Guessing Numbers of Odd Cycles

For a given number of colours, $s$, the guessing number of a graph is the base $s$ logarithm of the size of the largest family of colourings of the vertex set of the graph such that the colour of each vertex can be determined from the colours of the vertices in its neighbourhood. An upper bound for the guessing number of the $n$-vertex cycle graph $C_n$ is $n/2$. It is known that the guessing number equals $n/2$ whenever $n$ is even or $s$ is a perfect square \cite{Christofides2011guessing}. We show that, for any given integer $s\geq 2$, if $a$ is the largest factor of $s$ less than or equal to $\sqrt{s}$, for sufficiently large odd $n$, the guessing number of $C_n$ with $s$ colours is $(n-1)/2 + \log_s(a)$. This answers a question posed by Christofides and Markstr\"{o}m in 2011 \cite{Christofides2011guessing}. We also present an explicit protocol which achieves this bound for every $n$. Linking this to index coding with side information, we deduce that the information defect of $C_n$ with $s$ colours is $(n+1)/2 - \log_s(a)$ for sufficiently large odd $n$. Our results are a generalisation of the $s=2$ case which was proven in \cite{bar2011index}.Comment: 16 page

HEAD MOVEMENT IN OVERARM THROWING FOR CHILDREN WITH VARYING LEVELS OF MOTOR PROFICIENCY

The purpose of this study was to investigate head motion of children of varying levels of motor ability performing an overarm throw towards a forward facing target. Ten ten-yearold children were analysed using three-dimensional veideographic techniques. Angular motion of the head about its three axes was determined with respect to the external reference frame and with respect to the trunk. It was found that all subjects stabilised their heads during the performance of the throw. The angular velocity of the head immediately prior to release small despite large trunk angular velocity

HEAD STABILISATION DURING RUNNING IN PLACE OF CHILDREN WITH VARYING MOTOR PROFICIENCY LEVELS

The purpose of this study was to investigate head motion in children during stationary running. Participants in this study underwent a running proficiency test based on the Test for Gross Motor Proficiency (TGMD), and then underwent a one-minute trial of running on the spot while being videotaped. Head stabilisation for all subjects was found to remain well within the functional range of the vestibulo-occular reflex (VOR) and no significant relationship was found between running proficiency and head angular velocity. Proficient subjects moved the head independently of the trunk

Multiple prior concussions are associated with symptoms in high school athletes

Objectives: The purpose of this study was to evaluate the association of prior concussion on baseline computerized neurocognitive testing in a large cohort of high school athletes. Methods: This is a retrospective cohort study of student athletes from 49 Maine High Schools in 2010 who underwent baseline computerized neurocognitive evaluation with Immediate Post-Concussion Assessment and Cognitive Testing (ImPACT®). As part of the ImPACT®, subjects reported a prior history of concussion as well as demographic information and a symptom questionnaire. We used linear regression to evaluate the association of prior concussion with baseline: (1) ImPACT® composite scores; and (2) symptom scores. Results: Six thousand seventy-five subjects were included in the study, of whom 57% were boys. The majority of student athletes (85.3%) reported no prior history of concussion while 4.6% reported having sustained two or more prior concussions. On simple linear regression, increasing number of concussions was related to worse performance in verbal memory (P = 0.039) and greater symptoms scores (P < 0.001). On multivariate modeling, only the association with baseline symptoms remained (P < 0.001). Other factors associated with baseline symptom reporting in the multivariate model included mental health history, headache/migraine history, gender, developmental and/or learning problems, and number of prior concussions. Interpretation In this large-scale, retrospective survey study, history of multiple prior concussions was associated with higher symptom burden but not baseline computerized neurocognitive testing. The association between baseline symptom reporting and clinical and demographic factors was greater than the association with a history of multiple concussions

Combination of Statistical Shape Modeling and Statistical Parametric Mapping to Quantify Cartilage Contact Mechanics in Hip Dysplasia

Finite element models can predict subject-specific chondrolabral stresses and help to elucidate the effect of under-coverage and incongruency of the hip joint in patients with dysplasia. However, complex stress patterns are difficult to generalize and evaluate statistically. With an established correspondence across shapes from statistical shape modeling (SSM), statistical parametric mapping (SPM) allows for evaluation of local variability while preserving model subject-specificity. Herein, we evaluated the combined application of SSM and SPM to compare cartilage contact stress between control subjects and patients with dysplasia. Previously published hip joint contact stresses were mapped onto chondrolabral surface meshes and incorporated into an SSM. Principal component analysis (PCA) quantified shape variation. Contact stress values from heel-strike of stair ascent (AH), stair descent (DH), and level walking (WH) and mid-stance of level walking (WM) were evaluated. Using SPM, regions of significant contact stress variation were identified based on test statistics from general linear statistical models and corrected for multiple comparisons using Gaussian random fields. Shape differences of the femoral and acetabular cartilage with labrum were captured by two and one PCA mode, respectively. Contact stress differences were observed in anterosuperior regions of the femoral cartilage for AH and DH and of the acetabular cartilage for DH and WM (Figure 1). The SPM identified regions of varied contact stresses were small and likely would have been diluted through averaging or region-splitting using traditional analysis methods. The combined application of SSM and SPM provides a method to generalize and statistically-compare subject-specific mechanics and joint morphology

Categorization of compensatory motions in transradial myoelectric prosthesis users

Background: Prosthesis users perform various compensatory motions to accommodate for the loss of the hand and wrist as well as the reduced functionality of a prosthetic hand. Objectives: Investigate different compensation strategies that are performed by prosthesis users. Study Design: Comparative analysis Methods: 20 able-bodied subjects and 4 prosthesis users performed a set of bimanual activities. Movements of the trunk and head were recorded using a motion capture system, and a digital video recorder. Clinical motion angles were calculated to assess the compensatory motions made by the prosthesis users. The video recording also assisted in visually identifying the compensations. Results: Compensatory motions by the prosthesis users were evident in the tasks performed (slicing and stirring activities) as compared to the benchmark of able-bodied subjects. Compensations took the form of a measured increase in range of motion, an observed adoption of a new posture during task execution, and pre-positioning of items in the workspace prior to initiating a given task. Conclusion: Compensatory motions were performed by prosthesis users during the selected tasks. These can be categorized into three different types of compensations