The localization transition in SU(3) gauge theory

We study the Anderson-like localization transition in the spectrum of the Dirac operator of quenched QCD. Above the deconfining transition we determine the temperature dependence of the mobility edge separating localized and delocalized eigenmodes in the spectrum. We show that the temperature where the mobility edge vanishes and localized modes disappear from the spectrum, coincides with the critical temperature of the deconfining transition. We also identify topological charge related close to zero modes in the Dirac spectrum and show that they account for only a small fraction of localized modes, a fraction that is rapidly falling as the temperature increases.Comment: 7 pages, 5 figures, v3: additional data on finer lattice; final, published versio

Low-Rank Boolean Matrix Approximation by Integer Programming

Low-rank approximations of data matrices are an important dimensionality reduction tool in machine learning and regression analysis. We consider the case of categorical variables, where it can be formulated as the problem of finding low-rank approximations to Boolean matrices. In this paper we give what is to the best of our knowledge the first integer programming formulation that relies on only polynomially many variables and constraints, we discuss how to solve it computationally and report numerical tests on synthetic and real-world data

Binary matrix factorisation and completion via integer programming

Binary matrix factorisation is an essential tool for identifying discrete patterns in binary data. In this paper we consider the rank-k binary matrix factorisation problem (k-BMF) under Boolean arithmetic: we are given an n × m binary matrix X with possibly missing entries and need to find two binary matrices A and B of dimension n × k and k × m respectively, which minimise the distance between X and the Boolean product of A and B in the squared Frobenius distance. We present a compact and two exponential size integer programs (IPs) for k-BMF and show that the compact IP has a weak LP relaxation, while the exponential size IPs have a stronger equivalent LP relaxation. We introduce a new objective function, which differs from the traditional squared Frobenius objective in attributing a weight to zero entries of the input matrix that is proportional to the number of times the zero is erroneously covered in a rank-k factorisation. For one of the exponential size IPs we describe a computational approach based on column generation. Experimental results on synthetic and real word datasets suggest that our integer programming approach is competitive against available methods for k-BMF and provides accurate low-error factorisations

Binary Matrix Factorisation via Column Generation

Identifying discrete patterns in binary data is an important dimensionality reduction tool in machine learning and data mining. In this paper, we consider the problem of low-rank binary matrix factorisation (BMF) under Boolean arithmetic. Due to the NP-hardness of this problem, most previous attempts rely on heuristic techniques. We formulate the problem as a mixed integer linear program and use a large scale optimisation technique of column generation to solve it without the need of heuristic pattern mining. Our approach focuses on accuracy and on the provision of optimality guarantees. Experimental results on real world datasets demonstrate that our proposed method is effective at producing highly accurate factorisations and improves on the previously available best known results for 15 out of 24 problem instances

Metabolic Responses to a Battling Rope Protocol Performed in the Seated or Stance Positions

Achieving the recommended amount of physical activity is a greater challenge for individuals who are disabled than those who are not disabled. Battling ropes is an intense exercise that has been shown to promote elevation of heart rate and increased oxygen consumption, and may be a beneficial exercise alternative for individuals with disabilities. PURPOSE: To compare the levels of perceived exertion (RPE), oxygen consumption (VO2) and heart rate (HR) responses elicited by a treadmill (TM) or bicycle (BK) VO2 max test with a standing or sitting battling rope protocol (BRP) (TM vs. standing BRP) (BK vs. seated BRP). To examine the associations between the peak RPE, HR and VO2 responses elicited from the two BRP with their respective VO2 max tests (TM vs. standing BRP) (BK vs. seated BRP). METHODS: Forty healthy subjects, 24 females and 16 males, mean age 24.83 years, performed either a ramped VO2 max TM or BK test (respiratory exchange ratio \u3e 1.0). At least 3 days later, the subjects who performed the ramped TM test did the standing BRP, and the subjects who performed the ramped BK test did the sitting BRP. Each BRP (standing and sitting) consisted of 15 seconds of double arm swings, followed by 45 seconds of rest for 10 rounds. The highest recorded RPE, VO2 and HR values (RPE peak, VO2 peak and HR peak, respectively) were recorded after each round. RESULTS: Metabolic responses were significantly lower for the BRP in both sitting (VO2 peak and HR peak: p \u3c .001) and standing (VO2 peak and HR peak: p \u3c .001) as compared to the HR max and VO2 max values derived from the BK and TM tests. The subjects’ RPE peak elicited from the standing and seated BRP was not significantly different from the RPE peak of the TM or BK, respectively. The BRP produced a VO2 peak that was 71.87% (sitting) and 68.37% (standing) of the subjects\u27 VO2 max assessed via the BK and TM protocol, respectively. Moderate correlations were found between the VO2 during the seated (r = .61; p =.003) and standing (r =.43; p =.03) BRP and the BK and TM VO2 max tests, respectively. The HR peak obtained from the BRP done in sitting (r = .52; p = .009) and standing (r = .67; p = .001) had a moderate correlation with the HR max derived from the BK and TM tests. A moderate correlation was found between the RPE peak elicited from the seated BRP and the BK (r = .55; p = .008). A weak correlation was found between the RPE peak from the standing BRP and the TM (r = .18; p = .22). CONCLUSION: Both the seated and standing BRPs demonstrated the capacity to produce acute metabolic responses that may enhance aerobic capacity. Battling ropes may be a low cost, accessible option to improve cardiovascular endurance for individuals who cannot stand or move their lower extremities in a rhythmic manner to conduct aerobic exercise