The existence of small prime gaps in subsets of the integers

We consider the problem of finding small prime gaps in various sets of integers $\mathcal{C}$. Following the work of Goldston-Pintz-Yildirim, we will consider collections of natural numbers that are well-controlled in arithmetic progressions. Letting $q_n$ denote the $n$-th prime in $\mathcal{C}$, we will establish that for any small constant $\epsilon>0$, the set $\left\{q_n| q_{n+1}-q_n \leq \epsilon \log n \right\}$ constitutes a positive proportion of all prime numbers. Using the techniques developed by Maynard and Tao we will also demonstrate that $\mathcal{C}$ has bounded prime gaps. Specific examples, such as the case where $\mathcal{C}$ is an arithmetic progression have already been studied and so the purpose of this paper is to present results for general classes of sets

Random waves on T3\mathbb{T}^3: nodal area variance and lattice point correlations

We consider the ensemble of random Gaussian Laplace eigenfunctions on $\mathbb{T}^3=\mathbb{R}^3/\mathbb{Z}^3$ (`$3d$ arithmetic random waves'), and study the distribution of their nodal surface area. The expected area is proportional to the square root of the eigenvalue, or `energy', of the eigenfunction. We show that the nodal area variance obeys an asymptotic law. The resulting asymptotic formula is closely related to the angular distribution and correlations of lattice points lying on spheres.Comment: To appear in IMR

Performance Measurement Under Increasing Environmental Uncertainty In The Context of Interval Type-2 Fuzzy Logic Based Robotic Sailing

Performance measurement of robotic controllers based on fuzzy logic, operating under uncertainty, is a subject area which has been somewhat ignored in the current literature. In this paper standard measures such as RMSE are shown to be inappropriate for use under conditions where the environmental uncertainty changes significantly between experiments. An overview of current methods which have been applied by other authors is presented, followed by a design of a more sophisticated method of comparison. This method is then applied to a robotic control problem to observe its outcome compared with a single measure. Results show that the technique described provides a more robust method of performance comparison than less complex methods allowing better comparisons to be drawn.Comment: International Conference on Fuzzy Systems 2013 (Fuzz-IEEE 2013

The State, Society, Human Rights & Health: Ethical Challenges in the Development of New Interventions.

The theme of this presentation is that major impediments to the development of new interventions to improve global health comprise the combination of (1) an inadequate value system that is heavily dominated by selfish individualism, (2) an excessively downstream focus on health and (3) overriding emphasis on market forces. Moral imagination is required to move beyond the current impasse in which the lives of some seem to be of infinite value while the lives of others are apparently dispensable. A broader discourse on ethics and human rights coupled to demonstration of high moral standards by influential nations could facilitate the introduction of new interventions with the prospect of greatly improving population health.The theme of this presentation is that major impediments to the development of new interventions to improve global health comprise the combination of (1) an inadequate value system that is heavily dominated by selfish individualism, (2) an excessively downstream focus on health and (3) overriding emphasis on market forces. Moral imagination is required to move beyond the current impasse in which the lives of some seem to be of infinite value while the lives of others are apparently dispensable. A broader discourse on ethics and human rights coupled to demonstration of high moral standards by influential nations could facilitate the introduction of new interventions with the prospect of greatly improving population health.The theme of this presentation is that major impediments to the development of new interventions to improve global health comprise the combination of (1) an inadequate value system that is heavily dominated by selfish individualism, (2) an excessively downstream focus on health and (3) overriding emphasis on market forces. Moral imagination is required to move beyond the current impasse in which the lives of some seem to be of infinite value while the lives of others are apparently dispensable. A broader discourse on ethics and human rights coupled to demonstration of high moral standards by influential nations could facilitate the introduction of new interventions with the prospect of greatly improving population health

A comparison of non-stationary, type-2 and dual surface fuzzy control

Type-1 fuzzy logic has frequently been used in control systems. However this method is sometimes shown to be too restrictive and unable to adapt in the presence of uncertainty. In this paper we compare type-1 fuzzy control with several other fuzzy approaches under a range of uncertain conditions. Interval type-2 and non-stationary fuzzy controllers are compared, along with ‘dual surface’ type-2 control, named due to utilising both the lower and upper values produced from standard interval type-2 systems. We tune a type-1 controller, then derive the membership functions and footprints of uncertainty from the type-1 system and evaluate them using a simulated autonomous sailing problem with varying amounts of environmental uncertainty. We show that while these more sophisticated controllers can produce better performance than the type-1 controller, this is not guaranteed and that selection of Footprint of Uncertainty (FOU) size has a large effect on this relative performance

Urinary p75(ECD): A prognostic, disease progression, and pharmacodynamic biomarker in ALS.

OBJECTIVE: To evaluate urinary neurotrophin receptor p75 extracellular domain (p75(ECD)) levels as disease progression and prognostic biomarkers in amyotrophic lateral sclerosis (ALS). METHODS: The population in this study comprised 45 healthy controls and 54 people with ALS, 31 of whom were sampled longitudinally. Urinary p75(ECD) was measured using an enzyme-linked immunoassay and validation included intra-assay and inter-assay coefficients of variation, effect of circadian rhythm, and stability over time at room temperature, 4°C, and repeated freeze-thaw cycles. Longitudinal changes in urinary p75(ECD) were examined by mixed model analysis, and the prognostic value of baseline p75(ECD) was explored by survival analysis. RESULTS: Confirming our previous findings, p75(ECD) was higher in patients with ALS (5.6 ± 2.2 ng/mg creatinine) compared to controls (3.6 ± 1.4 ng/mg creatinine, p < 0.0001). Assay reproducibility was high, with p75(ECD) showing stability across repeated freeze-thaw cycles, at room temperature and 4°C for 2 days, and no diurnal variation. Urinary p75(ECD) correlated with the revised ALS Functional Rating Scale at first evaluation (r = -0.44, p = 0.008) and across all study visits (r = -0.36, p < 0.0001). p75(ECD) also increased as disease progressed at an average rate of 0.19 ng/mg creatinine per month (p < 0.0001). In multivariate prognostic analysis, bulbar onset (hazard ratio [HR] 3.0, p = 0.0035), rate of disease progression from onset to baseline (HR 4.4, p < 0.0001), and baseline p75(ECD) (HR 1.3, p = 0.0004) were predictors of survival. CONCLUSIONS: The assay for urinary p75(ECD) is analytically robust and shows promise as an ALS biomarker with prognostic, disease progression, and potential pharmacodynamic application. Baseline urinary p75(ECD) provides prognostic information and is currently the only biological fluid-based biomarker of disease progression

A comparison of non-stationary, type-2 and dual surface fuzzy control

Type-1 fuzzy logic has frequently been used in control systems. However this method is sometimes shown to be too restrictive and unable to adapt in the presence of uncertainty. In this paper we compare type-1 fuzzy control with several other fuzzy approaches under a range of uncertain conditions. Interval type-2 and non-stationary fuzzy controllers are compared, along with ‘dual surface’ type-2 control, named due to utilising both the lower and upper values produced from standard interval type-2 systems. We tune a type-1 controller, then derive the membership functions and footprints of uncertainty from the type-1 system and evaluate them using a simulated autonomous sailing problem with varying amounts of environmental uncertainty. We show that while these more sophisticated controllers can produce better performance than the type-1 controller, this is not guaranteed and that selection of Footprint of Uncertainty (FOU) size has a large effect on this relative performance