Tropical Duality in $$(d+2)$$-Angulated Categories

Let $$\mathscr {C}$$ C be a 2-Calabi–Yau triangulated category with two cluster tilting subcategories $$\mathscr {T}$$ T and $$\mathscr {U}$$ U . A result from Jørgensen and Yakimov (Sel Math (NS) 26:71–90, 2020) and Demonet et al. (Int Math Res Not 2019:852–892, 2017) known as tropical duality says that the index with respect to $$\mathscr {T}$$ T provides an isomorphism between the split Grothendieck groups of $$\mathscr {U}$$ U and $$\mathscr {T}$$ T . We also have the notion of c-vectors, which using tropical duality have been proven to have sign coherence, and to be recoverable as dimension vectors of modules in a module category. The notion of triangulated categories extends to the notion of $$(d+2)$$ ( d + 2 ) -angulated categories. Using a higher analogue of cluster tilting objects, this paper generalises tropical duality to higher dimensions. This implies that these basic cluster tilting objects have the same number of indecomposable summands. It also proves that under conditions of mutability, c-vectors in the $$(d+2)$$ ( d + 2 ) -angulated case have sign coherence, and shows formulae for their computation. Finally, it proves that under the condition of mutability, the c-vectors are recoverable as dimension vectors of modules in a module category.

Highly efficient MoIV3⋯SbIII cluster frustrated Lewis pair hydrogenation

A new concept of a cluster frustrated Lewis pair (CFLP) featuring extensive cooperative effects both within Lewis acidic/basic cluster components and between Lewis cluster partners has been proposed.

Mixed cobinary trees

We develop basic cluster theory from an elementary point of view using a variation of binary trees which we call mixed cobinary trees (MCTs). We show that the number of isomorphism classes of such trees is given by the Catalan number [Formula: see text] where [Formula: see text] is the number of internal nodes. We also consider the corresponding quiver [Formula: see text] of type [Formula: see text]. As a special case of more general known results about the relation between [Formula: see text]-vectors, representations of quivers and their semi-invariants, we explain the bijection between MCTs and the vertices of the generalized associahedron corresponding to the quiver [Formula: see text]. These results are extended to [Formula: see text]-clusters in the next paper. We give one application: a new short proof of a conjecture of Reineke using MCTs.

Inferring Companies Similarities from Brazilian Government Expenditure Data

A graph-based method is proposed for inferring similarities among companies from their affiliations in the context of expenditure financial transactions in the Brazilian Federal Government. There are trusted and untrusted companies. We performed a basic cluster analysis in the companies network to verify whether clusters (connected components) are discriminative concerning companies trustworthiness. Results show evidences that this is true, reinforcing the following hypotheses: (1) there are suppliers associations, which evidences the formation of cartels; and (2) public agencies and agents play an important role in the legality of financial transactions.

A Basic Cluster in the N Terminus of Yellow Fever Virus NS2A Contributes to Infectious Particle Production

ABSTRACTThe flavivirus NS2A protein is involved in the assembly of infectious particles. To further understand its role in this process, a charged-to-alanine scanning analysis was performed on NS2A encoded by an infectious cDNA clone of yellow fever virus (YFV). Fifteen mutants containing single, double, or triple charged-to-alanine changes were tested. Five of them did not produce infectious particles, whereas efficient RNA replication was detectable for two of the five NS2A mutants (R22A-K23A-R24A and R99A-E100A-R101A mutants). Prolonged cultivation of transfected cells resulted in the recovery of pseudorevertants. Besides suppressor mutants in NS2A, a compensating second-site mutation in NS3 (D343G) arose for the NS2A R22A-K23A-R24A mutant. We found this NS3 mutation previously to be suppressive for the NS2Aα cleavage site Q189S mutant, also deficient in virion assembly. In this study, the subsequently suggested interaction between NS2A and NS3 was proven by coimmunoprecipitation analyses. Using selectively permeabilized cells, we could demonstrate that the regions encompassing R22A-K23A-R24A and Q189S in NS2A are localized to the cytoplasm, where NS3 is also known to reside. However, the defect in particle production observed for the NS2A R22A-K23A-R24A and Q189S mutants was not due to a defect in physical interaction between NS2A and NS3, as the NS2A mutations did not interrupt NS3 interaction. In fact, a region just upstream of R22-K23-R24 was mapped to be critical for NS2A-NS3 interaction. Taken together, these data support a complex interplay between YFV NS2A and NS3 in virion assembly and identify a basic cluster in the NS2A N terminus to be critical in this process.IMPORTANCEDespite an available vaccine, yellow fever remains endemic in tropical areas of South America and Africa. To control the disease, antiviral drugs are required, and an understanding of the determinants of virion assembly is central to their development. In this study, we identified a basic cluster of amino acids in the N terminus of YFV NS2A which inhibited virion assembly upon mutation. The defect was rescued by a spontaneously occurring mutation in NS3. Our study proves an interaction between NS2A and NS3, which, remarkably, was maintained for the NS2A mutant in the presence and absence of the NS3 mutation. This suggests a role for other viral and/or cellular proteins in virion assembly. Residues important for YFV virion production reported here only partially coincided with those reported for other flaviviruses, suggesting that the determinants for particle production are virus specific. Reconstruction of a YFV encoding tagged NS2A paves the way to identify further NS2A interaction partners.

CLUSTERS IN SPIN GLASSES AND THE FRUSTRATED PERCOLATION MODEL

The basic cluster formalism which maps the Ising model into a correlated percolation model will be discussed. More recently the extension of the cluster approach to frustrated systems, like spin glasses, has lead to the definition of a new geometrical model, called frustrated percolation. The properties of this model will be reviewed.