Conjugacy in Baumslag's group, generic case complexity, and division in power circuits

The conjugacy problem belongs to algorithmic group theory. It is the following question: given two words x, y over generators of a fixed group G, decide whether x and y are conjugated, i.e., whether there exists some z such that zxz^{-1} = y in G. The conjugacy problem is more difficult than the word problem, in general. We investigate the complexity of the conjugacy problem for two prominent groups: the Baumslag-Solitar group BS(1,2) and the Baumslag(-Gersten) group G(1,2). The conjugacy problem in BS(1,2) is TC^0-complete. To the best of our knowledge BS(1,2) is the first natural infinite non-commutative group where such a precise and low complexity is shown. The Baumslag group G(1,2) is an HNN-extension of BS(1,2). We show that the conjugacy problem is decidable (which has been known before); but our results go far beyond decidability. In particular, we are able to show that conjugacy in G(1,2) can be solved in polynomial time in a strongly generic setting. This means that essentially for all inputs conjugacy in G(1,2) can be decided efficiently. In contrast, we show that under a plausible assumption the average case complexity of the same problem is non-elementary. Moreover, we provide a lower bound for the conjugacy problem in G(1,2) by reducing the division problem in power circuits to the conjugacy problem in G(1,2). The complexity of the division problem in power circuits is an open and interesting problem in integer arithmetic.Comment: Section 5 added: We show that an HNN extension G = < H, b | bab^-1 = {\phi}(a), a \in A > has a non-amenable Schreier graph with respect to the base group H if and only if A \neq H \neq

Can MR Imaging Diagnose Adult-Onset Alexander Disease?

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Harmonisation and diagnostics of MIPAS ESA CH4 and N2O profiles using data assimilation

This paper discusses assimilation experiments of methane (CH4) and nitrous oxide (N2O) profiles retrieved from the Michelson Interferometer for Passive Atmospheric Sounding (MIPAS). Here we focus on data versions 6 and 7 provided by the ESA processor. These data sets have been assimilated by the Belgian Assimilation System for Chemical ObsErvations (BASCOE). The CH4 and N2O retrieved profiles can oscillate, especially in the tropical lower stratosphere. Using the averaging kernels of the observations and a background error covariance matrix, which has previously been calibrated, allows the system to partly remedy this issue and provide assimilated fields that are more regular vertically. In general, there is a good agreement between the BASCOE analyses and independent observations from ACE–FTS (CH4 and N2O) and MLS (N2O), demonstrating the general good quality of CH4 and N2O retrievals provided by MIPAS ESA. Nevertheless, this study also identifies two issues in these data sets. First, time series of the observations show unexpected discontinuities due to an abrupt change in the gain of MIPAS band B, generally occurring after the instrument decontamination. Since the calibration is performed weekly, the abrupt change in the gain affects the measurements until the subsequent calibration is performed. Second, the correlations between BASCOE analyses and independent observations are poor in the lower stratosphere, especially in the tropics, probably due to the presence of outliers in the assimilated data. In this region, we recommend using MIPAS CH4 and N2O retrievals with caution

Prevalence of resistance mutations related to integrase inhibitor S/GSK1349572 in HIV-1 subtype B raltegravir-naive and -treated patients

Objectives To compare the frequency of previously in vitro-selected integrase mutations (T124A, T124A/S153F, S153Y, T124A/S153Y and L101I/T124A/S153Y) conferring resistance to S/GSK1349572 between HIV-1 subtype B integrase inhibitor (INI)-naive and raltegravir-treated patients. Methods Integrase sequences from 650 INI-naive patients and 84 raltegravir-treated patients were analysed. Results The T124A mutation alone and the combination T124A/L101I were more frequent in raltegravir-failing patients than in INI-naive patients (39.3% versus 24.5%, respectively, P = 0.005 for T124A and 20.2% versus 10.0%, respectively, P = 0.008 for T124A/L101I). The S153Y/F mutations were not detected in any integrase sequence (except for S153F alone, only detected in one INI-naive patient). Conclusions T124A and T124A/L101I, more frequent in raltegravir-treated patients, could have some effect on raltegravir response and their presence could play a role in the selection of other mutations conferring S/GSK1349572 resistance. The impact of raltegravir-mediated changes such as these on the virological response to S/GSK1349572 should be studied further