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In this work, using some tools of Lie theory, we compute the Lie algebra of the Shearlet group regarding as a 3-fold semidirect product Lie group. As we will see, it is a 3-fold semidirect sum of Lie algebras. [ABSTRACT FROM AUTHOR]

Let G be a finite group. We denote by ψ ′ (G) the integer ∏ g ∈ G o (g) , where o (g) denotes the order of g ∈ G. It is verified that some finite groups are uniquely determined by their orders and their product of element orders. In this paper, we prove that, if p , q , r , t are different prime numbers, then ℤ p 3 , ℤ p q , ℤ p 2 q and ℤ p q r t are characterized by ψ ′ (G) and if ψ ′ (G) = ψ ′ (ℤ p q r) , then G ≅ A 5 or ℤ p q r . [ABSTRACT FROM AUTHOR]

Let |$\alpha $| and |$\beta $| be uniformly random permutations of orders |$2$| and |$3$|⁠ , respectively, in |$S_{N}$|⁠ , and consider, say, the permutation |$\alpha \beta \alpha \beta ^{-1}$|⁠. How many fixed points does this random permutation have on average? The current paper studies questions of this kind and relates them to surprising topological and algebraic invariants of elements in free products of groups. Formally, let |$\Gamma =G_{1}*\ldots *G_{k}$| be a free product of groups where each of |$G_{1},\ldots ,G_{k}$| is either finite, finitely generated free, or an orientable hyperbolic surface group. For a fixed element |$\gamma \in \Gamma $|⁠ , a |$\gamma $| -random permutation in the symmetric group |$S_{N}$| is the image of |$\gamma $| through a uniformly random homomorphism |$\Gamma \to S_{N}$|⁠. In this paper we study local statistics of |$\gamma $| -random permutations and their asymptotics as |$N$| grows. We first consider |$\mathbb{E}\big [\textrm{fix}_{\gamma }\big (N\big)\big ]$|⁠ , the expected number of fixed points in a |$\gamma $| -random permutation in |$S_{N}$|⁠. We show that unless |$\gamma $| has finite order, the limit of |$\mathbb{E}\big [\textrm{fix}_{\gamma }\big (N\big)\big ]$| as |$N\to \infty $| is an integer, and is equal to the number of subgroups |$H\le \Gamma $| containing |$\gamma $| such that |$H\cong \mathbb{Z}$| or |$H\cong C_{2}*C_{2}$|⁠. Equivalently, this is the number of subgroups |$H\le \Gamma $| containing |$\gamma $| and having (rational) Euler characteristic zero. We also prove there is an asymptotic expansion for |$\mathbb{E}\big [\textrm{fix}_{\gamma }\big (N\big)\big ]$| and determine the limit distribution of the number of fixed points as |$N\to \infty $|⁠. These results are then generalized to all statistics of cycles of fixed lengths. [ABSTRACT FROM AUTHOR]

Breakable semihypergroups, defined by a simple property: every non-empty subset of them is a subsemihypergroup. In this paper, we introduce a class of semihypergroups, in which every hyperproduct of n elements is equal to a subset of the factors, called πn-semihypergroups. Then, we prove that every semihypergroup of type π2k, (k ≥ 2) is breakable and every semihypergroup of type π2k+1 is of type π3. Furthermore, we obtain a decomposition of a semihypergroup of type πn into the cyclic group of order 2 and a breakable semihypergroup. Finally, we give a characterization of semi-symmetric semihypergroups of type πn. [ABSTRACT FROM AUTHOR]

The key value proposition of supply chain segmentation is to differentiate supply chains through a reasonable number of segments in order to gain a level of standardisation and avoid managerial complexity incurred in fully customised supply chains. The decision on how products are grouped into segments is at the core of a successful implementation. A fundamental trade-off in this decision-making process is between higher differentiation by having small group sizes and higher standardisation from a smaller number of groups. In this manuscript, we implement segmentation on supply chain configurations and investigate the trade-off by analysing several network scenarios. We use optimisation models for each scenario to align decisions of segment formation and supply chain configurations. We show that divergences in demand characteristics, geographic difference, and cost synergy such as pooling effect have impacts on the balance of standardisation and differentiation. [ABSTRACT FROM AUTHOR]

Planners and designers of production systems must frequently evaluate the similarities in the manufacturing process of various products to achieve efficient and economical production. The purpose is to reasonably arrange mixed production. The similarity of manufacturing technique processes is an important basis for classifying product manufacturing processes. This study proposes an innovative method based on process constituent elements model to objectively calculate the similarity of product manufacturing processes. Firstly, the similarity model of the manufacturing technique process is established on the basis of six dimensions of process constituent elements, namely, input, output, resource, environment, value-added processing activity and quality control & inspection. Secondly, the characteristics of the six dimensions of the process constituent elements are determined. Corresponding methods are applied to solve the similarity of process constituent elements in each dimension on the basis of the characteristic attributes of process constituent elements of different dimensions. Thirdly, extended interval analytic hierarchy process and quantitative calculation method based on angle information entropy are integrated to determine the weights of dimensions of the six elements. Lastly, the proposed method is applied to the process classification of a group of product manufacturing technique processes, and the feasibility and effectiveness of the method are proven. [ABSTRACT FROM AUTHOR]

This paper considers a two-stage assembly flowshop scheduling problem with distinct due windows to minimise the sum of weighted earliness and tardiness. There are several identical parallel machines which produce parts in the first stage. When the required parts are available, a single assembly machine can group these parts into products in the second stage. It is assumed that a part can be split into sub-parts which can be processed independently on the parallel machines in the first stage. Setup is also considered. A mathematical model is established to describe and define the proposed problem. A new decoding method is developed by extending an existing decoding method. Two novel operators, named part splitting (PS) and optimal idle time insertion (ITI), are incorporated into the decoding procedure for improving the quality of the solution. A rule named Priority of Earliness and Tardiness (PET) and a Complete Immunoglobulin-based Artificial Immune System (C-IAIS) algorithm are proposed for solving the problem. To evaluate PET and C-IAIS algorithm, several existing algorithms are used in the experiments. Computational results show that C-IAIS algorithm performs better than other algorithms for solving the proposed problem. [ABSTRACT FROM AUTHOR]

The first author of this article together with Gould studied the closure properties of graph products of monoids with respect to abundancy and Fountanicity (which may be viewed as weaker forms of regularity). In that article the notion of graph products of semigroups is introduced. The construction of graph products of semigroups is essentially different from that for monoids and groups, since semigroups are algebras with a reduced signature. Inspired by the work of Barkauskas on centralizers of elements in graph products of groups, we calculate centralizers of elements in graph products of semigroups. We introduce notions of decomposable and indecomposable words, which turn out to be key to the whole discussion. We show that every element of a graph product of semigroups can be written as a product of elements which are represented by indecomposable words. [ABSTRACT FROM AUTHOR]

Graph Laplacian based algorithms for data lying on a manifold have been proven effective for tasks such as dimensionality reduction, clustering, and denoising. In this work, we consider data sets whose data points lie on a manifold that is closed under the action of a known unitary matrix Lie group G. We propose to construct the graph Laplacian by incorporating the distances between all the pairs of points generated by the action of G on the data set. We deem the latter construction the " G -invariant Graph Laplacian" (G -GL). We show that the G -GL converges to the Laplace-Beltrami operator on the data manifold, while enjoying a significantly improved convergence rate compared to the standard graph Laplacian which only utilizes the distances between the points in the given data set. Furthermore, we show that the G -GL admits a set of eigenfunctions that have the form of certain products between the group elements and eigenvectors of certain matrices, which can be estimated from the data efficiently using FFT-type algorithms. We demonstrate our construction and its advantages on the problem of filtering data on a noisy manifold closed under the action of the special unitary group S U (2). [ABSTRACT FROM AUTHOR]

Let G be a simple algebraic group over an algebraically closed field k of characteristic p. The classification of the conjugacy classes of unipotent elements of G(k) and nilpotent orbits of G on Lie (G) is well-established. One knows there are representatives of every unipotent class as a product of root group elements and every nilpotent orbit as a sum of root elements. We give explicit representatives in terms of a Chevalley basis for the eminent classes. A unipotent (resp. nilpotent) element is said to be eminent if it is not contained in any subsystem subgroup (resp. subalgebra), or a natural generalization if G is of type Dn. From these representatives, it is straightforward to generate representatives for any given class. Along the way we also prove recognition theorems for identifying both the unipotent classes and nilpotent orbits of exceptional algebraic groups. [ABSTRACT FROM AUTHOR]

Coal spontaneous combustion (CSC) characteristics were identified by their different types of coal. In this study, the samples of lignite, non-caking coal, lean coal, and anthracite were chosen to investigate the variations of gaseous products and main active groups during the low-temperature oxidation. The law of indicator gases was obtained by temperature-programmed experiment, and active groups of four coal samples had specific and different probabilities of spontaneous combustion via Fourier transform infrared spectrometer (FTIR). The relationship between the indicator gases at different stages and the main functional groups of coal were explored by Grey correlation analysis. The results indicated that the characteristic temperatures of water evaporation and gases desorption period of four types of coal showed the similar variation with those in thermal storage stage (Stage 1). Moreover, temperature ranges (TRs) in Stage 1 of lignite, non-caking coal, lean coal, and anthracite were 20–60, 20–60, 20–60, and 20–80°C, respectively; TRs in Stage 2 were 60–100, 60–140, 60–120, and 80–200°C, respectively; TRs in Stage 3 were 100–180, 140–180, 120–180, and 200–240°C, respectively. Based on the correlation analysis, the hydroxyl, carboxyl, and methylene had a sound temperature correlation with indicator gases emission. Eventually, the prediction system for CSC during the low-temperature oxidation was revealed. [ABSTRACT FROM AUTHOR]

We study finiteness conditions and closure properties for the box-tensor product G ⊠ H and related constructions. The content of this paper extends some results concerning the non-abelian tensor product of groups G ⊗ H. In particular, we deduce a quantitative version of the finiteness criterion for the non-abelian tensor product. Moreover, we obtain finiteness conditions for some functors that arise out of the non-abelian tensor square of groups, such as the second homology group H 2 (G) , the non-abelian exterior square G ∧ G and the second stable homotopy group of an Eilenberg-MacLane space π 2 S (K (G , 1)). [ABSTRACT FROM AUTHOR]

For a given Jacobi-Jordan algebra A and a vector space V over a field k , a non-abelian cohomological type object H A 2 (V , A) is constructed: it classifies all Jacobi-Jordan algebras containing A as a subalgebra of codimension equal to dim k (V). Any such algebra is isomorphic to a so-called unified product A ♮ V. Furthermore, we introduce the bicrossed (semi-direct, crossed, or skew crossed) product A ⋈ V associated to two Jacobi-Jordan algebras as a special case of the unified product. Several examples and applications are provided: the Galois group of the extension A ⊆ A ⋈ V is described as a subgroup of the semidirect product of groups GL k (V) ⋊ Hom k (V , A) and an Artin type theorem for Jacobi-Jordan algebra is proven. [ABSTRACT FROM AUTHOR]

• BPA exposure were 0.0297 and 0.0507 μg/kg bw/d due to urine creatinine and excretion. • Dietary BPA intake values were 0.1009 and 0.9165 μg/kg bw/day due to 24-HDR and FFQ. • Exposure to BPA was within the EFSA's recommended daily intake (4 μg/kg bw/d). • The estimated BPA hazard ratio was within the international references' values. Bisphenol A (BPA) is an artificial chemical, and one of the significant external routes of daily BPA exposure is diet. Dietary BPA exposure can be calculated by urinary BPA concentration and dietary recall data. This cross-sectional study investigates exclusively breastfeeding women's BPA exposure by urinary total BPA concentration and nutritional records, including the 24 h Dietary Recall (HDR) and Food Frequency Questionnaire (FFQ). In this study, we included exclusively breastfeeding, healthy women volunteers (n = 80; 18–40 years), collected spot-morning urine samples and conducted a comprehensive face-to-face survey. Moreover, the women's urine BPA concentration was adjusted according to their urine creatinine concentrations. We assessed dietary BPA intake with the 24HDR and FFQ. Estimated daily BPA exposure according to urinary output volume and urinary creatinine concentration median values were 0.0507 and 0.06 μg/kg bw/day, respectively. Moreover, dietary BPA daily intake was found to be 0.17 and 0.95 μg/kg bw/day according to 24HDR data and FFQ data. The milk and dairy product group's and soft drinks group's contributions to the daily intake of BPA were 55.9 % and 25.92 %, respectively. The hazard ratio for BPA exposure was within limits according to references, including US EPA, Health Canada, and EFSA. This study indicates that BPA exposure, based on both total urinary BPA concentration and dietary recall data, was within the recommended daily intake level (4 μg/kg bw/day). However, further studies are required to understand the influence of seasonal, multicentre, and socioeconomic differences on BPA exposure. [ABSTRACT FROM AUTHOR]

The paper aims to present the main tendencies on the global and European natural polymer markets in recent years from the point of view of the requirements and achievements of the European Commission that have been expressed in recent programs and Strategies, with special reference to the Circular Economy New Action Plan and the European Green Deal—the EU Strategies for Climate Neutrality. Natural polymers play an important role among biodegradable products whose role in the production and international trade has been systematically increasing, especially since the middle of the last decade of the new century (2015). Natural polymers are also recommended by the EU from the point of view of their specific importance in the group of biodegradable products. [ABSTRACT FROM AUTHOR]

Given a group G and an endomorphism φ of G, two elements x , y ∈ G are said to be φ -conjugate if x = g y φ (g) − 1 for some g ∈ G. The number of equivalence classes for this relation is the Reidemeister number R (φ) of φ. The set { R (ψ) | ψ ∈ Aut (G) } is called the Reidemeister spectrum of G. We investigate Reidemeister numbers and spectra on direct products of finitely many groups and determine what information can be derived from the individual factors. [ABSTRACT FROM AUTHOR]

We provide a new large class of countable icc groups A for which the product rigidity result from Chifan et al. (Geom Funct Anal 26(1): 136–159, 2016) holds: if Γ 1 , ... , Γ n ∈ A and Λ is any group such that L (Γ 1 × ⋯ × Γ n) ≅ L (Λ) , then there exists a product decomposition Λ = Λ 1 × ⋯ × Λ n such that L (Λ i) is stably isomorphic to L (Γ i) , for any 1 ≤ i ≤ n . Class A consists of groups Γ for which L (Γ) admits an s-malleable deformation in the sense of Sorin Popa and it includes all non-amenable groups Γ such that either (a) Γ admits an unbounded 1-cocycle into its left regular representation, or (b) Γ is an arbitrary wreath product group with amenable base. As a byproduct of these results, we obtain new examples of W ∗ -superrigid groups and new rigidity results in the C ∗ -algebra theory. [ABSTRACT FROM AUTHOR]

In this article we discuss the factorisations of semigroups and monoids in the context of direct, semidirect and Zappa–Szép products addressing the question of uniqueness. An equivalence between external and internal Zappa–Szép product of groups and monoids is known, but no such correspondence exists for semigroups in general. We prove the equivalence between external and internal Zappa–Szép product of semigroups subject to certain conditions in this article. We end with some illustrative examples of the Zappa–Szép product of the bisimple inverse monoids. [ABSTRACT FROM AUTHOR]

In climate footprints of countries, but also of companies, the use of goods that come from outside makes a considerable contribution. For companies, capturing these greenhouse gas emissions is difficult because they occur in the supply chain, which can usually only be traced to the first supplier. Input‐output analysis methods can be used to estimate emissions in supply chains. They provide indications of the origin of emissions by country and industry sector or product group. With recursive methods in the supply chain, these calculations can be continuously improved. [ABSTRACT FROM AUTHOR]

A highly stereoselective synthesis of (E)-δ-boryl-anti-homoallylic alcohols is developed. In the presence of a Lewis acid, aldehyde allylation with α-boryl-(E)-crotylboronate gave δ-boryl-anti-homoallylic alcohols in good yields with excellent E-selectivity. The E-vinylboronate group in the products provides a useful handle for cross-coupling reactions as illustrated in the fragment synthesis of chaxamycins. [ABSTRACT FROM AUTHOR]

Asymmetric allylic alkylation (AAA) is a powerful method for the formation of highly useful, non‐racemic allylic compounds. Here we present a complementary enantioselective process that generates allylic lactones via π‐acid catalysis. More specifically, a catalytic enantioselective dehydrative lactonization of allylic alcohols using a novel PdII‐catalyst containing the imidazole‐based P,N‐ligand (S)‐StackPhos is reported. The high‐yielding reactions are operationally simple to perform with enantioselectivities up to 99 % ee. This strategy facilitates the replacement of a poor leaving group with what would ostensibly be a better leaving group in the product avoiding complications arising from racemization by equilibration. [ABSTRACT FROM AUTHOR]

Purpose: Because of short life cycle and fluctuating greatly in total sales volumes (TSV), it is difficult to accumulate enough sales data and mine an attribute set reflecting the common needs of all consumers for a kind of experiential product with short life cycle (EPSLC). Methods for predicting TSV of long-life-cycle products may not be suitable for EPSLC. Furthermore, point prediction cannot obtain satisfactory prediction results because information available before production is inadequate. Thus, this paper aims at proposing and verifying a novel interval prediction method (IPM). Design/methodology/approach: Because interval prediction may satisfy requirements of preproduction investment decision-making, interval prediction was adopted, and then the prediction difficult was converted into a classification problem. The classification was designed by comparing similarities in attribute relationship patterns between a new EPSLC and existing product groups. The product introduction may be written or obtained before production and thus was designed as primary source information. IPM was verified by using data of crime movies released in China from 2013 to 2017. Findings: The IPM is valid, which uses product introduction as input, classifies existing products into three groups with different TSV intervals, mines attribute relationship patterns using content and association analyses and compares similarities in attribute relationship patterns – to predict TSV interval of a new EPSLC before production. Originality/value: Different from other studies, the IPM uses product introduction to mine attribute relationship patterns and compares similarities in attribute relationship patterns to predict the interval values. It has a strong applicability in data content and structure and may realize rolling prediction. [ABSTRACT FROM AUTHOR]

Access to highly disaggregated trade data allows for a more nuanced investigation of different margins of trade, and the factors known to influence them. In this paper, the number of importers and shipments to each importer is investigated together with the more traditional margins. Potential explanatory factors of these trade margins are combined from three literature strands in addition to the standard gravity variables; firm productivity, per-unit shipment costs and country-specific trade costs. The empirical results show, not unexpectedly, that insights from all these different strands of literature influence trade margins significantly. In particular, the number of shipments per importer increases with distance, degree of remoteness and per-shipment cost, and the number of importers decreases with the distance, remoteness and per-unit shipping cost. This indicates that increased trade costs make exporters economize in existing networks. Finally, disaggregating the data into three main product categories using Rauch's classification, trade patterns are shown to vary by product group. [ABSTRACT FROM AUTHOR]

Nature forms S-S bonds by oxidizing two sulfhydryl groups, and no enzyme installing an intact hydropersulfide (-SSH) group into a natural product has been identified to date. The leinamycin (LNM) family of natural products features intact S-S bonds, and previously we reported an SH domain (LnmJ-SH) within the LNM hybrid nonribosomal peptide synthetase (NRPS)-polyketide synthase (PKS) assembly line as a cysteine lyase that plays a role in sulfur incorporation. Here we report the characterization of an S-adenosyl methionine (SAM)-dependent hydropersulfide methyltransferase (GnmP) for guangnanmycin (GNM) biosynthesis, discovery of hydropersulfides as the nascent products of the GNM and LNM hybrid NRPS-PKS assembly lines, and revelation of three SH domains (GnmT-SH, LnmJ-SH, and WsmR-SH) within the GNM, LNM, and weishanmycin (WSM) hybrid NRPS-PKS assembly lines as thiocysteine lyases. Based on these findings, we propose a biosynthetic model for the LNM family of natural products, featuring thiocysteine lyases as PKS domains that directly install a -SSH group into the GNM, LNM, or WSM polyketide scaffold. Genome mining reveals that SH domains are widespread in Nature, extending beyond the LNM family of natural products. The SH domains could also be leveraged as biocatalysts to install an -SSH group into other biologically relevant scaffolds. Enzymes installing an intact hydropersulfide (-SSH) group into natural products have so far not been identified. Here, the authors report the characterization of an S-adenosyl methionine-dependent hydropersulfide methyltransferase (GnmP) for guangnanmycin biosynthesis, and identification of three SH domains within several NRPS-PKS assembly lines as thiocysteine lyases. [ABSTRACT FROM AUTHOR]

We give a positive answer to the problem of power conjugacy words in a tree product of free groups with cyclic amalgamation. It is a generalization of the well-known result obtained by S. Lipschutz for a free product of two free groups with cyclic amalgamation. [ABSTRACT FROM AUTHOR]

Using a hyperbolic ring based formalism, it is shown that special unitary groups correspond to thermo-algebras for closed quantum systems, while for open systems, the full unitary algebra is the suitable underlying structure. Specifically the groups U (2) and U (1, 1) are realized as the dynamical symmetry groups for open quantum systems through continuous tensorial products; these unitary groups are obtained by mean of abelian extensions through U (1)-factors, from the standard thermo-algebras SU (2), and SU (1, 1). [ABSTRACT FROM AUTHOR]

We prove that any regular Casimir in 3D magnetohydrodynamics (MHD) is a function of the magnetic helicity and cross-helicity. In other words, these two helicities are the only independent regular integral invariants of the coadjoint action of the MHD group |$\textrm{SDiff}(M)\ltimes \mathfrak X^*(M)$|⁠ , which is the semidirect product of the group of volume-preserving diffeomorphisms and the dual space of its Lie algebra. [ABSTRACT FROM AUTHOR]

DNA‐encoded compound libraries are a widely used small molecule screening technology. One important aim in library design is the coverage of chemical space through structurally diverse molecules. Yet, the chemical reactivity of native DNA barcodes limits the toolbox of reactions for library design. Substituting the chemically vulnerable purines by 7‐deazaadenine, which exhibits tautomerization stability similar to natural adenine with respect to the formation of stable Watson–Crick pairs, yielded ligation‐competent, amplifiable, and readable DNA barcodes for encoded chemistry with enhanced stability against protic acid‐ and metal ion‐promoted depurination. The barcode stability allowed for straightforward translation of 16 exemplary reactions that included isocyanide multicomponent reactions, acid‐promoted Pictet–Spengler and Biginelli reactions, and metal‐promoted pyrazole syntheses on controlled pore glass‐coupled barcodes for diverse DEL design. The Boc protective group of reaction products offered a convenient handle for encoded compound purification. [ABSTRACT FROM AUTHOR]

In this study, it is aimed to determine the organic product consumption habits of the students who go to the exam preparation centers (private teaching center). For this reason, face to face surveys were applied to 227 people and a 24-question questionnaire was applied to them. Surveys were conducted in 2019. In the study, the factors affecting the consumption preference of organic products were the not harmful to health and being natural price, being safe, nutritional value. In the study, the factors affecting the price of organic products are the low cost of production (41.7%), the low amount of production (33.3%) and the difficult production (25.0%); the factors affecting the price of organic products were found to be difficult in production (25.0%), low production cost (41.7%) and low production (33.3%) in the study. In the study, it was determined that the logo (58.3%), label (22.2%), packaging (19.5%) were looked at when buying organic products; the expiration date of the products, information on the packaging, promotion and discounts and advertisements played an active role when consuming organic products. In the study, organic fresh vegetables and fruits (38.9%), organic tomato paste and olive oil (36.1%) are in the first place in terms of consuming organic product groups. [ABSTRACT FROM AUTHOR]

This work focused on the evaluation of co-combustion of sewage sludge (SS) and rice husk (RH) using thermogravimetric–Fourier transform infrared spectrometry method (TG–FTIR). The thermal behavior was assessed through the combustion characteristic, interaction, kinetics, and gaseous product characteristics. The (co-)combustion process was divided into three stages. Index D exponentially increased as RH blending ratio increased and index S also increased from 10 to 70% RH blending ratio, indicating that the combustion behavior was improved by the RH blending. The activation energy was evaluated by two model-free iso-conversional methods: Flynn–Wall–Ozawa (FWO) and Kissinger–Akahira–Sunose (KAS). The minimum value of average activation energy among blends was obtained at sample with 30% RH blending, which was 103.31 kJ/mol by FWO and 97.81 kJ/mol by KAS. Several typical gaseous products and functional groups were detected by FTIR spectrum. The results showed that CO2 was the main volatile and its yield increased with the RH blending ratio. [ABSTRACT FROM AUTHOR]

Kitaev's lattice models are usually defined as representations of the Drinfeld quantum double D(H) = H ⋈ H*op, as an example of a double cross product quantum group. We propose a new version based instead on M(H) = Hcop ⧑ H as an example of Majid's bicrossproduct quantum group, related by semidualisation or 'quantum Born reciprocity' to D(H). Given a finite-dimensional Hopf algebra H, we show that a quadrangulated oriented surface defines a representation of the bicrossproduct quantum group Hcop ⧑ H. Even though the bicrossproduct has a more complicated and entangled coproduct, the construction of this new model is relatively natural as it relies on the use of the covariant Hopf algebra actions. Working locally, we obtain an exactly solvable Hamiltonian for the model and provide a definition of the ground state in terms of a tensor network representation. [ABSTRACT FROM AUTHOR]

Let Z2 denote a cyclic group of 2 order and Z22 = Z2 × Z2 the direct product of groups. Suppose that (M, Φ) is a closed and smooth manifold M with a smooth Z22-action whose fixed point set is the disjoint union of two real projective spaces with the same dimension. In this paper, the authors give a sufficient condition on the fixed data of the action for (M, Φ) bounding equivariantly. [ABSTRACT FROM AUTHOR]

We study congruences on the partial automorphism monoid of a finite rank free group action. We determine a decomposition of a congruence on this monoid into a Rees congruence, a congruence on a Brandt semigroup and an idempotent separating congruence. The constituent parts are further described in terms of subgroups of direct and semidirect products of groups. We utilize this description to demonstrate how the number of congruences on the partial automorphism monoid depends on the group and the rank of the action. [ABSTRACT FROM AUTHOR]

For convex domains with C1,ε boundary we give a precise description of the automorphism group: if an orbit of the automorphism group accumulates on at least two different closed complex faces of the boundary, then the automorphism group has finitely many components and the connected component of the identity is the almost direct product of a compact group and a non‐compact connected simple Lie group with real rank one and finite center. In this case, we also show the limit set is homeomorphic to a sphere and prove a gap theorem: either the domain is biholomorphic to the unit ball (and the limit set is the entire boundary) or the limit set has co‐dimension at least two in the boundary. [ABSTRACT FROM AUTHOR]

The construction sector is progressively becoming more circular by reducing waste, re-using building materials and adopting regenerative solutions for energy production and biodiversity protection. The implications of circularity on construction activities are complex and require the careful evaluation of impacts to select the appropriate path forward. Evaluations of circular solutions and their environmental effectiveness are often performed based on various types of life cycle-based impact assessments. This paper uses systemic thinking to map and evaluate different impact assessment methodologies and their implications for a shift to more circular solutions. The following systemic levels are used to group the methodologies: product (material life cycle declarations and building assessments), organisation (certification and management schemes) and system (policies, standards and regulations). The results confirm that circular economy is integrated at all levels. However, development and structure are not coordinated or governed unidirectionally, but rather occur simultaneously at different levels. This recursive structure is positive if the methods are applied in the correct context, thus providing both autonomy and cohesion in decision making. Methods at lower systemic levels may then improve production processes and stimulate the market to create circular and innovative building solutions, whereas methods at higher systemic levels can be used, for example, by real estate builders, trade organisations and governments to create incentives for circular development and innovation in a broader perspective. Use of the performance methods correctly within an actor network is therefore crucial for successful and effective implementation of circular economy in the construction sector. [ABSTRACT FROM AUTHOR]

The development and manufacturing of high-precision micro-mechatronic systems (MMS) is a challenging task, and the high demand for individualized products complicates the engineering design process (EDP) in particular. The established EDP for MMS is not designed for individualized products. This article gives an overview of the challenges (critical factors) in product development and manufacturing of individualized MMS (iMMS), a novel definition of iMMS, and describes a new qualitative methodology in order to tailor an EDP based on use cases, so-called "Tailored EDP-Methodology" (TEDP-Methodology). This TEDP-Methodology allows creating use-case-based product groups through the abstraction of the use cases and evaluating the requirements, which is essential to tailor or develop a new EDP. For the development of this new approach, a literature review and qualitative content analysis are prefaced. The TEDP-Methodology is critically examined and validated with a real case study for the development and manufacturing of an iMMS. This study shows critical points within the EDP. It shows fields of action for innovative tools to support the development process of iMMS and requirements for different product groups within iMMS. This article has both theoretical and practical implications. [ABSTRACT FROM AUTHOR]

Given a discrete group G , for any integer r ≥ 0 we consider the family of all virtually abelian subgroups of G of rank at most r. We give an upper bound for the Bredon cohomological dimension of G for this family for a certain class of groups acting on CAT (0) spaces. This covers the case of Coxeter groups, Right-angled Artin groups, fundamental groups of special cube complexes and graph products of finite groups. Our construction partially answers a question of Lafont. [ABSTRACT FROM AUTHOR]

We investigate to what extent a nilpotent Lie group is determined by its C ∗ -algebra. We prove that, within the class of exponential Lie groups, direct products of Heisenberg groups with abelian Lie groups are uniquely determined even by their unitary dual, while nilpotent Lie groups of dimension ≤ 5 are uniquely determined by the Morita equivalence class of their C ∗ -algebras. We also find that this last property is shared by the filiform Lie groups and by the 6 -dimensional free two-step nilpotent Lie group. [ABSTRACT FROM AUTHOR]

Let ℍ n be the (2 n + 1) -dimensional Heisenberg group and let n be the n -dimensional torus acting on ℍ n by automorphisms. We consider the semidirect product group G n : = n ⋉ ℍ n. The cortex, cor (G n) , of G n is the set of all unitary irreducible representations π in the unitary dual G n ̂ of G n that cannot be Hausdorff separated from the identity representation 1 G n of G n. In this paper, we describe explicitly the cortex (cor (G n)) of G n using the coadjoint orbits of the group. [ABSTRACT FROM AUTHOR]

The complementarity and substitutability between products are essential concepts in retail and marketing. Qualitatively, two products are said to be substitutable if a customer can replace one product by the other, while they are complementary if they tend to be bought together. In this article, we take a network perspective to help automatically identify complements and substitutes from sales transaction data. Starting from a bipartite product-purchase network representation, with both transaction nodes and product nodes, we develop appropriate null models to infer significant relations, either complements or substitutes, between products, and design measures based on random walks to quantify their importance. The resulting unipartite networks between products are then analysed with community detection methods, in order to find groups of similar products for the different types of relationships. The results are validated by combining observations from a real-world basket dataset with the existing product hierarchy, as well as a large-scale flavour compound and recipe dataset. [ABSTRACT FROM AUTHOR]

In this paper, we construct group trees of tame Polish groups of the form ∏i∈ωHi, where Hi are countable abelian. Our first example where all Hi are torsion reaches the known upper bound ω·2. Next we give examples having height ω·2+N for any N∈ω, close to the known bound ω·3. [ABSTRACT FROM AUTHOR]

For every Lie group G , we compute the maximal n such that an n -fold product of nonabelian free groups embeds into G. [ABSTRACT FROM AUTHOR]

We herein report an asymmetric protocol to access a series of orthogonally functionalized acyclic chiral target molecules containing a quaternary stereogenic center by carrying out the enantioselective α‐alkylation of novel orthogonally functionalized dioxolane‐containing cyanoacetates under chiral ammonium salt catalysis. By using just 1 mol % of Maruoka's spirocyclic ammonium salt catalysts enantioselectivities up to e.r.=97.5 : 2.5 could be achieved and further functional group manipulations of the products were carried out as well. [ABSTRACT FROM AUTHOR]

This paper proves that the infinite dihedral group is only such a free product of two finite groups that its group algebra over a field K is a CS-ring in case the orders of two groups are not zero in K. Furthermore, it is shown that the group algebra of any free product of two finite cyclic groups does not satisfy the condition (C 3). [ABSTRACT FROM AUTHOR]

We use bicyclic units to give an explicit construction of a subgroup of U Z G isomorphic to the free product of two free abelian groups of rank two, assuming that G is a finite nilpotent group and it contains an element g of odd prime order such that the subgroup 〈 g 〉 is not normal in G. To do this we first construct a subgroup isomorphic to the desired free product inside GL (2 , C) and then we find a nontrivial matrix representation of a subgroup of U Z G generated by some bicyclic units and their conjugations under the involution of Z G. We show that for an arbitrary finite group G our construction need not lead to a free product. At the end we shortly discuss possibility of constructing subgroups isomorphic to the free product of two free abelian groups of rank p − 1 for p > 3 in a similar way. [ABSTRACT FROM AUTHOR]

Endoperoxide-containing natural products are a group of compounds with structurally unique cyclized peroxide moieties. Although numerous endoperoxide-containing compounds have been isolated, the biosynthesis of the endoperoxides remains unclear. NvfI from Aspergillus novofumigatus IBT 16806 is an endoperoxidase that catalyzes the formation of fumigatonoid A in the biosynthesis of novofumigatonin. Here, we describe our structural and functional analyses of NvfI. The structural elucidation and mutagenesis studies indicate that NvfI does not utilize a tyrosyl radical in the reaction, in contrast to other characterized endoperoxidases. Further, the crystallographic analysis reveals significant conformational changes of two loops upon substrate binding, which suggests a dynamic movement of active site during the catalytic cycle. As a result, NvfI installs three oxygen atoms onto a substrate in a single enzyme turnover. Based on these results, we propose a mechanism for the NvfI-catalyzed, unique endoperoxide formation reaction to produce fumigatonoid A. Many endoperoxide-containing natural products have been isolated, but the biosynthesis of the endoperoxides remains unclear. Here, the authors report the structural and functional analysis of the NvfI endoperoxidase that catalyzes the formation of fumigatonoid A in the biosynthesis of novofumigatonin, and show that it does not employ tyrosyl radical in the reaction. [ABSTRACT FROM AUTHOR]

Recently, there has been an increasing demand for fresh and high quality food as a result of healthy eating awareness. Depending on the risk status of the product, consumer health and safety get into danger as well as breaking consumer reliance and satisfaction due to the foods which have started to become unfresh and lose quality at the time of purchasing. Nutritious and economically valuable foods such as meat has been one of the major product groups which is needed to be checked for its freshness against time. However, classical methods require much labor and chemical usage as well as being time consuming. Therefore, there has been a need for alternative methods which can eliminate present disadvantages. Moreover, it is aimed to make possible real time detection of meat freshness including consumer based controls. This review evaluates new detection methods such as color indicators, sensors and biosensors to determine freshness of several kinds of meat. [ABSTRACT FROM AUTHOR]