational equivalence classes of curves is captured by a group scheme that is a generalization of the Prym variety of Δ → Δ. This generalizes Beauville's result that the algebraically trivial curve classes on Xk are parametrized by the Prym variety. We apply our structural result on curve classes to study the refined intermediate Jacobian torsor (IJT) obstruction to rationality introduced by Hassett-Tschinkel and Benoist-Wittenberg. The first case of interest is where W = P² and Δ is a smooth plane quartic. In this case, we show that the IJT obstruction characterizes rationality when the ground field has less arithmetic complexity (precisely, when the 2-torsion in the Brauer group of the ground field is trivial). We also show that a hypothesis of this form is necessary by constructing, over any k ⊂ R, a conic bundle threefold with Δ a smooth quartic where the IJT obstruction vanishes, yet X is irrational over k. [ABSTRACT FROM AUTHOR]

We have studied initial value problems for Caputo fractional differential equations with singular nonlinearities involving the p-Laplacian operator. We have given a precise mathematical analysis of the equivalence of the fractional differential equations and Volterra integral equations studied in this paper. A theorem for the global existence of the solution was proven. In addition, an example was given at the end of the article as an application of the results found in this paper. [ABSTRACT FROM AUTHOR]

This paper initiates a teaching sequence that focuses on building up equivalent definitions to the standard ones for the limit concept in Real Analysis. It comprises two parts: The first provides a classroom assignment where students, guided by Analysis lecturers, are led to develop an alternative definition to the $ \varepsilon - \delta $ ϵ − δ one for limits of one-variable real functions that are based on the realisation of the interval of $ \delta $ δ 's. In the second one, students are directed by their instructor to restore equivalence between two distinct definitions of limit for a function mapping a subset of $ {R^2} $ R 2 into $ R $ R. This is done by adding a condition to one of the definitions, resulting in a third definition that is equivalent to the other one. [ABSTRACT FROM AUTHOR]

Solving mathematical equations faster and more efficiently has been a Holy Grail for centuries for scientists and engineers across all disciplines. While electronic digital circuits have revolutionized equation solving in recent decades, it has become apparent that performance gains from brute-force approaches of compute-solvers are quickly saturating over time. Instead, paradigms that leverage the universes' natural tendency to minimize a system's free energy, such as annealers or Ising Machines, are being sought after due to favorable complexity scaling. Here, we introduce a programmable analog solver leveraging the formal mathematical equivalence between Maxwell's equations and photonic circuitry. It features a mesh network of nanophotonic beams to find solutions to partial differential equations. As an example, we designed, fabricated, and demonstrated a novel application-specific photonic integrated circuit comprised of electro-optically reconfigurable nodes and experimentally validated 90 % accuracy with respect to a commercial solver. Finally, we tested this photonic integrated chip performance by simulating thermal diffusion on a spacecraft's heat shield during re-entry to a planet's atmosphere. The programmable light-circuitry presented herein offers a facile route for solving complex problems and thus will have profound potential applications across many scientific and engineering fields. [ABSTRACT FROM AUTHOR]

The paper presents a critical review and new accomplishments on the equivalence of the first-order displacement-based zigzag theories for laminated composite and sandwich structures. Zigzag theories (ZZTs) have widely spread among researchers over the last few decades thanks to their accuracy in predicting the response of multilayered composite and sandwich structures while retaining the simplicity of their underlying equivalent single-layer (ESL) theory. The displacement field consists of two main contributions: the global one, able to describe the overall structural behaviour, and the local layer-wise one that considers the transverse shear continuity at the layer interfaces that describe the "zigzag" displacement pattern typical of multilayered structures. In the framework of displacement-based linear ZZTs, various assumptions have been made on the local contribution, and different theories have been deduced. This paper aims to provide a unified formulation for first-order ZZTs, highlighting some common aspects and underlying equivalencies with existing formulations. The mathematical demonstrations and the numerical examples prove the equivalence of the approaches to characterising local zigzag enrichment. Finally, it is demonstrated that the kinematic assumptions are the discriminants of the ZZTs' accuracy. [ABSTRACT FROM AUTHOR]

This paper addresses the recent body of research in algebra and algebraic thinking from 2018 to 2022. We reviewed 74 journal articles and identified four clusters of content areas: (a) literal symbols and symbolizing, (b) equivalence and the equal sign, (c) equations and systems, and (d) functions and graphing. We present the research on each of these content clusters, and we discuss insights on effective teaching practices and the social processes supporting algebraic reasoning. The research base shows that incorporating algebraic thinking into the elementary grades, emphasizing analytic and structural thinking processes, and emphasizing covariational reasoning supports students' meaningful learning of core algebraic ideas. We close with a discussion of the major theoretical contributions emerging from the past five years, offering suggestions for future research. [ABSTRACT FROM AUTHOR]

The operator of the strong zero mode for the one-dimensional system of interacting fermions without pairing is presented. It is conjectured that the strong zero mode is related to the Majorana edge eigenstate, which is shown to exist (using the exact Bethe ansatz study) in this system. The results are robust with respect to the sign randomness of hopping amplitudes (and if the pairing amplitudes are nonzero, similar results exist for equal sign randomness of hopping and pairing amplitudes). [ABSTRACT FROM AUTHOR]

This paper studies grade 7 (age 13) students' expressed understanding about the equal sign/notion of equivalence in order to investigate what aspects of the concept that could be seen as an established knowledge at lower secondary school/middle school. Using items from different instruments and combining these to a new one that covered a broad spectrum of procedural and conceptual knowledge, we collected data from 159 students. The different statistical tests showed that if focusing only on separate items, it could confirm that students could be seen either having operational or relational understanding of the equal sign. However, when taking all results into account using several analyses, students' understanding appear to be much more complex. Instead of a dichotomized view, students' expressed knowledge of mathematical equivalence should be seen as a continuum. [ABSTRACT FROM AUTHOR]

This article presents a methodology for solving one-variable inequalities, specifically focusing on inequalities involving radicals. The author highlights the lack of connections between different types of inequalities in algebra textbooks and suggests best practices for implementing this methodology in the classroom. The text also mentions the author's work on a generalized approach to graphing inequalities of two variables. Overall, this article provides a useful resource for educators and students studying inequalities in mathematics. [Extracted from the article]

The notion of von Neumann equivalence (vNE), which encapsulates both measure equivalence and W*-equivalence, was introduced recently by Ishan, Peterson, and Ruth (2019). They have shown that many analytic properties, such as amenability, property (T), the Haagerup property, and proper proximality are preserved under von Neumann equivalence. In this article, we expand on the list of properties that are stable under von Neumann equivalence, and prove that weak amenability, weak Haagerup property, and the approximation property (AP) are von Neumann equivalence invariants. In particular, we get that AP is stable under measure equivalence. Furthermore, our techniques give an alternate proof for vNE-invariance of the Haagerup property. [ABSTRACT FROM AUTHOR]

This paper analyses the abstractionist account of quantity championed by Leibniz, especially in the 1680s. Leibniz introduced the notion of quantity in an indirect way, via an abstraction principle. In the first part of the paper, I identify the context in which this approach arose in light of Leibniz's criticism of his earlier dream of an 'alphabet of human thought'. Recognising the impossibility of such a project led him to realise that, when dealing with terms referring to abstract objects, we should always consider them within the true sentences in which they occur. In the second part, I describe this approach in detail. This allows us to look at some key concepts of Leibniz's theory of quantity. In particular, I raise the problem of the relationship between the two sides of the abstraction principle: how should we think of the relation between the claim that a and b are equal, and the claim that the quantity of a is identical to the quantity of b? I argue that we can find a positive answer to this problem in Leibniz. [ABSTRACT FROM AUTHOR]

Rosenfeld defined a fuzzy subgroup of a given group as a fuzzy subset with two special conditions and Mustafa Demirci proposed the idea of fuzzifying the operations on a group through a fuzzy equality and a fuzzy equivalence relation. This paper mainly focuses on fuzzy subsets and vague sets of monoids with several extended algebraic properties. Firstly, we generalize some algebraic properties of t-norms to fuzzy t-norms, this allows for a broader analysis and classification of fuzzy t-norms, enabling their wider application. Furthermore, we explore specific research on the properties of vague t-norms. Finally, selected conclusions about fuzzy t-norms are extended to bounded lattices. [ABSTRACT FROM AUTHOR]

In this paper, we study the classic submodular maximization problem subject to a group equality constraint under both nonadaptive and adaptive settings. It is shown that the utility function of many machine learning applications, including data summarization, influence maximization in social networks, and personalized recommendation, satisfies the property of submodularity. Hence, maximizing a submodular function subject to various constraints can be found at the heart of many of those applications. On a high level, submodular maximization aims to select a group of most representative items (e.g., data points). However, the design of most existing algorithms does not incorporate the fairness constraint, leading to underrepresentation or overrepresentation of some particular groups. This motivates us to study the submodular maximization problem with group equality, in which we aim to select a group of items to maximize a (possibly nonmonotone) submodular utility function subject to a group equality constraint. To this end, we develop the first constant-factor approximation algorithm for this problem. The design of our algorithm is robust enough to be extended to solving the submodular maximization problem under a more complicated adaptive setting. Moreover, we further extend our study to incorporating a global cardinality constraint and other fairness notations. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Supplemental Material: The online supplement is available at https://doi.org/10.1287/ijoc.2022.0384. [ABSTRACT FROM AUTHOR]

The straightening–unstraightening correspondence of Grothendieck–Lurie provides an equivalence between cocartesian fibrations between $(\infty, 1)$ -categories and diagrams of $(\infty, 1)$ -categories. We provide an alternative proof of this correspondence, as well as an extension of straightening–unstraightening to all higher categorical dimensions. This is based on an explicit combinatorial result relating two types of fibrations between double categories, which can be applied inductively to construct the straightening of a cocartesian fibration between higher categories. [ABSTRACT FROM AUTHOR]

theory is examined, highlighting his approach to teaching the subject and the impact it had on the field of mathematics. The author discusses Klein's emphasis on the historical development of the theory and his use of concrete examples to illustrate abstract concepts. The study provides insight into the pedagogical methods employed by Klein and their influence on the teaching of mathematics. [Extracted from the article]

In a cluster randomized trial clusters of persons, for instance, schools or health centers, are assigned to treatments, and all persons in the same cluster get the same treatment. Although less powerful than individual randomization, cluster randomization is a good alternative if individual randomization is impossible or leads to severe treatment contamination (carry-over). Focusing on cluster randomized trials with a pretest and post-test of a quantitative outcome, this paper shows the equivalence of four methods of analysis: a three-level mixed (multilevel) regression for repeated measures with as levels cluster, person, and time, and allowing for unstructured between-cluster and within-cluster covariance matrices; a two-level mixed regression with as levels cluster and person, using change from baseline as outcome; a two-level mixed regression with as levels cluster and time, using cluster means as data; a one-level analysis of cluster means of change from baseline. Subsequently, similar equivalences are shown between a constrained mixed model and methods using the pretest as covariate. All methods are also compared on a cluster randomized trial on mental health in children. From these equivalences follows a simple method to calculate the sample size for a cluster randomized trial with baseline measurement, which is demonstrated step-by-step. [ABSTRACT FROM AUTHOR]

This document is a collection of postage stamps from around the world that feature mathematicians, mathematical concepts, and related topics. The stamps are presented in chronological order and provide a visual representation of the history of mathematics. Each stamp is accompanied by a brief description of the mathematician or concept depicted. The collection showcases the global recognition of mathematics and its impact on society, with stamps from various countries and time periods. The document also includes stamps that depict computer art, op art, mathematical olympiads, geometry teaching, and other mathematical themes. Additional reading materials and online resources are suggested for further exploration. [Extracted from the article]

Angular equivalence of norms, introduced by Kikianty and Sinnamon (2017), is a notion of norm equivalence that is more attuned to the geometry of the norms. For certain geometrical properties and two angularly equivalent norms, it is the case that if one of the norms has a property, then so does the other. In this paper, we show further results in this direction, namely angularly equivalent norms share the property of non-squareness; and that the exposed points of the unit balls are in the same direction, under the condition that these points are assumed to be smooth with respect to both norms. A discussion on (the angular equivalence of) the dual norms of angularly equivalent norms is also given, giving a partial answer to an open problem as stated in the paper by Kikianty and Sinnamon (2017). [ABSTRACT FROM AUTHOR]

The presented paper is a compendium or a kind of précis of relationships between the four classical theorems of mathematical analysis. More precisely, the first aim of this paper is to present the equivalence of the four classical theorems: the Hahn–Banach theorem, the Mazur–Orlicz theorem, the Markov–Kakutani fixed-point theorem and the von Neumann theorem on amenability of Abelian groups. The second purpose is to prove these equivalences in Zermelo–Fraenkel set theory with the axiom of choice replaced by the Boolean prime ideal theorem, from which we can also prove the Hahn–Banach theorem. [ABSTRACT FROM AUTHOR]

In this paper we obtain some refinements and reverses of Callebaut's inequality for isotonic functionals via a result of Young's inequality due to Cartwright and Field. [ABSTRACT FROM AUTHOR]

We study the (z,k)-equivalence of the pairs of matrices over the ring of Gaussian integers and their reducibility to standard forms. It is shown that the number of standard forms of the pairs of matrices over this ring is finite. The classes of the pairs of matrices with minimum and maximum numbers of standard forms are presented. [ABSTRACT FROM AUTHOR]

It has been suggested that equivalence testing (otherwise known as negligible effect testing) should be used to evaluate model fit within structural equation modelling (SEM). In this study, we propose novel variations of equivalence tests based on the popular root mean squared error of approximation and comparative fit index fit indices. Using Monte Carlo simulations, we compare the performance of these novel tests to other existing equivalence testing‐based fit indices in SEM, as well as to other methods commonly used to evaluate model fit. Results indicate that equivalence tests in SEM have good Type I error control and display considerable power for detecting well‐fitting models in medium to large sample sizes. At small sample sizes, relative to traditional fit indices, equivalence tests limit the chance of supporting a poorly fitting model. We also present an illustrative example to demonstrate how equivalence tests may be incorporated in model fit reporting. Equivalence tests in SEM also have unique interpretational advantages compared to other methods of model fit evaluation. We recommend that equivalence tests be utilized in conjunction with descriptive fit indices to provide more evidence when evaluating model fit. [ABSTRACT FROM AUTHOR]

Although children learn more when teachers gesture, it is not clear how gesture supports learning. Here, we sought to investigate the nature of the memory processes that underlie the observed benefits of gesture on lasting learning. We hypothesized that instruction with gesture might create memory representations that are particularly resistant to interference. We investigated this possibility in a classroom study with 402 second‐ and third‐grade children. Participants received classroom‐level instruction in mathematical equivalence using videos with or without accompanying gesture. After instruction, children solved problems that were either visually similar to the problems that were taught, and consistent with an operational interpretation of the equal sign (interference), or visually distinct from equivalence problems and without an equal sign (control) in order to assess the role of gesture in resisting interference after learning. Gesture facilitated learning, but the effects of gesture and interference varied depending on type of problem being solved and the strategies that children used to solve problems prior to instruction. Some children benefitted from gesture, while others did not. These findings have implications for understanding the mechanisms underlying the beneficial effect of gesture on mathematical learning, revealing that gesture does not work via a general mechanism like enhancing attention or engagement that would apply to children with all forms of prior knowledge. [ABSTRACT FROM AUTHOR]

We prove a sharp stability result for the Brunn–Minkowski inequality for A,B⊂R2. Assuming that the Brunn–Minkowski deficit δ=∣A+B∣1/2/(∣A∣1/2/(∣A∣1/2/(∣A∣1/2+∣B∣1/2/(∣A∣1/2)−1 is sufficiently small in terms of t=∣A∣1/2/(∣A∣1/2/(∣A∣1/2/(∣A∣1/2+∣B∣1/2/(∣A∣1/2), there exist homothetic convex sets KA​⊃A and KB​⊃B such that ∣((KA​)/A)∣​+((∣BKB​))/B∣​≤Ct−1/2/(∣A∣1/2δ1/2/(∣A∣1/2. The key ingredient is to show for every, if δ is sufficiently small then ∣co(A+B)/(A+B)∣≤(1+ϵ)(∣co(A)/A∣+∣co(B)/B∣). [ABSTRACT FROM AUTHOR]

We consider the Brans–Dicke theory in non-metricity gravity, which belongs to the family of symmetric teleparallel scalar–tensor theories. Our focus lies in exploring the implications of the conformal transformation, as we derive the conformal equivalent theory in the Einstein frame, distinct from the minimally coupled scalar field theory. The fundamental principle of the conformal transformation suggests the mathematical equivalence of the related theories. However, to thoroughly analyze the impact on physical variables, we investigate the spatially flat Friedmann–Lemaître–Robertson–Walker geometry, defining the connection in the non-coincidence gauge. We construct exact solutions for the cosmological model in one frame and compare the physical properties in the conformal related frame. Surprisingly, we find that the general physical properties of the exact solutions remain invariant under the conformal transformation. Finally, we construct, for the first time, an analytic solution for the symmetric teleparallel scalar–tensor cosmology. [ABSTRACT FROM AUTHOR]

We establish the necessary and sufficient conditions for the equivalence of surfaces with respect to the action of a special pseudo-orthogonal group. [ABSTRACT FROM AUTHOR]

In this paper we introduce the notion of 2-primal semimodule related to a Morita context < R, S,R PS,S QR, θ, ϕ >. We obtain some characterizations of 2-primal semimodule. [ABSTRACT FROM AUTHOR]