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Motivated by the fact that the Hopf-cyclic (co)homologies of function algebras over Lie groups and universal enveloping algebras over Lie algebras capture the Lie group and Lie algebra (co)homologies, we hereby upgrade the classical van Est isomorphism to ones between the Hopf-cyclic (co)homologies of quantized algebras of functions and quantized universal enveloping algebras, both in h-adic and q-deformation frameworks. [ABSTRACT FROM AUTHOR]

Summary: In this study, we demonstrate that engineered topological features can distinguish time series sampled from different stochastic processes with different noise characteristics, in both balanced and unbalanced sampling schemes. We compare our classification results against the results of the same classification on features coming from descriptive statistics and the wavelet transform. We conclude that machine learning models built on engineered topological features alone perform consistently better than those built on the standard statistical and wavelet features for time series classification tasks. We also apply dimension reduction techniques to our engineered features and compare the result of our classification models before and after dimensionality reduction. Finally, we also show that in our calculations of the engineered topological features, employing parallel computing methods does yield significant improvements in run time and memory footprint. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to determine the variables of high importance affecting the mathematics achievement of the students of 12 Asia-Pacific countries participating in the Program for International Student Assessment (PISA) 2018. For this purpose, we used random forest (RF), logistic regression (LR) and support vector machine (SVM) models to classify student achievement in mathematics. The variables affecting the student achievement in mathematics were examined by the feature importance method. We observed that the variables with the highest importance for all of the 12 Asia-Pacific countries we considered are the educational status of the parents, having access to educational resources, age, the time allocated to weekly lessons, and the age of starting kindergarten. Then we applied two different clustering analysis by using the variable importance values and socio-economic variables of these countries. We observed that Korea, Japan and Taipei form one group of Asia-Pacific countries, while Thailand, China, Indonesia, and Malaysia form another meaningful group in both clustering analyses. The results we obtained strongly suggest that there is a quantifiable relationship between the educational attainment and socio-economic levels of these 12 Asia-Pacific countries. [ABSTRACT FROM AUTHOR]

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Fetva collections are important sources for Islamic legal history. However, few scholars have considered a particular collection of fetvas or the fetvas of an individual jurist as specific areas of legal and historical exploration. Instead, most researchers use fetvas selectively and instrumentally, that is in (at best) small groups, and in their explorations of various other topics. This article proposes computational methodologies that could characterize the contents of a 6,000-fetva corpus by an important Ottoman jurist, Şeyhülislam Ebussuud Efendi (d. 1574), to reveal its substantive composition and range. The article conceptualizes a previously uncharted textual space in a way similar to how a map depicts a geographical one. By doing so, it also provides insights into Ebussuud's jurisprudential legacy and the major socio-legal concerns and anxieties in the Ottoman Empire during the sixteenth century. [ABSTRACT FROM AUTHOR]

We show that faithfully flat smooth extensions of associative unital algebras are reduced flat, and therefore, fit into the Jacobi-Zariski exact sequence in Hochschild homology and cyclic (co)homology even when the algebras are noncommutative or infinite dimensional. We observe that such extensions correspond to étale maps of affine schemes, and we propose a definition for generic noncommutative fibrations using distributive laws and homological properties of the induction and restriction functors. Then we show that Galois fibrations do produce the right exact sequence in homology. We then demonstrate the versatility of our model on a geometro-combinatorial example. For a connected unramified covering of a connected graph , we construct a smooth Galois fibration and calculate the homology of the corresponding local coefficient system. [ABSTRACT FROM AUTHOR]

We compute the continuous Hochschild cohomology of four reduced incidence algebras: the algebra of formal power series, the algebra of exponential power series, the algebra of Eulerian power series, and the algebra of formal Dirichlet series. We achieve the result by carrying out the computation for the coalgebra Cotor-groups of their pre-dual coalgebras. [ABSTRACT FROM AUTHOR]

We show that if G is a compact Lie group and $$\mathfrak {g}$$ is its Lie algebra, then there is a map from the Hopf-cyclic cohomology of the quantum enveloping algebra $$U_q(\mathfrak {g})$$ to the twisted cyclic cohomology of quantum group algebra $${\mathcal O}(G_q)$$ . We also show that the Schmüdgen-Wagner index cocycle associated with the volume form of the differential calculus on the standard Podleś sphere $${\mathcal O}(S^2_q)$$ is in the image of this map. [ABSTRACT FROM AUTHOR]

In our paper “Jacobi–Zariski exact sequence for Hochschild homology and cyclic (co)homology” [Homology Homotopy Appl., 14(1):65–78, 2012], in Theorem 2.3 and in all remaining results based on this theorem, it is stated that the Jacobi–Zariski long exact sequence works for the range p⩾0. The range should be corrected to p⩾1. [ABSTRACT FROM AUTHOR]

Copyright of Continuity & Change is the property of Cambridge University Press and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We prove that for an inclusion of unital associative but not necessarily commutative k-algebras B ⊆ A we have long exact sequences in Hochschild homology and cyclic (co)homology akin to the Jacobi-Zariski sequence in André-Quillen homology, provided that the quotient B-module A=B is flat. We also prove that for an arbitrary r-flat morphism φ : B → A with an H-unital kernel, one can express the Wodzicki excision sequence and our Jacobi-Zariski sequence in Hochschild homology and cyclic (co)homology as a single long exact sequence. [ABSTRACT FROM AUTHOR]

For module algebras and module coalgebras over an arbitrary bialgebra, we define two types of bivariant cyclic cohomology groups called bivariant Hopf cyclic cohomology and bivariant equivariant cyclic cohomology. These groups are defined through an extension of Connes' cyclic category Λ. We show that, in the case of module coalgebras, bivariant Hopf cyclic cohomology specializes to Hopf cyclic cohomology of Connes and Moscovici and its dual version by fixing either one of the variables as the ground field. We also prove an appropriate version of Morita invariance for both of these theories. [ABSTRACT FROM AUTHOR]

We construct several pairings in Hopf-cyclic cohomology of (co)module (co)algebras with arbitrary coefficients. As a special case of one of these pairings, we recover the Connes-Moscovici characteristic map in Hopf-cyclic cohomology. We also prove that this particular pairing, along with a few others, would stay the same if we replace the derived category of (co)cyclic modules with the homotopy category of (special) towers of super complexes, or the derived category of mixed complexes. [ABSTRACT FROM AUTHOR]

We define a version of Hochschild homology and cohomology suitable for a class of algebras admitting compatible actions of bialgebras, called module algebras. We show that this (co)homology, called Hopf-Hochschild (co)homology, can also be defined as a derived functor on the category of representations of an equivariant analogue of the enveloping algebra of a crossed product algebra. We investigate the relationship of our theory with Hopf cyclic cohomology and also prove Morita invariance of the Hopf-Hochschild (co)homology. [ABSTRACT FROM AUTHOR]