Advancing Mathematical Activity: A Practice-Oriented View of Advanced Mathematical Thinking

The purpose of this article is to contribute to the dialogue about the notion of advanced mathematical thinking by offering an alternative characterization for this idea, namely advancing mathematical activity. We use the term advancing (versus advanced) because we emphasize the progression and evolution of students' reasoning in relation to their previous activity. We also use the term activity, rather than thinking. This shift in language reflects our characterization of progression in mathematical thinking as acts of participation in a variety of different socially or culturally situated mathematical practices. For these practices, we emphasize the changing nature of students' mathematical activity and frame the process of progression in terms of multiple layers of horizontal and vertical mathematizing.     

Connectivity and minimal distance spectral radius of graphs

In this article, we study how the distance spectral radius behaves when the graph is perturbed by grafting edges. As applications, we also determine the graph with k cut vertices (respectively, k cut edges) with the minimal distance spectral radius.     

New Interfaces and Approaches to Machine Learning When Classifying Gestures within Music

Interactive music uses wearable sensors (i.e., gestural interfaces—GIs) and biometric datasets to reinvent traditional human–computer interaction and enhance music composition. In recent years, machine learning (ML) has been important for the artform. This is because ML helps process complex biometric datasets from GIs when predicting musical actions (termed performance gestures). ML allows musicians to create novel interactions with digital media. Wekinator is a popular ML software amongst artists, allowing users to train models through demonstration. It is built on the Waikato Environment for Knowledge Analysis (WEKA) framework, which is used to build supervised predictive models. Previous research has used biometric data from GIs to train specific ML models. However, previous research does not inform optimum ML model choice, within music, or compare model performance. Wekinator offers several ML models. Thus, we used Wekinator and the Myo armband GI and study three performance gestures for piano practice to solve this problem. Using these, we trained all models in Wekinator and investigated their accuracy, how gesture representation affects model accuracy and if optimisation can arise. Results show that neural networks are the strongest continuous classifiers, mapping behaviour differs amongst continuous models, optimisation can occur and gesture representation disparately affects model mapping behaviour; impacting music practice.     

Diffusion tensor parameters and principal eigenvector coherence: Relation to b-value intervals and field strength

Diffusion-weighted MRI images acquired at b-value greater than 1000 s mm^− 2 measure the diffusion of a restricted pool of water molecules. High b-value images are accompanied by a reduction in signal-to-noise ratio (SNR) due to the application of large diffusion gradients. By fitting the diffusion tensor model to data acquired at incremental b-value intervals, we determined the effect of SNR on tensor parameters in normal human brains, in vivo. In addition, we also investigated the impact of field strength on the diffusion tensor model. Data were acquired at 1.5 and 3 T, at b-values 0, 1000, 2000 and 3000 s mm^− 2 in twenty diffusion-sensitised directions. Fractional anisotropy (FA), mean diffusivity (MD) and principal eigenvector coherence (κ) were calculated from diffusion tensors fitted between datasets with b-values 0–1000, 0–2000, 0–3000, 1000–2000 and 2000–3000 s mm^− 2. Field strength and b-value effects on diffusion parameters were analysed in white and grey matter regions of interest. Decreases in FA, κ and MD were found with increasing b-value in white matter. Univariate analysis showed a significant increase in FA with increasing field strength in highly organised white matter. These results suggest there are significant differences in diffusion parameters at 1.5 and 3 T and that the optimal results, in terms of the highest values of FA in white matter, are obtained at 3 T with a maximum b = 1000 s mm^− 2.     

Evaluation of aortic stenosis severity using 4D flow jet shear layer detection for the measurement of valve effective orifice area

Aims

The objective of this study was to evaluate the potential of 4D flow MRI to assess valve effective orifice area (EOA) in patients with aortic stenosis as determined by the jet shear layer detection (JSLD) method.

Methods and Results

An in-vitro stenosis phantom was used for validation and in-vivo imaging was performed in 10 healthy controls and 40 patients with aortic stenosis. EOA was calculated by the JSLD method using standard 2D phase contrast MRI (PC-MRI) and 4D flow MRI measurements (EOA_JSLD-2D and EOA_JSLD-4D, respectively). As a reference standard, the continuity equation was used to calculate EOA (EOA_CE) with the 2D PC-MRI velocity field and compared to the EOA_JSLD measurements. The in-vitro results exhibited excellent agreement between flow theory (EOA = 0.78 cm²) and experimental measurement (EOA_JSLD-4D = 0.78 ± 0.01 cm²) for peak velocities ranging from 0.9 to 3.7 m/s. In-vivo results showed good correlation and agreement between EOA_JSLD-2D and EOA_CE (r = 0.91, p < 0.001; bias: − 0.01 ± 0.38 cm²; agreement limits: 0.75 to − 0.77 cm²), and between EOA_JSLD-4D and EOA_CE (r = 0.95, p < 0.001; bias: − 0.09 ± 0.26 cm²; limits: 0.43 to − 0.62 cm²).

Conclusion

This study demonstrates the feasibility of measuring EOA_JSLD using 4D flow MRI. The technique allows for optimization of the EOA measurement position by visualizing the 3D vena contracta, and avoids potential sources of EOA_CE measurement variability.     

High P–T Raman study of transitions in relaxor multiferroic Pb(Fe0.5Nb0.5)O3

The vibrational and structural properties of Pb(Fe_0.5Nb_0.5)O₃ have been investigated using Raman spectroscopy up to 40 GPa at 300 K and from 300 to 415 K at selected pressures. The measurements reveal three phase transitions, at 5.5, 8.7, and 24 GPa at room temperature. The temperature dependences of the spectra indicate transitions at 1.5 GPa, at 335 and 365 K. The results are consistent with the appearance of an intermediate tetragonal P4mm phase between the ferroelectric R3m and paraelectric Pm‐3m phases. A P–T phase diagram is proposed that allows further insight into the magnetoelectric coupling present in this material. Copyright © 2015 John Wiley & Sons, Ltd.     

Representation in the (Artificial) Immune System

Much of contemporary research in Artificial Immune Systems (AIS) has partitioned into either algorithmic machine learning and optimisation, or, modelling biologically plausible dynamical systems, with little overlap between. We propose that this dichotomy is somewhat to blame for the lack of significant advancement of the field in either direction and demonstrate how a simplistic interpretation of Perelson’s shape-space formalism may have largely contributed to this dichotomy. In this paper, we motivate and derive an alternative representational abstraction. To do so we consider the validity of shape-space from both the biological and machine learning perspectives. We then take steps towards formally integrating these perspectives into a coherent computational model of notions such as life-long learning, degeneracy, constructive representations and contextual recognition—rhetoric that has long inspired work in AIS, while remaining largely devoid of operational definition.     

Enhanced Ratiometric pH Sensing Using SNAFL-2 on Silver Island Films: Metal-enhanced Fluorescence Sensing

We have explored the opportunities for enhanced ratiometric pH sensing using the well-known carboxy seminaphthofluorescein (SNAFL-2) and silver island films (SiFs). Our results show that the metallic surfaces can provide up to a 40-fold increase in probe fluorescence intensity as compared to nonmetallic surfaces with the same probe coverage. However, while the S/N is significantly better for pH sensing, the emission wavelength ratiometric values are similar to that obtained in solution, due to the fact that the emission of both the acidic and basic forms of the probe are enhanced to similar extents. To the best of our knowledge this is the first report of enhanced ratiometric fluorescence sensing on metallic surfaces.