Optimal False Discovery Control of Minimax Estimator

In the analysis of high dimensional regression models, there are two important objectives: statistical estimation and variable selection. In literature, most works focus on either optimal estimation, e.g., minimax $L_2$ error, or optimal selection behavior, e.g., minimax Hamming loss. However in this study, we investigate the subtle interplay between the estimation accuracy and selection behavior. Our result shows that an estimator's $L_2$ error rate critically depends on its performance of type I error control. Essentially, the minimax convergence rate of false discovery rate over all rate-minimax estimators is a polynomial of the true sparsity ratio. This result helps us to characterize the false positive control of rate-optimal estimators under different sparsity regimes. More specifically, under near-linear sparsity, the number of yielded false positives always explodes to infinity under worst scenario, but the false discovery rate still converges to 0; under linear sparsity, even the false discovery rate doesn't asymptotically converge to 0. On the other side, in order to asymptotically eliminate all false discoveries, the estimator must be sub-optimal in terms of its convergence rate. This work attempts to offer rigorous analysis on the incompatibility phenomenon between selection consistency and rate-minimaxity observed in the high dimensional regression literature

Effects of Low-Level Laser Therapy and Eccentric Exercises in the Treatment of Patellar Tendinopathy

The study aims to investigate if low-level laser therapy (LLLT) combined with eccentric exercises could more effectively treat patellar tendinopathy than LLLT alone and eccentric exercises alone. Twenty-one patients with patellar tendinopathy were randomized to three groups: laser alone, exercise alone, or laser plus exercise, with seven in each group. Laser irradiations were administered at the inferior pole of the patella and the two acupoints of Extra 36 (Xiyan) with the intensity of 1592 mW/cm2. Eccentric training program consisted of three sets of 15 repetitions of unilateral squat on level ground. All patients received six treatments per week for four weeks. Knee pain and function and quadriceps muscle strength and endurance were evaluated at baseline and the end of treatment. After the 4-week intervention, all groups showed significant improvements in all the outcomes (P<0.01). The laser + exercise group had significantly greater improvements in all the outcomes than the other two groups (P<0.05), except nonsignificant difference in pain relief between the laser + exercise group and the laser group. In conclusion, LLLT combined with eccentric exercises is superior to LLLT alone and eccentric exercises alone to reduce pain and improve function in patients with patellar tendinopathy

Strangelets at finite temperature: nucleon emission rates, interface and shell effects

We investigate the properties of strangelets at finite temperature $T$, where an equivparticle model is adopted with both the linear confinement and leading-order perturbative interactions accounted for using density-dependent quark masses. The shell effects are examined by solving the Dirac equations for quarks in the mean-field approximation, which diminish with temperature as the occupation probability of each single-particle levels fixed by the Fermi-Dirac statistics, i.e., shell dampening. Consequently, instead of decreasing with temperature, the surface tension extracted from a liquid-drop formula increases with $T$ until reaching its peak at $T\approx 20$-40 MeV with vanishing shell corrections, where the formula roughly reproduces the free energy per baryon of all strangelets. The curvature term, nevertheless, decreases with $T$ despite the presence of shell effects. The neutron and proton emission rates are fixed microscopically according to the external nucleon gas densities that are in equilibrium with strangelets, which generally increase with $T$ ($\lesssim 50$ MeV) for stable strangelets but decrease for those that are unstable against nucleon emission at $T=0$. The energy, free energy, entropy, charge-to-mass ratio, strangeness per baryon, and root-mean-square radius of $\beta$-stable strangelets obtained with various parameter sets are presented as well. The results indicated in this work are useful for understanding the products of binary compact star mergers and heavy-ion collisions

Expression quantitative trait locus studies in the era of single-cell omics

Genome-wide association studies have revealed that the regulation of gene expression bridges genetic variants and complex phenotypes. Profiling of the bulk transcriptome coupled with linkage analysis (expression quantitative trait locus (eQTL) mapping) has advanced our understanding of the relationship between genetic variants and gene regulation in the context of complex phenotypes. However, bulk transcriptomics has inherited limitations as the regulation of gene expression tends to be cell-type-specific. The advent of single-cell RNA-seq technology now enables the identification of the cell-type-specific regulation of gene expression through a single-cell eQTL (sc-eQTL). In this review, we first provide an overview of sc-eQTL studies, including data processing and the mapping procedure of the sc-eQTL. We then discuss the benefits and limitations of sc-eQTL analyses. Finally, we present an overview of the current and future applications of sc-eQTL discoveries