Central Invariants and Higher Indicators for Semisimple Quasi-Hopf Algebras

In this paper, we define the higher Frobenius-Schur (FS-)indicators for finite-dimensional modules $V$ of a semisimple quasi-Hopf algebra $H$ via the categorical counterpart developed in \cite{NS05}. We prove that this definition of higher FS-indicators coincides with the higher indicators introduced by Kashina, Sommerh\"auser, and Zhu when $H$ is a Hopf algebra. We also obtain a sequence of canonical central elements of $H$, which is invariant under gauge transformations, whose values, when evaluated by the character of an $H$-module $V$, are the higher Frobenius-Schur indicators of $V$. As an application, we show that FS-indicators are sufficient to distinguish the four gauge equivalence classes of semisimple quasi-Hopf algebras of dimension eight corresponding to the four fusion categories with certain fusion rules classified by Tambara and Yamagami. Three of these categories correspond to well-known Hopf algebras, and we explicitly construct a quasi-Hopf algebra corresponding to the fourth. We also derive explicit formulae for FS-indicators for some quasi-Hopf algebras associated to group cocycles.Comment: The higher Frobenius-Schur indicators for certain quasi-Hopf algebras associated with finite groups and their 3-cocycles have been computed in section

Higher Frobenius-Schur Indicators for Pivotal Categories

We define higher Frobenius-Schur indicators for objects in linear pivotal monoidal categories. We prove that they are category invariants, and take values in the cyclotomic integers. We also define a family of natural endomorphisms of the identity endofunctor on a $k$-linear semisimple rigid monoidal category, which we call the Frobenius-Schur endomorphisms. For a $k$-linear semisimple pivotal monoidal category -- where both notions are defined --, the Frobenius-Schur indicators can be computed as traces of the Frobenius-Schur endomorphisms.Comment: A paragraph which describes the organization of the paper has been added to the introduction. Some observations have been added to Theorems 5.1 and 7.

Distributed Gaussian Processes

To scale Gaussian processes (GPs) to large data sets we introduce the robust Bayesian Committee Machine (rBCM), a practical and scalable product-of-experts model for large-scale distributed GP regression. Unlike state-of-the-art sparse GP approximations, the rBCM is conceptually simple and does not rely on inducing or variational parameters. The key idea is to recursively distribute computations to independent computational units and, subsequently, recombine them to form an overall result. Efficient closed-form inference allows for straightforward parallelisation and distributed computations with a small memory footprint. The rBCM is independent of the computational graph and can be used on heterogeneous computing infrastructures, ranging from laptops to clusters. With sufficient computing resources our distributed GP model can handle arbitrarily large data sets.Comment: 10 pages, 5 figures. Appears in Proceedings of ICML 201

Investigation of the speed-up of a dual microcontroller parallel processing system in the execution of a mathematical operation

An investigation of the performance of a two microcontroller parallel processing system is presented. A twomicrocontroller parallel processing is developed using low end microcontrollers (PIC 16F877). An 8x8 bit multiply operation and a 16x16 bit multiply operation are executed on a single microcontroller and on the proposed dual microcontroller parallel processing system in order to assess the performance of the proposed system. Results presented show poor performance for the 8x8 bit multiply with an average speed up factor of 0.82 This is due to the time required to transfer data around the dual microcontroller system being significant in comparison to the time required to complete the multiply operation, thus nullifying the potential advantage that might be expected of a dual microcontroller system. The 16x16 multiplier exhibited good performance, with results showing a maximum average speed up factor of 1.7 and an average speed up factor of 1.5. The 16x16 multiplication requires longer time to compute and the data transfer time between microcontrollers whilst still having an impact on the overall computation time is significantly less than for the 8x8 multiplier A formula has been developed to provide an estimate of the possible speed up within a system in relation to the process execution time and the time required to communicate data around the proposed system. The proposed system was developed and tested using the Proteus simulation software

Managing Exchange Rate Volatility: A Comparative Counterfactual Analysis of Singapore 1994 to 2003

The objective of this paper is see how well Singapore’s exchange rate regime has coped with exchange rate volatility before and after the Asian financial crisis by comparing the performance of Singapore’s actual regime in minimising the volatility of the nominal effective exchange rate (NEER) and the bilateral rate against the US$ against some counterfactual regimes and the corresponding performance of eight other East Asian countries. In contrast to previous counterfactual exercises, such as Williamson (1998a) and Ohno (1999) which compute the weights for effective exchange rates on the basis of simple bloc aggregates, we apply a more disaggregated methodology using a larger number of trade partners. We also utilize ARCH/GARCH techniques to obtain estimates of heteroskedastic variances to better capture the time-varying characteristics of volatility for the actual and simulated exchange rate regimes. Our findings confirm that Singapore’s managed floating exchange rate system has delivered relatively low currency volatility. Although there are gains in volatility reduction for all countries in the sample from the adoption of either a unilateral or common basket peg, particularly post-crisis, these gains are relatively low for Singapore, largely because low actual volatility. Finally, there are additional gains for nondollar peggers from stabilizing intra-EA exchange rates against the dollar if they were to adopt a basket peg, especially post-crisis, but the gains for Singapore are again relatively modest.East Asia, exchange rates, counterfactuals.

Memory effects in biochemical networks as the natural counterpart of extrinsic noise

We show that in the generic situation where a biological network, e.g. a protein interaction network, is in fact a subnetwork embedded in a larger "bulk" network, the presence of the bulk causes not just extrinsic noise but also memory effects. This means that the dynamics of the subnetwork will depend not only on its present state, but also its past. We use projection techniques to get explicit expressions for the memory functions that encode such memory effects, for generic protein interaction networks involving binary and unary reactions such as complex formation and phosphorylation, respectively. Remarkably, in the limit of low intrinsic copy-number noise such expressions can be obtained even for nonlinear dependences on the past. We illustrate the method with examples from a protein interaction network around epidermal growth factor receptor (EGFR), which is relevant to cancer signalling. These examples demonstrate that inclusion of memory terms is not only important conceptually but also leads to substantially higher quantitative accuracy in the predicted subnetwork dynamics