Non-polynomial Worst-Case Analysis of Recursive Programs

We study the problem of developing efficient approaches for proving worst-case bounds of non-deterministic recursive programs. Ranking functions are sound and complete for proving termination and worst-case bounds of nonrecursive programs. First, we apply ranking functions to recursion, resulting in measure functions. We show that measure functions provide a sound and complete approach to prove worst-case bounds of non-deterministic recursive programs. Our second contribution is the synthesis of measure functions in nonpolynomial forms. We show that non-polynomial measure functions with logarithm and exponentiation can be synthesized through abstraction of logarithmic or exponentiation terms, Farkas' Lemma, and Handelman's Theorem using linear programming. While previous methods obtain worst-case polynomial bounds, our approach can synthesize bounds of the form $\mathcal{O}(n\log n)$ as well as $\mathcal{O}(n^r)$ where $r$ is not an integer. We present experimental results to demonstrate that our approach can obtain efficiently worst-case bounds of classical recursive algorithms such as (i) Merge-Sort, the divide-and-conquer algorithm for the Closest-Pair problem, where we obtain $\mathcal{O}(n \log n)$ worst-case bound, and (ii) Karatsuba's algorithm for polynomial multiplication and Strassen's algorithm for matrix multiplication, where we obtain $\mathcal{O}(n^r)$ bound such that $r$ is not an integer and close to the best-known bounds for the respective algorithms.Comment: 54 Pages, Full Version to CAV 201

Conformational studies of peptides representing a segment of TM7 from H+-VO-ATPase in SDS micelles

The conformation of a transmembrane peptide, sMTM7, encompassing the cytoplasmic hemi-channel domain of the seventh transmembrane section of subunit a from V-ATPase from Saccharomyces cerevisiae solubilized in SDS solutions was studied by circular dichroism (CD) spectroscopy and fluorescence spectroscopy of the single tryptophan residue of this peptide. The results show that the peptide adopts an α-helical conformation or aggregated β-sheet depending on the peptide-to-SDS ratio used. The results are compared with published data about a longer version of the peptide (i.e., MTM7). It is concluded that the bulky, positively charged arginine residue located in the center of both peptides has a destabilizing effect on the helical conformation of the SDS-solubilized peptides, leading to β-sheet formation and subsequent aggregation

Orbit Closures and Invariants

The first author would like to thank Sebastian Herpel for the conversations we had which led to the first iteration of some of the ideas in this paper, and also Stephen Donkin for some very helpful nudges towards the right literature. All three authors acknowledge the funding of EPSRC grant EP/L005328/1. We would like to thank the anonymous referee for their very insightful comments and for pointing out a subtle gap in the proof of Theorem 1.1.Peer reviewedPublisher PD

Percutaneous closure of atrial septal defects leads to normalisation of atrial and ventricular volumes

Background: Percutaneous closure of atrial septal defects (ASDs) should potentially reduce right heart volumes by removing left-to-right shunting. Due to ventricular interdependence, this may be associated with impaired left ventricular filling and potentially function. Furthermore, atrial changes post-ASD closure have been poorly understood and may be important for understanding risk of atrial arrhythmia post-ASD closure. Cardiovascular magnetic resonance (CMR) is an accurate and reproducible imaging modality for the assessment of cardiac function and volumes. We assessed cardiac volumes pre- and post-percutaneous ASD closure using CMR. Methods: Consecutive patients (n = 23) underwent CMR pre- and 6 months post-ASD closure. Steady state free precession cine CMR was performed using contiguous slices in both short and long axis views through the ASD. Data was collected for assessment of left and right atrial, ventricular end diastolic volumes (EDV) and end systolic volumes (ESV). Data is presented as mean ± SD, volumes as mL, and paired t-testing performed between groups. Statistical significance was taken as p < 0.05. Results: There was a significant reduction in right ventricular volumes at 6 months post-ASD closure (RVEDV: 208.7 ± 76.7 vs. 140.6 ± 60.4 mL, p < 0.0001) and RVEF was significantly increased (RVEF 35.5 ± 15.5 vs. 42.0 ± 15.2%, p = 0.025). There was a significant increase in the left ventricular volumes (LVEDV 84.8 ± 32.3 vs. 106.3 ± 38.1 mL, p = 0.003 and LVESV 37.4 ± 20.9 vs. 46.8 ± 18.5 mL, p = 0.016). However, there was no significant difference in LVEF and LV mass post-ASD closure. There was a significant reduction in right atrial volumes at 6 months post-ASD closure (pre-closure 110.5 ± 55.7 vs. post-closure 90.7 ± 69.3 mL, p = 0.019). Although there was a trend to a decrease in left atrial volumes post-ASD closure, this was not statistically significant (84.5 ± 34.8 mL to 81.8 ± 44.2 mL, p = NS). Conclusion: ASD closure leads to normalisation of ventricular volumes and also a reduction in right atrial volume. Further follow-up is required to assess how this predicts outcomes such as risk of atrial arrhythmias after such procedures.Karen SL Teo, Benjamin K Dundon, Payman Molaee, Kerry F Williams, Angelo Carbone, Michael A Brown, Matthew I Worthley, Patrick J Disney, Prashanthan Sanders and Stephen G Worthle