Witten's Invariants of Rational Homology Spheres at Prime Values of KK and Trivial Connection Contribution

We establish a relation between the coefficients of asymptotic expansion of trivial connection contribution to Witten's invariant of rational homology spheres and the invariants that T.~Ohtsuki extracted from Witten's invariant at prime values of $K$. We also rederive the properties of prime $K$ invariants discovered by H.~Murakami and T.~Ohtsuki. We do this by using the bounds on Taylor series expansion of the Jones polynomial of algebraically split links, studied in our previous paper. These bounds are enough to prove that Ohtsuki's invariants are of finite type. The relation between Ohtsuki's invariants and trivial connection contribution is verified explicitly for lens spaces and Seifert manifolds.Comment: 32 pages, no figures, LaTe

25 Years of Self-Organized Criticality: Numerical Detection Methods

The detection and characterization of self-organized criticality (SOC), in both real and simulated data, has undergone many significant revisions over the past 25 years. The explosive advances in the many numerical methods available for detecting, discriminating, and ultimately testing, SOC have played a critical role in developing our understanding of how systems experience and exhibit SOC. In this article, methods of detecting SOC are reviewed; from correlations to complexity to critical quantities. A description of the basic autocorrelation method leads into a detailed analysis of application-oriented methods developed in the last 25 years. In the second half of this manuscript space-based, time-based and spatial-temporal methods are reviewed and the prevalence of power laws in nature is described, with an emphasis on event detection and characterization. The search for numerical methods to clearly and unambiguously detect SOC in data often leads us outside the comfort zone of our own disciplines - the answers to these questions are often obtained by studying the advances made in other fields of study. In addition, numerical detection methods often provide the optimum link between simulations and experiments in scientific research. We seek to explore this boundary where the rubber meets the road, to review this expanding field of research of numerical detection of SOC systems over the past 25 years, and to iterate forwards so as to provide some foresight and guidance into developing breakthroughs in this subject over the next quarter of a century.Comment: Space Science Review series on SO

Detection and Interpretation Of Long-Lived X-Ray Quasi-Periodic Pulsations in the X-Class Solar Flare On 2013 May 14

Quasi-periodic pulsations (QPP) seen in the time derivative of the GOES soft X-ray light curves are analyzed for the near-limb X3.2 event on 14 May 2013. The pulsations are apparent for a total of at least two hours from the impulsive phase to well into the decay phase, with a total of 163 distinct pulses evident to the naked eye. A wavelet analysis shows that the characteristic time scale of these pulsations increases systematically from $\sim$25 s at 01:10 UT, the time of the GOES peak, to $\sim$100 s at 02:00 UT. A second ridge in the wavelet power spectrum, most likely associated with flaring emission from a different active region, shows an increase from $\sim$40 s at 01:40 UT to $\sim$100 s at 03:10 UT. We assume that the QPP that produced the first ridge result from vertical kink-mode oscillations of the newly formed loops following magnetic reconnection in the coronal current sheet. This allows us to estimate the magnetic field strength as a function of altitude given the density, loop length, and QPP time scale as functions of time determined from the GOES light curves and RHESSI images. The calculated magnetic field strength of the newly formed loops ranges from about $\sim$500 G at an altitude of 24 Mm to a low value of $\sim$10 G at 60 Mm, in general agreement with the expected values at these altitudes. Fast sausage mode oscillations are also discussed and cannot be ruled out as an alternate mechanism for producing the QPP

A note on the sign (unit root) ambiguities of Gauss sums in index 2 and 4 cases

Recently, the explicit evaluation of Gauss sums in the index 2 and 4 cases have been given in several papers (see [2,3,7,8]). In the course of evaluation, the sigh (or unit root) ambiguities are unavoidably occurred. This paper presents another method, different from [7] and [8], to determine the sigh (unit root) ambiguities of Gauss sums in the index 2 case, as well as the ones with odd order in the non-cyclic index 4 case. And we note that the method in this paper are more succinct and effective than [8] and [7]

On Tractable Exponential Sums

We consider the problem of evaluating certain exponential sums. These sums take the form $\sum_{x_1,...,x_n \in Z_N} e^{f(x_1,...,x_n) {2 \pi i / N}}$, where each x_i is summed over a ring Z_N, and f(x_1,...,x_n) is a multivariate polynomial with integer coefficients. We show that the sum can be evaluated in polynomial time in n and log N when f is a quadratic polynomial. This is true even when the factorization of N is unknown. Previously, this was known for a prime modulus N. On the other hand, for very specific families of polynomials of degree \ge 3, we show the problem is #P-hard, even for any fixed prime or prime power modulus. This leads to a complexity dichotomy theorem - a complete classification of each problem to be either computable in polynomial time or #P-hard - for a class of exponential sums. These sums arise in the classifications of graph homomorphisms and some other counting CSP type problems, and these results lead to complexity dichotomy theorems. For the polynomial-time algorithm, Gauss sums form the basic building blocks. For the hardness results, we prove group-theoretic necessary conditions for tractability. These tests imply that the problem is #P-hard for even very restricted families of simple cubic polynomials over fixed modulus N

Properties of dense partially random graphs

We study the properties of random graphs where for each vertex a {\it neighbourhood} has been previously defined. The probability of an edge joining two vertices depends on whether the vertices are neighbours or not, as happens in Small World Graphs (SWGs). But we consider the case where the average degree of each node is of order of the size of the graph (unlike SWGs, which are sparse). This allows us to calculate the mean distance and clustering, that are qualitatively similar (although not in such a dramatic scale range) to the case of SWGs. We also obtain analytically the distribution of eigenvalues of the corresponding adjacency matrices. This distribution is discrete for large eigenvalues and continuous for small eigenvalues. The continuous part of the distribution follows a semicircle law, whose width is proportional to the "disorder" of the graph, whereas the discrete part is simply a rescaling of the spectrum of the substrate. We apply our results to the calculation of the mixing rate and the synchronizability threshold.Comment: 14 pages. To be published in Physical Review

Associations between e-cigarette access and smoking and drinking behaviours in teenagers

Background: Public health concerns regarding e-cigarettes and debate on appropriate regulatory responses are focusing on the need to prevent child access to these devices. However, little is currently known about the characteristics of those young people that are accessing e-cigarettes. Methods: Using a cross-sectional survey of 14-17 year old school students in North West England (n = 16,193) we examined associations between e-cigarette access and demographics, conventional smoking behaviours, alcohol consumption, and methods of accessing cigarettes and alcohol. Access to e-cigarettes was identified through a question asking students if they had ever tried or purchased e-cigarettes. Results: One in five participants reported having accessed e-cigarettes (19.2%). Prevalence was highest among\ud smokers (rising to 75.8% in those smoking >5 per day), although 15.8% of teenagers that had accessed e-cigarettes had never smoked conventional cigarettes (v.13.6% being ex-smokers). E-cigarette access was independently associated with male gender, having parents/guardians that smoke and students’ alcohol use. Compared with non-drinkers, teenagers that drank alcohol at least weekly and binge drank were more likely to have accessed e-cigarettes (adjusted odds ratio [AOR] 1.89, P < 0.001), with this association particularly strong among never-smokers (AOR 4.59, P < 0.001). Among drinkers, e-cigarette access was related to: drinking to get drunk, alcohol-related violence, consumption of spirits; self-purchase of alcohol from shops or supermarkets; and accessing alcohol by recruiting adult proxy purchasers outside shops. Conclusions: There is an urgent need for controls on the promotion and sale of e-cigarettes to children. Findings suggest that e-cigarettes are being accessed by teenagers more for experimentation than smoking cessation. Those most likely to access e-cigarettes may already be familiar with illicit methods of accessing age-restricted substances

Business model design–performance relationship under external and internal contingencies: evidence from SMEs in an emerging economy

We examined Business Model (BM) designs – performance relationship and the moderating effects of firm age and external environment on this relationship. Furthermore, we investigated the impact of simultaneously operating dual BM designs (i.e. novelty and efficiency) on firm performance and contingent effect of firm age on this relationship. Based on data from 241 Indian SMEs, our findings highlighted that BM novelty was of greater benefit to younger SMEs compared to mature SMEs, while BM efficiency was of greater benefit to more mature SMEs. The environmental dynamism positively moderated the relationship between BM novelty and performance but it negatively moderated the relationship between BM efficiency and performance. We also found that BM efficiency is more beneficial in a low, rather than a high, munificent environment but we found environmental munificence did not moderate the BM novelty and SME performance relationship. Finally, we found simultaneous deployment of BM novelty and BM efficiency resulted in an enhancement of performance among mature SMEs compared to younger SMEs. Our study not only adds to the limited literature on BMs in SMEs but also helps practicing managers and entrepreneurs to make informed choices about their BMs