Non-self-similar blow-up in the heat flow for harmonic maps in higher dimensions

We analyze the finite-time blow-up of solutions of the heat flow for $k$-corotational maps $\mathbb R^d\to S^d$. For each dimension $d>2+k(2+2\sqrt{2})$ we construct a countable family of blow-up solutions via a method of matched asymptotics by glueing a re-scaled harmonic map to the singular self-similar solution: the equatorial map. We find that the blow-up rates of the constructed solutions are closely related to the eigenvalues of the self-similar solution. In the case of $1$-corotational maps our solutions are stable and represent the generic blow-up.Comment: 26 pages, 5 figure

Association of socio-economic and demographic factors with physical activity of males and females aged 20–69 years

Objective To assess the physical activity of working residents of Warsaw aged 20–69 years, as well as to identify the socio-demographic factors associated with their levels of physical activity. Material and Methods The study involved 2,544 working residents of Warsaw aged 20–69 years. The short version of the IPAQ was applied and four physical activity levels (insufficient, sufficient, augmented, high) were distinguished. The relationships between physical activity and gender, age, BMI, education, economic and martial status as well as participation in recreation were determined. Results High levels of physical activity were reached by 8% of respondents, 22% achieved augmented level, 32% were sufficiently and 32% insufficiently active. Out of 2544 studied subjects, 6% declared complete sedentariness. Females were, as compared to males, more frequently (p<0.05) insufficiently active (35.9 vs. 31.9%). In obese and overweight subjects insufficient physical activity predominated (42.9 and 36.2%, respectively) and was significantly more frequent than in subjects with normal BMI (31.0%). Moreover, the subjects living in partner relationships were significantly (p<0.05) more frequently insufficiently active than those staying single (36.3 vs. 30.3%). Respondents who declared regular participation in leisure activities were less frequently insufficiently active (20.0%) and more frequently met the criteria of sufficient (37.6%), augmented (28.0%) or high (14.4%) level of physical activity. No significant effects were found with respect to education of respondents. Conclusions Prophylactic schedules associated with the improvement of physical activity level should be addressed particularly to females, people taking up recreation occasionally or to those not involved in recreation at all, living in partner relationships, youngest (21–30 years), in obese and overweight and in the lowest economic category

Shrinkers, expanders, and the unique continuation beyond generic blowup in the heat flow for harmonic maps between spheres

Using mixed analytical and numerical methods we investigate the development of singularities in the heat flow for corotational harmonic maps from the $d$-dimensional sphere to itself for $3\leq d\leq 6$. By gluing together shrinking and expanding asymptotically self-similar solutions we construct global weak solutions which are smooth everywhere except for a sequence of times $T_1<T_2<...<T_k<\infty$ at which there occurs the type I blow-up at one of the poles of the sphere. We show that in the generic case the continuation beyond blow-up is unique, the topological degree of the map changes by one at each blow-up time $T_i$, and eventually the solution comes to rest at the zero energy constant map.Comment: 24 pages, 8 figures, minor corrections, matches published versio

Machine learning on the road to unlocking microbiota's potential for boosting immune checkpoint therapy

The intestinal microbiota is a complex and diverse ecological community that fulfills multiple functions and substantially impacts human health. Despite its plasticity, unfavorable conditions can cause perturbations leading to so-called dysbiosis, which have been connected to multiple diseases. Unfortunately, understanding the mechanisms underlying the crosstalk between those microorganisms and their host is proving to be difficult. Traditionally used bioinformatic tools have difficulties to fully exploit big data generated for this purpose by modern high throughput screens. Machine Learning (ML) may be a potential means of solving such problems, but it requires diligent application to allow for drawing valid conclusions. This is especially crucial as gaining insight into the mechanistic basis of microbial impact on human health is highly anticipated in numerous fields of study. This includes oncology, where growing amounts of studies implicate the gut ecosystems in both cancerogenesis and antineoplastic treatment outcomes. Based on these reports and first signs of clinical benefits related to microbiota modulation in human trials, hopes are rising for the development of microbiome-derived diagnostics and therapeutics. In this mini-review, we're inspecting analytical approaches used to uncover the role of gut microbiome in immune checkpoint therapy (ICT) with the use of shotgun metagenomic sequencing (SMS) data