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The structure of Borel reducibility for equivalence relations on \\( _1\\) is determined.

In this paper we study the theory Q. We prove a basic result that says that, in a sense explained in the paper, Q can be split into two parts. We prove some consequences of this result. (i) Q is not a poly-pair theory. This means that, in a strong sense, pairing cannot be defined in Q. (ii) Q does not have the Pudlák Property. This means that there two interpretations of S21 in Q which do not have a definably isomorphic cut. (iii) Q is not sententially equivalent with PA-. This tells us that we cannot do much better than mutual faithful interpretability as a measure of sameness of Q and PA-. We briefly consider the idea of characterizing Q as the minimal-in-some-sense theory of some kind modulo some equivalence relation. We show that at least one possible road towards this aim is closed.

In this research, we mainly conduct research on three types of curvatures—namely the isotropic mean Berwald curvature, relatively isotropic Landsberg curvature, and relatively isotropic mean Landsberg curvature—with respect to the Minkowskian product Finsler metric F . Firstly, we showed that the F possesses isotropic mean Berwald curvature equals that it can be qualified as a weakly Berwald metric. Secondly, we obtained that F possessing relatively isotropic Landsberg curvature is equivalent to that it is a Landsberg metric. Ultimately, this study presented the equivalent conditions that ensure the Minkowskian product Finsler metric constructed from two Finsler metrics, each possessing relatively isotropic mean Landsberg curvature, also exhibits the property that it qualifies as relatively isotropic mean Landsberg curvature.

Data equivalence refers to the extent to which the elements of a research design have the same meaning, and can be applied in the same way, in different cultural contexts. Failure to establish data equivalence in cross-cultural studies may bias empirical results and theoretical inferences. Although several authors have encouraged researchers to ensure high levels of data equivalence, no study has assessed the status of the field in relation to compliance with data equivalence standards. Accordingly, this study examines three aspects of data equivalence (construct equivalence, measurement equivalence, and data collection equivalence) within 167 studies that involve cross-cultural data published in the Journal of International Business Studies, Management International Review, Journal of World Business, Strategic Management Journal and the Academy of Management Journal from 1995 to 2005. The findings indicate that international business researchers report insufficient information in relation to data equivalence issues, thus limiting confidence in the findings of many cross-cultural studies. To enhance future research, a guideline for procedures for researchers to follow and report in establishing data equivalence is offered.

A primary research area within the field of international business (IB) is to establish the extent to which concepts, theories, and findings identified in one country are applicable to other contexts and which are unique and cannot be found in other contexts. Researchers in IB acknowledge the importance of the context in their studies, but the practice of assessing equivalence (or invariance) is not widely diffused within the community. We first discuss the components of equivalence (construct, method, and item equivalence), and we offer a three-step approach to address equivalence in the writing and revision of a paper. We aim to help editors, reviewers, and researchers produce more reliable research and navigate the tension between generalizable relationships and context-specific ones, both theoretically and empirically, before performing analysis and hypothesis testing. We then apply equivalence to the construct of firm economic performance as a case study, but the same logic can be applied to other constructs as well.

The purpose of this paper is to describe the formulation of quaternion algebra by means of exterior algebra and calculus, in a three dimensional time-like spacetime. A formal structure is provided, corroborating the equivalence with existing concepts and formulas known in literature. A first application is thus presented by depicting the description of rotations expressed with exterior-algebraic quaternionic notation. Secondly, a formal equivalence between exterior-algebraic quaternions and the classical theory of electromagnetism is recovered, too.

Equivalence-based instruction (EBI) involves strategic selection of a small number of stimulus relations to teach, leaving others to emerge without instruction. This translational study aimed to compare efficiency of acquisition and longer-term maintenance in EBI to a control condition that did not include the defining feature of the EBI arrangement. College students ( N  = 61) were randomly assigned to three training conditions—many-to-one EBI, linear series EBI, and comprehensive instruction—to establish three 4-member classes. On average, participants in all three groups required a similar number of trials to pass a comprehensive posttest, and maintenance after 2 weeks did not differ significantly between groups. The linear-series group, however, showed differential loss of transitivity and equivalence relations compared to baseline and symmetry. Limitations notwithstanding, the results underscore a need to systematically investigate conditions under which the defining features of EBI result in efficient learning and maintenance.

Stimulus equivalence paradigm has been used to investigate symbolic behavior over the last 50 years. Still, little attention has been directed in the study of equivalence classes maintenance. This bibliographic review, based on the PRISMA Protocol, provides an overview of the effects of different variables on the stability of equivalence classes over time. Searches were conducted on four databases (PsychINFO, Elsevier Scopus, Web of Science, and SciELO) and returned 95 results. Excluding the duplicated papers and applying the inclusion and exclusion criteria resulted in 14 papers. The references of each paper were examined for papers about maintenance of equivalence relations that were not found in the database search and 10 more articles were included. Therefore, 24 papers were selected in this bibliographic review. The papers were categorized considering the variable investigated, that is: class size, training and testing parameters, nature of stimuli, generalized relations, and equivalence based instruction. Future studies should focus on the study of variables that affect the maintenance of equivalence classes, considering the small number of papers in this field and the importance of this knowledge in both experimental and applied settings.

This paper explores the impulse equivalent method for constructing voltage source converter (VSC) model in large power grids containing multiple converters. It addresses the challenge of simulation cycle mismatch with device switching moments and proposes a cross-cycle correction method. The method enhances the VSC electromagnetic transient (EMT) simulation model, making it suitable for large-scale power grid simulations. The approach is validated through case studies, confirming its accuracy and computational efficiency for complex power electronic networks.