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

A comparative study of managerial stress in greater China : the direct and indirect effects of coping strategies and work locus of control

The present study aims at comparing the direct and indirect effects of coping strategies (control and support coping) and work locus of control (externality) on the stress-strain relationships among managers in Greater China (Peopled Republic of China [PRC], Hong Kong, and Taiwan) using the Occupational Stress Indicator (OSI). Self-administered survey method was employed to collect data from 249 (164 males, 85 females) managers in China, 280 (159 males, 120 females, 1 unclassified) managers in Hong Kong, and 347 (191 males, 151 females, 5 unclassified) managers in Taiwan. The direct and indirect effects of control coping, support coping, and locus of control on some stress-strain relationships were demonstrated in the studied samples. The similarities and differences in managerial stress in Greater China were discussed according to economic and socio-cultural variations

A Genome-Wide Analysis of Small Regulatory RNAs in the Human Pathogen Group A Streptococcus

The coordinated regulation of gene expression is essential for pathogens to infect and cause disease. A recently appreciated mechanism of regulation is that afforded by small regulatory RNA (sRNA) molecules. Here, we set out to assess the prevalence of sRNAs in the human bacterial pathogen group A Streptococcus (GAS). Genome-wide identification of candidate GAS sRNAs was performed through a tiling Affymetrix microarray approach and identified 40 candidate sRNAs within the M1T1 GAS strain MGAS2221. Together with a previous bioinformatic approach this brings the number of novel candidate sRNAs in GAS to 75, a number that approximates the number of GAS transcription factors. Transcripts were confirmed by Northern blot analysis for 16 of 32 candidate sRNAs tested, and the abundance of several of these sRNAs were shown to be temporally regulated. Six sRNAs were selected for further study and the promoter, transcriptional start site, and Rho-independent terminator identified for each. Significant variation was observed between the six sRNAs with respect to their stability during growth, and with respect to their inter- and/or intra-serotype-specific levels of abundance. To start to assess the contribution of sRNAs to gene regulation in M1T1 GAS we deleted the previously described sRNA PEL from four clinical isolates. Data from genome-wide expression microarray, quantitative RT-PCR, and Western blot analyses are consistent with PEL having no regulatory function in M1T1 GAS. The finding that candidate sRNA molecules are prevalent throughout the GAS genome provides significant impetus to the study of this fundamental gene-regulatory mechanism in an important human pathogen

A Markov chain model for studying suicide dynamics: an illustration of the Rose theorem

Background: High-risk strategies would only have a modest effect on suicide prevention within a population. It is best to incorporate both high-risk and population-based strategies to prevent suicide. This study aims to compare the effectiveness of suicide prevention between high-risk and population-based strategies. Methods: A Markov chain illness and death model is proposed to determine suicide dynamic in a population and examine its effectiveness for reducing the number of suicides by modifying certain parameters of the model. Assuming a population with replacement, the suicide risk of the population was estimated by determining the final state of the Markov model. Results: The model shows that targeting the whole population for suicide prevention is more effective than reducing risk in the high-risk tail of the distribution of psychological distress (i.e. the mentally ill). Conclusions: The results of this model reinforce the essence of the Rose theorem that lowering the suicidal risk in the population at large may be more effective than reducing the high risk in a small population

Age differences in work adjustment : a study of male and female managerial stress, coping strategies and locus of control in Hong Kong

The present study is a 15 month project which involved data collection from Hong Kong managers at three points to examine mechanisms by which age would relate to work well-being. A total of 634 managers, both male and female, was drawn by random sampling and purposive sampling methods. The results showed that age was negatively related to job strains and quitting intention, and positively related to job satisfaction. Furthermore, older managers reported fewer sources of stress, better control coping, and a more internal work locus of control. Multiple regression analyses suggested that relations of age with job satisfaction and well-being can be attributed to the better coping, greater internality, lower sources of stress, higher organizational level, and longer tenure of older managers, although differences occurred as a function of gender and the specific well-being measure

On the Classification of Weakly Integral Modular Categories

We classify all modular categories of dimension $4m$, where $m$ is an odd square-free integer, and all ranks $6$ and $7$ weakly integral modular categories. This completes the classification of weakly integral modular categories through rank $7$. Our results imply that all integral modular categories of rank at most $7$ are pointed (that is, every simple object has dimension $1$). All strictly weakly integral (weakly integral but non-integral) modular categories of ranks $6$ and $7$ have dimension $4m$, with $m$ an odd square free integer, so their classification is an application of our main result. The classification of rank $7$ integral modular categories is facilitated by an analysis of two actions on modular categories: the Galois group of the field generated by the entries of the $S$-matrix and the group of isomorphism classes of invertible simple objects. The interplay of these two actions is of independent interest, and we derive some valuable arithmetic consequences from their actions.Comment: Version 2: fixed missing metadata, version 3: corrected incomplete introduction and added theorem numbers, version 4: 32 pages, significant cosmetic revisions, version 5: final revision pre-submissio

Classification of Super-Modular Categories by Rank

We pursue a classification of low-rank super-modular categories parallel to that of modular categories. We classify all super-modular categories up to rank = 6, and spin modular categories up to rank = 11. In particular, we show that, up to fusion rules, there is exactly one non-split super-modular category of rank 2, 4 and 6, namely PSU(2)4k+ 2 for k = 0,1 and 2. This classification is facilitated by adapting and extending well-known constraints from modular categories to super-modular categories, such as Verlinde and Frobenius-Schur indicator formulae