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Higher category theory is generally regarded as technical and forbidding, but part of it is considerably more tractable: the theory of infinity-categories, higher categories in which all higher morphisms are assumed to be invertible. InHigher Topos Theory, Jacob Lurie presents the foundations of this theory, using the language of weak Kan complexes introduced by Boardman and Vogt, and shows how existing theorems in algebraic topology can be reformulated and generalized in the theory's new language. The result is a powerful theory with applications in many areas of mathematics. The book's first five chapters give an exposition of the theory of infinity-categories that emphasizes their role as a generalization of ordinary categories. Many of the fundamental ideas from classical category theory are generalized to the infinity-categorical setting, such as limits and colimits, adjoint functors, ind-objects and pro-objects, locally accessible and presentable categories, Grothendieck fibrations, presheaves, and Yoneda's lemma. A sixth chapter presents an infinity-categorical version of the theory of Grothendieck topoi, introducing the notion of an infinity-topos, an infinity-category that resembles the infinity-category of topological spaces in the sense that it satisfies certain axioms that codify some of the basic principles of algebraic topology. A seventh and final chapter presents applications that illustrate connections between the theory of higher topoi and ideas from classical topology.

A fuzzified matrix space consists of a collection of matrices with a fuzzy structure, modeling the cases of uncertainty on the part of values of different matrices, including the uncertainty of the very existence of matrices with the given values. The fuzzified matrix space also serves as a test for the admissibility of certain approximate solutions to matrix equations, as well as a test for the approximate validity of certain laws. We introduce quotient structures derived from the original fuzzified matrix space and demonstrate the transferability of certain fuzzy properties from the fuzzified matrix space to its associated quotient structures. These properties encompass various aspects, including the solvability and unique solvability of equations of a specific type, the (unique) solvability of individual equations, as well as the validity of identities such as associativity. While the solvability and unique solvability of a single equation in a matrix space are equivalent to the solvability and unique solvability in a certain quotient structure, we proved that the (unique) solvability of a whole type of equations, as well as the validity of a certain algebraic law, are equivalent to the (unique) solvability and validity in all the quotient structures. Consequently, these quotient structures serve as an effective tool for evaluating whether specific properties hold within a given fuzzified matrix space.

The COVID-19 pandemic has imposed significant challenges on global health, emphasizing the persistent threat of large-scale infectious diseases in the future. This study addresses the need to enhance pooled testing efficiency for large populations. The common approach in pooled testing involves consolidating multiple test samples into a single tube to efficiently detect positivity at a lower cost. However, what is the optimal number of samples to be grouped together in order to minimize costs? i.e. allocating ten individuals per group may not be the most cost-effective strategy. In response, this paper introduces the hierarchical quotient space, an extension of fuzzy equivalence relations, as a method to optimize group allocations. In this study, we propose a cost-sensitive multi-granularity intelligent decision model to further minimize testing costs. This model considers both testing and collection costs, aiming to achieve the lowest total cost through optimal grouping at a single layer. Building upon this foundation, two multi-granularity models are proposed, exploring hierarchical group optimization. The experimental simulations were conducted using MATLAB R2022a on a desktop with Intel i5-10500 CPU and 8G RAM, considering scenarios with a fixed number of individuals and fixed positive probability. The main findings from our simulations demonstrate that the proposed models significantly enhance the efficiency and reduce the overall costs associated with pooled testing. For example, testing costs were reduced by nearly half when the optimal grouping strategy was applied, compared to the traditional method of grouping ten individuals. Additionally, the multi-granularity approach further optimized the hierarchical groupings, leading to substantial cost savings and improved testing efficiency.

We show that the uniform measure-theoretic ergodic decomposition of a countable Borel equivalence relation $(X, E)$ may be realized as the topological ergodic decomposition of a continuous action of a countable group $\\Gamma \\curvearrowright X$ generating E. We then apply this to the study of the cardinal algebra $\\mathcal {K}(E)$ of equidecomposition types of Borel sets with respect to a compressible countable Borel equivalence relation $(X, E)$ . We also make some general observations regarding quotient topologies on topological ergodic decompositions, with an application to weak equivalence of measure-preserving actions.

Teaching and learning fractions has traditionally been one of the most problematic areas in primary school mathematics. Several studies have suggested that one of the main factors contributing to this complexity is that fractions comprise a multifaceted notion encompassing five interrelated subconstructs (i.e., part-whole, ratio, operator, quotient, and measure). Kieren was the first to establish that the concept of fractions is not a single construct, but consists of several interrelated subconstructs. Later on, in the early 1980s, Behr et al. built on Kieren's conceptualization and suggested a theoretical model linking the five subconstructs of fractions to the operations of fractions, fraction equivalence, and problem solving. In the present study we used this theoretical model as a reference point to investigate students' constructions of the different subconstructs of fractions. In particular, using structural equation modeling techniques to analyze data of 646 fifth and sixth graders' performance on fractions, we examined the associations among the different subconstructs of fractions as well as the extent to which these subconstructs explain students' performance on fraction operations and fraction equivalence. To a great extent, the data provided support to the associations included in the model, although, they also suggested some additional associations between the notions of the model. We discuss these findings taking into consideration the context in which the study was conducted and we provide implications for the teaching of fractions and suggestions for further research.

Given a smooth partial action α of a Lie groupoid G on a smooth manifold M ,  we provide necessary and sufficient conditions for α to be globalizable with smooth globalization. As an application, we provide results on the differentiable structure of orbit and stabilizer spaces induced by α , which leads to other criteria for its globalization in terms of its orbit maps in the case that α is free and transitive. Further, under the assumption that α is free and proper, we prove that there exists exactly one differentiable structure on the quotient structure of the orbit equivalence space M / G such that the quotient map π : M → M / G is a submersion.

We study ultrafilters on ω² produced by forcing with the quotient of P(ω²) by the Fubini square of the Fréchet filter on ω. We show that such an ultrafilter is a weak P-point but not a P-point and that the only nonprincipal ultrafilters strictly below it in the Rudin-Keisler order are a single isomorphism class of selective ultrafilters. We further show that it enjoys the strongest square-bracket partition relations that are possible for a non-P-point. We show that it is not basically generated but that it shares with basically generated ultrafilters the property of not being at the top of the Tukey ordering. In fact, it is not Tukey-above [ω₁]<ω, and it has only continuum many ultrafilters Tukey-below it. A tool in our proofs is the analysis of similar (but not the same) properties for ultrafilters obtained as the sum, over a selective ultrafilter, of nonisomorphic selective ultrafilters.

The detection of redundant or irrelevant variables (attributes) in datasets becomes essential in different frameworks, such as in Formal Concept Analysis (FCA). However, removing such variables can have some impact on the concept lattice, which is closely related to the algebraic structure of the obtained quotient set and their classes. This paper studies the algebraic structure of the induced equivalence classes and characterizes those classes that are convex sublattices of the original concept lattice. Particular attention is given to the reductions removing FCA’s unnecessary attributes. The obtained results will be useful to other complementary reduction techniques, such as the recently introduced procedure based on local congruences.

Western musicians traditionally classify pitch sequences by disregarding the effects of five musical transformations: octave shift, permutation, transposition, inversion, and cardinality change. We model this process mathematically, showing that it produces 32 equivalence relations on chords, 243 equivalence relations on chord sequences, and 32 families of geometrical quotient spaces, in which both chords and chord sequences are represented. This model reveals connections between music-theoretical concepts, yields new analytical tools, unifies existing geometrical representations, and suggests a way to understand similarity between chord types.

We prove that a certain homological epimorphism between two algebras induces a triangle equivalence between their singularity categories. Applying the result to a construction of matrix algebras, we describe the singularity categories of some non-Gorenstein algebras.