Rank-finiteness for modular categories

We prove a rank-finiteness conjecture for modular categories: up to equivalence, there are only finitely many modular categories of any fixed rank. Our technical advance is a generalization of the Cauchy theorem in group theory to the context of spherical fusion categories. For a modular category $\mathcal{C}$ with $N=ord(T)$, the order of the modular $T$-matrix, the Cauchy theorem says that the set of primes dividing the global quantum dimension $D^2$ in the Dedekind domain $\mathbb{Z}[e^{\frac{2\pi i}{N}}]$ is identical to that of $N$.Comment: 25 pages (last version). Version 2: removed weakly integral rank 6 and integral rank 7 section, improved rank 5 classification up to monoidal equivalence. Version 3: removed rank 5 classification (note title change)--this will be published separately. Significantly improved expositio

On classification of modular categories by rank

The feasibility of a classification-by-rank program for modular categories follows from the Rank-Finiteness Theorem. We develop arithmetic, representation theoretic and algebraic methods for classifying modular categories by rank. As an application, we determine all possible fusion rules for all rank=$5$ modular categories and describe the corresponding monoidal equivalence classes.Comment: arXiv admin note: substantial text overlap with arXiv:1310.705

Late Eighteenth-Century English Violin Concertos: A Genre in Transition

Classical violin concertos by English composers are relatively obscure nowadays, as the genre is largely monopolized by Mozart’s last three concertos. This study explores the compositional and violinistic traits of ten English concertos from the late eighteenth century, as well as the social and cultural circumstances under which they were written. These concertos are challenging violinistically, suggesting that they were primarily intended as virtuosic showpieces. In addition, a number of the concertos display musical eccentricities that hint at the quirky personalities of their composers. In some respects, these concertos are unadventurous, particularly in terms of harmony and thematic contrasts. However, they contain a number of unique compositional features that are worthy of our attention. The most notable of these is the incorporation of Baroque features in a large number of the concertos, despite their general adherence to the new galant style. All evidence suggests that the two styles were combined deliberately as a compositional technique, and this is perhaps the most distinctly “English” characteristic of these concertos

COROSOLIC ACID: A SYNOPSIS ON ITS ANTICANCER PROPERTIES

Corosolic acid (CA) or 2β-hydroxyursolic acid is an ursane-type pentacyclic triterpene with a molecular formula of C30H48O4 and molecular weight of 473 g/mol. The 30-carbon skeleton and five six-membered rings (A−E) of CA are structurally similar to those of ursolic acid, asiatic acid, and 23-hydroxyl CA. CA was first isolated from the leaf of Lagerstroemia speciosa and later from the fruit of Crataegus pinnatifida. Although L. speciosa (Lythraceae) remains the most important source of CA, Rosaceae and Lamiaceae are the dominant families. This synopsis is focused on the anticancer properties of CA as recent studies have generated new and additional knowledge on its oncology. CA has antitumor, antiproliferative, and apoptotic activities against many types of human cancer cells (including some murine cancer cells), which are inhibited through different molecular mechanisms. Non-apoptotic cell death has also been reported. Depending on the type of cancer cells, the cytotoxicity of CA is comparable to ursolic acid, its analog. Currently, there are no studies on the structure-activity relationship of CA. In ursolic acid, which is structurally similar to CA, the −OH group at C-3 and the −COOH group at C-28 exhibited cytotoxic activity