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There is no single canonical polynomial-time version of the Axiom of Choice (AC); several statements of AC that are equivalent in Zermelo-Fraenkel (ZF) set theory are already inequivalent from a constructive point of view, and are similarly inequivalent from a complexity-theoretic point of view. In this paper we show that many classical formulations of AC, when restricted to polynomial time in natural ways, are equivalent to standard complexity-theoretic hypotheses, including several that were of interest to Selman. This provides a unified view of these hypotheses, and we hope provides additional motivation for studying some of the lesser-known hypotheses that appear here. Additionally, because several classical forms of AC are formulated in terms of cardinals, we develop a theory of polynomial-time cardinality. Nerode & Remmel (Contemp. Math. 106, 1990 and Springer Lec. Notes Math. 1432, 1990) developed a related theory, but restricted to unary sets. Downey (Math. Reviews MR1071525) suggested that such a theory over larger alphabets could have interesting connections to more standard complexity questions, and we illustrate some of those connections here. The connections between AC, cardinality, and complexity questions also allow us to highlight some of Selman's work. We hope this paper is more of a beginning than an end, introducing new concepts and raising many new questions, ripe for further research. [ABSTRACT FROM AUTHOR]

In this paper, I present what I call the symmetry conception of God within 1st order, extensional, non-well-founded set theory. The symmetry conception comes in two versions. According to the first, God is that unique being that is universally symmetrical with respect to set membership. According to the second, God is the universally symmetrical set of all sets that are universally symmetrical with respect to set membership. I present a number of theorems, most importantly that any universally symmetrical set is identical to its essence, that show that the two symmetry conceptions intersect with some dominant theological conceptions of God. The theorems also show that both of the symmetry conceptions of God entail that God has a fractal like structure. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new concept of sets and a new class of functions called E-univex sets and E-univex functions, respectively. For an E-differentiable function, the concept of E-univexity is introduced by generalizing several concepts of generalized convexity earlier defined into optimization theory. In addition, some properties of E-differentiable E-univex functions are investigated. Further, also concepts of E-differentiable generalized E-univexity are introduced. Then, the sufficiency of the so-called E-Karush–Kuhn–Tucker necessary optimality conditions are proved for an E-differentiable nonlinear optimization problem in which the involved functions are E-univex and/or generalized E-univex. [ABSTRACT FROM AUTHOR]

Research question: This paper examines the overarching question of to what extent different analytic choices may influence the inference about country-specific cross-sectional and trend estimates in international large-scale assessments. We take data from the assessment of PISA mathematics proficiency from the four rounds from 2003 to 2012 as a case study. Methods: In particular, four key methodological factors are considered as analytical choices in the rescaling and analysis of the data: (1) The selection of country sub-samples for item calibration differing at three factor levels. (2) The item sample refering to two sets of mathematics items used within PISA. (3) The estimation method used for item calibration: marginal maximum likelihood estimation method as implemented in R package TAM or an pairwise row averaging approach as implemented in the R package pairwise. (4) The type of linking method: concurrent calibration or separate calibration with successive chain linking. Findings: It turned out that analytical decisions for scaling did affect the PISA outcomes. The factors of choosing different calibration samples, estimation method and linking method tend to show only small effects on the country-specific cross-sectional and trend estimates. However, the selection of different link items seems to have a decisive influence on country ranking and development trends between and within countries. [ABSTRACT FROM AUTHOR]

Students' negative affective performance in mathematics learning is a global problem, and the situation is especially challenging in high-achieving East Asian countries. Taiwan's Just Do Math (JDM) program was launched to resolve this problem and to serve as an example of successful scaling up with regard to implementation of innovation. We developed a framework consisting of a synthesis of the literature on essentials for successful implementation and for scaling up, and applied this framework for the purpose of identification of the factors contributing to the success of the JDM program. The identified factors include the following: the development of low-complexity sources for teachers' use and the low degree of adaptation required to sustain fidelity; the consideration of characteristics of adopters to promote innovation outside schools; construction of communication channels to ensure the dissemination of the core ideas of JDM; the feedback mechanism through which participating teachers can acknowledge the relative advantages of innovation as realized in the program; the consideration of the influence of historical, social, political, and economic factors on program implementation; and the efforts to ensure teachers' in-depth learning, acknowledgment of the promoted principles, and the sustainability of implementation through implementation of JDM activities in regular mathematics classes. [ABSTRACT FROM AUTHOR]

In this study, we examine "what went wrong" in our professional development program for encouraging cognitively demanding instruction, focusing on the difficulties we encountered in using an observational tool for evaluating this type of instruction and reaching inter-rater reliability. We do so through the lens of a discursive theory of teaching and learning. Data consisted of 10 coders' coding sheets while learning to apply the Coding Rubric for Video Observations tool on a set of recorded mathematics lessons. We show that some discrepancies between novice coders and experts were found in narratives about valued actions relating to social aspects of teaching, such as teacher explanations and students' struggle and discussion. These were relatively easy to detect from the written evaluations in the coding sheets. More problematic to pinpoint were places where discrepancies were found between novice and experts' stories about the mathematical objects discussed. These required re-observing the recorded lessons. The analysis revealed that novice coders referred to objects about which the observed teacher mainly explained calculations, whereas the experts searched for (often, signaling the absence of) mathematical objects that invited exploration. We discuss these findings based on a theory of different Pedagogical Discourses that form the background for expert and novice coders' interpretations of the coding manual. We conclude with practical implications for the process of achieving inter-rater reliability on observation tools for cognitively demanding instruction. [ABSTRACT FROM AUTHOR]

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Equivalence relations and orderings are key concepts of mathematics. For those two relations, the formulation in fuzzy relations has been presented in the early days of fuzzy set theory. Utilizing the existing equivalence relations, we are unable to prove that fuzzy Zorn's lemma and fuzzy axiom of choice are equivalent. The main objective of this research is to redefine the fuzzy-order relation with some fundamental properties. Also, introduce fuzzy equivalence relation based on our developed fuzzy-order relation and proved that any equivalence relation is equivalent to fuzzy equivalence relation. Furthermore, introduce the fuzzy versions of the axiom of choice, Zorn's lemma, and well-ordering principle, such as the fuzzy axiom of choice, fuzzy Zorn's lemma, and fuzzy well-ordering principle and discuss the relations between them. We proposed a fuzzy version of the Hausdorff maximal principle. Moreover, we give the application of fuzzy Zorn's lemma in fuzzy ideals and fuzzy hausdroff maximal principle in fuzzy filters. [ABSTRACT FROM AUTHOR]