Averages Along the Primes: Improving and Sparse Bounds

Consider averages along the prime integers ℙ given by{\mathcal{A}_N}f(x) = {N^{ - 1}}\sum\limits_{p \in \mathbb{P}:p \le N} {(\log p)f(x - p).} These averages satisfy a uniform scale-free ℓp-improving estimate. For all 1 < p < 2, there is a constant Cp so that for all integer N and functions f supported on [0, N], there holds{N^{ - 1/p'}}{\left\| {{\mathcal{A}_N}f} \right\|_{\ell p'}} \le {C_p}{N^{ - 1/p}}{\left\| f \right\|_{\ell p}}.The maximal function 𝒜*f = supN|𝒜N f | satisfies (p, p) sparse bounds for all 1 < p < 2. The latter are the natural variants of the scale-free bounds. As a corollary, 𝒜* is bounded on ℓp(w), for all weights w in the Muckenhoupt 𝒜p class. No prior weighted inequalities for 𝒜* were known.

Using the “Uniform Scale” to facilitate meta-analysis where exposure variables are qualitative and vary between studies – methodology, examples and software

Meta-analyses often combine covariate-adjusted effect estimates (odds ratios or relative risks) and confidence intervals relating a specified endpoint to a given exposure. Standard techniques are available to do this where the exposure is a simple presence/absence variable, or can be expressed in defined units. However, where the definition of exposure is qualitative and may vary between studies, meta-analysis is less straightforward. We introduce a new “Uniform Scale” approach allowing expression of effect estimates in a consistent manner, comparing individuals with the most and least possible exposure. In 2008, we presented methodology and made available software to obtain estimates for specific pairwise comparisons of exposure, such as any versus none, where the source paper provides estimates for multiple exposure categories, expressed relative to a common reference group. This methodology takes account of the correlation between the effect estimates for the different levels. We have now extended our software, available in Excel, SAS and R, to obtain effect estimates per unit of exposure, whether the exposure is defined or is to be expressed in the “Uniform Scale”. Examples of its use are presented.

New Bayesian Single Index Quantile Regression Based on Uniform Scale Mixture

To scale back the dimensionality while holding a lot of flexibility of a nonparametric model Wu, et al. (2010) proposed a single index conditional quantile regression model. In this paper, a new Bayesian lasso for single index quantile regression model is proposed based on a scale mixture uniform. In addition, we construct an efficient and sampling Gibbs algorithm for posterior inference based on a uniform scale mixture representation for Laplace distribution. Simulation study have considered to evaluate our proposed method compare to the existing methods. The results of simulations indicate that the new Bayesian algorithm performs well.

International Standards for the 3‐Minute Burpee Test: High‐ Intensity Motor Performance

The aim of this study was to develop international standards for evaluating strength endurance with the use of the 3‐Minute Burpee Test. The results of 3862 women (Poland – 2502, Great Britain – 500, Hungary – 412, Serbia – 448) and 5971 men (Poland – 4517, Great Britain – 500, Hungary – 451, Serbia – 503) aged 18‐25 (mean age of 20.36 ± 0.94 and 20.05 ± 1.25 y, respectively) were collated between 2004 and 2018. The students’ strength endurance was evaluated in the 3‐Minute Burpee Test. The results were expressed on a uniform scale with the 3‐sigma rule which was used to develop the T‐score scale for the 3‐Minute Burpee Test. Men completed 56.69 cycles/3 min and women – 48.84/3 min on average. The best male participant completed 82 burpees, and the best female participant – 73 burpees. The majority of male and female participants (66.71% and 68.18%, respectively) were characterized by average strength endurance in the 3‐Minute Burpee Test (range of scores: 47‐66 and 37‐60 cycles/3 min, respectively). Very good strength endurance (76‐85 and 72‐83 cycles/3 min, respectively) was noted in the smallest percentage of male and female participants (0.52% and 0.26%, respectively). Similar studies should be carried out in other countries and in different age groups to develop objective international classification standards for variously‐aged individuals.

Adaptive GridMap Building Algorithm for Mobile Robot Based on Multiway Tree

In order to solve the problem that uniform grid map occupies large storage space in the process of large-scale map creation, a new adaptive grid map building algorithm based on multiway tree is proposed. The environment map is divided into nine grids. The algorithm determines whether the grid is completely occupied, partially occupied or vacant. Then the part occupied gridsare further subdivided into smaller grids. The algorithm continues to determine whether the smaller grid is completely occupied, partially occupied or vacant.Repeat the above segmentation process until the entire map search is completed and the accuracy requirements are met. Finally, the scale of grid map does not depend on human experience but presenting adaptive characteristics. The algorithm is compared with the uniform grid method.The simulation results show that the algorithm has convergence.Andcompared with the uniform scale raster map, itgreatly saves the storage space.