Top Hat (Canada)

Head acceleration events (HAEs) can potentially have adverse consequences for athlete brain health. In sports, in which head injuries have the highest incidence, identifying strategies to reduce HAE frequency and magnitude is a priority. Neck training is a potential strategy to mitigate against the magnitude of HAEs. This two-part study aimed to (1) systematically review the literature of neck training interventions in sport and (2) undertake an expert Delphi consensus on the best practices for neck training implementation to reduce HAEs in sport. Part I: a systematic search of four databases was undertaken from the earliest records to September 2024. The PRISMA (Preferred Reporting Items for Systematic Review and Meta-Analysis) guidelines were followed, and a quality assessment was completed using a modified Downs and Black assessment tool and the GRADE (Grading of Recommendations Assessment, Development and Evaluation). Papers were eligible if they both (1) implemented a reproducible exercise intervention targeting the neck within collision, combat or motor sport, and (2) assessed outcomes relating to either: the physical profile of the neck; head/neck injury incidence; and/or HAEs. Part II: 18 international experts, with experience in research and/or applied practice of neck exercise training, concussion and/or HAEs, reviewed the part I findings before completing a three-round Delphi consensus process. Part I included 21 papers, highlighting the heterogeneity of existing interventions. Part II resulted in 57 statements coded into five categories: contextual factors (n=17), neck training periodisation (n=12), training adaptations (n=10), neck training content (n=15) and athlete adherence (n=3). This study presents recommendations for neck exercise training aiming to reduce HAEs in sport, supporting both practice and future research.

Physical and mathematical applications of the recently invented fractional Poisson probability distribution have been presented. As a physical application, a new family of quantum coherent states has been introduced and studied. As mathematical applications, we have developed the fractional generalization of Bell polynomials, Bell numbers, and Stirling numbers of the second kind. The appearance of fractional Bell polynomials is natural if one evaluates the diagonal matrix element of the evolution operator in the basis of newly introduced quantum coherent states. Fractional Stirling numbers of the second kind have been introduced and applied to evaluate the skewness and kurtosis of the fractional Poisson probability distribution function. A representation of the Bernoulli numbers in terms of fractional Stirling numbers of the second kind has been found. In the limit case when the fractional Poisson probability distribution becomes the Poisson probability distribution, all of the above listed developments and implementations turn into the well-known results of the quantum optics and the theory of combinatorial numbers.