Integer colorings with forbidden rainbow sums

For a set of positive integers $A \subseteq [n]$, an $r$-coloring of $A$ is rainbow sum-free if it contains no rainbow Schur triple. In this paper we initiate the study of the rainbow Erd\H{o}s-Rothchild problem in the context of sum-free sets, which asks for the subsets of $[n]$ with the maximum number of rainbow sum-free $r$-colorings. We show that for $r=3$, the interval $[n]$ is optimal, while for $r\geq8$, the set $[\lfloor n/2 \rfloor, n]$ is optimal. We also prove a stability theorem for $r\geq4$. The proofs rely on the hypergraph container method, and some ad-hoc stability analysis.Comment: 20 page

Gait phase classification for in-home gait assessment

With growing ageing population, acquiring joint measurements with sufficient accuracy for reliable gait assessment is essential. Additionally, the quality of gait analysis relies heavily on accurate feature selection and classification. Sensor-driven and one-camera optical motion capture systems are becoming increasingly popular in the scientific literature due to their portability and cost-efficacy. In this paper, we propose 12 gait parameters to characterise gait patterns and a novel gait-phase classifier, resulting in comparable classification performance with a state-of-the-art multi-sensor optical motion system. Furthermore, a novel multi-channel time series segmentation method is proposed that maximizes the temporal information of gait parameters improving the final classification success rate after gait event reconstruction. The validation, conducted over 126 experiments on 6 healthy volunteers and 9 stroke patients with handlabelled ground truth gait phases, demonstrates high gait classification accuracy

Human upper limb motion analysis for post-stroke impairment assessment using video analytics

Stroke is a worldwide healthcare problem which often causes long-term motor impairment, handicap, and disability. Optical motion analysis systems are commonly used for impairment assessment due to high accuracy. However, the requirement of equipment-heavy and large laboratory space together with operational expertise, makes these systems impractical for local clinic and home use. We propose an alternative, cost-effective and portable, decision support system for optical motion analysis, using a single camera. The system relies on detecting and tracking markers attached to subject's joints, data analytics for calculating relevant rehabilitation parameters, visualization, and robust classification based on graph-based signal processing. Experimental results show that the proposed decision support system has the potential to offer stroke survivors and clinicians an alternative, affordable, accurate and convenient impairment assessment option suitable for home healthcare and tele-rehabilitation

Shift-enabled graphs : graphs where shift-invariant filters are representable as polynomials of shift operations

In digital signal processing, a shift-invariant filter can be represented as a polynomial expansion of a shift operation, that is, the Z-transform representation. When extended to Graph Signal Processing (GSP), this would mean that a shift-invariant graph filter can be represented as a polynomial of the shift matrix of the graph. Prior work shows that this holds under the shift-enabled condition that the characteristic and minimum polynomials of the shift matrix are identical. While the shiftenabled condition is often ignored in the literature, this letter shows that this condition is essential for the following reasons. First, we prove that this condition is not just sufficient but also necessary for any shift-invariant filter to be representable by the shift matrix. Moreover, we provide a counterexample showing that given a filter that commutes with a non-shift-enabled graph, it is generally impossible to convert the graph to a shift-enabled graph with a shift matrix still commuting with the original filter. The result provides a deeper understanding of shift-invariant filters when applied in GSP and shows that further investigation of shift-enabled graphs is needed to make them applicable to practical scenarios

Making sense of Wnt signaling—linking hair cell regeneration to development

Wnt signaling is a highly conserved pathway crucial for development and homeostasis of multicellular organisms. Secreted Wnt ligands bind Frizzled receptors to regulate diverse processes such as axis patterning, cell division, and cell fate specification. They also serve to govern self-renewal of somatic stem cells in several adult tissues. The complexity of the pathway can be attributed to the myriad of Wnt and Frizzled combinations as well as its diverse context-dependent functions. In the developing mouse inner ear, Wnt signaling plays diverse roles, including specification of the otic placode and patterning of the otic vesicle. At later stages, its activity governs sensory hair cell specification, cell cycle regulation, and hair cell orientation. In regenerating sensory organs from non-mammalian species, Wnt signaling can also regulate the extent of proliferative hair cell regeneration. This review describes the current knowledge of the roles of Wnt signaling and Wnt-responsive cells in hair cell development and regeneration. We also discuss possible future directions and the potential application and limitation of Wnt signaling in augmenting hair cell regeneration