Near-Optimal Induced Universal Graphs for Bounded Degree Graphs

A graph $U$ is an induced universal graph for a family $F$ of graphs if every graph in $F$ is a vertex-induced subgraph of $U$. For the family of all undirected graphs on $n$ vertices Alstrup, Kaplan, Thorup, and Zwick [STOC 2015] give an induced universal graph with $O\!\left(2^{n/2}\right)$ vertices, matching a lower bound by Moon [Proc. Glasgow Math. Assoc. 1965]. Let $k= \lceil D/2 \rceil$. Improving asymptotically on previous results by Butler [Graphs and Combinatorics 2009] and Esperet, Arnaud and Ochem [IPL 2008], we give an induced universal graph with $O\!\left(\frac{k2^k}{k!}n^k \right)$ vertices for the family of graphs with $n$ vertices of maximum degree $D$. For constant $D$, Butler gives a lower bound of $\Omega\!\left(n^{D/2}\right)$. For an odd constant $D\geq 3$, Esperet et al. and Alon and Capalbo [SODA 2008] give a graph with $O\!\left(n^{k-\frac{1}{D}}\right)$ vertices. Using their techniques for any (including constant) even values of $D$ gives asymptotically worse bounds than we present. For large $D$, i.e. when $D = \Omega\left(\log^3 n\right)$, the previous best upper bound was ${n\choose\lceil D/2\rceil} n^{O(1)}$ due to Adjiashvili and Rotbart [ICALP 2014]. We give upper and lower bounds showing that the size is ${\lfloor n/2\rfloor\choose\lfloor D/2 \rfloor}2^{\pm\tilde{O}\left(\sqrt{D}\right)}$. Hence the optimal size is $2^{\tilde{O}(D)}$ and our construction is within a factor of $2^{\tilde{O}\left(\sqrt{D}\right)}$ from this. The previous results were larger by at least a factor of $2^{\Omega(D)}$. As a part of the above, proving a conjecture by Esperet et al., we construct an induced universal graph with $2n-1$ vertices for the family of graphs with max degree $2$. In addition, we give results for acyclic graphs with max degree $2$ and cycle graphs. Our results imply the first labeling schemes that for any $D$ are at most $o(n)$ bits from optimal

Long COVID symptoms and duration in SARS-CoV-2 positive children - a nationwide cohort study

Most children have a mild course of acute COVID-19. Only few mainly non-controlled studies with small sample size have evaluated long-term recovery from SARS-CoV-2 infection in children. The aim of this study was to evaluate symptoms and duration of ‘long COVID’ in children. A nationwide cohort study of 37,522 children aged 0–17 years with RT-PCR verified SARS-CoV-2 infection (response rate 44.9%) and a control group of 78,037 children (response rate 21.3%). An electronic questionnaire was sent to all children from March 24th until May 9th, 2021. Symptoms lasting > 4 weeks were common among both SARS-CoV-2 children and controls. However, SARS-CoV-2 children aged 6–17 years reported symptoms more frequently than the control group (percent difference 0.8%). The most reported symptoms among pre-school children were fatigue Risk Difference (RD) 0.05 (CI 0.04–0.06), loss of smell RD 0.01 (CI 0.01–0.01), loss of taste RD 0.01 (CI 0.01–0.02) and muscle weakness RD 0.01 (CI 0.00–0.01). Among school children the most significant symptoms were loss of smell RD 0.12 (CI 0.12–0.13), loss of taste RD 0.10 (CI 0.09–0.10), fatigue RD 0.05 (CI 0.05–0.06), respiratory problems RD 0.03 (CI 0.03–0.04), dizziness RD 0.02 (CI 0.02–0.03), muscle weakness RD 0.02 (CI 0.01–0.02) and chest pain RD 0.01 (CI 0.01–0.01). Children in the control group experienced significantly more concentration difficulties, headache, muscle and joint pain, cough, nausea, diarrhea and fever than SARS-CoV-2 infected. In most children ‘long COVID’ symptoms resolved within 1–5 months. Conclusions: Long COVID in children is rare and mainly of short duration. SUPPLEMENTARY INFORMATION: The online version contains supplementary material available at 10.1007/s00431-021-04345-z

Breakdown of disordered media by surface loads

We model an interface layer connecting two parts of a solid body by N parallel elastic springs connecting two rigid blocks. We load the system by a shear force acting on the top side. The springs have equal stiffness but are ruptured randomly when the load reaches a critical value. For the considered system, we calculate the shear modulus, G, as a function of the order parameter, \phi, describing the state of damage, and also the ``spalled'' material (burst) size distribution. In particular, we evaluate the relation between the damage parameter and the applied force and explore the behaviour in the vicinity of material breakdown. Using this simple model for material breakdown, we show that damage, caused by applied shear forces, is analogous to a first-order phase transition. The scaling behaviour of G with \phi is explored analytically and numerically, close to \phi=0 and \phi=1 and in the vicinity of \phi_c, when the shear load is close but below the threshold force that causes material breakdown. Our model calculation represents a first approximation of a system subject to wear induced loads.Comment: 15 pages, 7 figure

Impaired transmission in the corticospinal tract and gait disability in spinal cord injured persons

Rehabilitation following spinal cord injury is likely to depend on recovery of corticospinal systems. Here we investigate whether transmission in the corticospinal tract may explain foot drop (inability to dorsiflex ankle) in persons with spinal cord lesion. The study was performed in 24 persons with incomplete spinal cord lesion (C1 to L1) and 15 healthy controls. Coherence in the 10- to 20-Hz frequency band between paired tibialis anterior muscle (TA) electromyographic recordings obtained in the swing phase of walking, which was taken as a measure of motor unit synchronization. It was significantly correlated with the degree of foot drop, as measured by toe elevation and ankle angle excursion in the first part of swing. Transcranial magnetic stimulation was used to elicit motor-evoked potentials (MEPs) in the TA. The amplitude of the MEPs at rest and their latency during contraction were correlated to the degree of foot drop. Spinal cord injured participants who exhibited a large foot drop had little or no MEP at rest in the TA muscle and had little or no coherence in the same muscle during walking. Gait speed was correlated to foot drop, and was the lowest in participants with no MEP at rest. The data confirm that transmission in the corticospinal tract is of importance for lifting the foot during the swing phase of human gait

Treatment with glucagon-like peptide-1 receptor agonists and incidence of dementia:Data from pooled double-blind randomized controlled trials and nationwide disease and prescription registers

INTRODUCTION: People with type 2 diabetes have increased risk of dementia. Glucagon‐like peptide‐1 (GLP‐1) receptor agonists (RAs) are among the promising therapies for repurposing as a treatment for Alzheimer's disease; a key unanswered question is whether they reduce dementia incidence in people with type 2 diabetes. METHODS: We assessed exposure to GLP‐1 RAs in patients with type 2 diabetes and subsequent diagnosis of dementia in two large data sources with long‐term follow‐up: pooled data from three randomized double‐blind placebo‐controlled cardiovascular outcome trials (15,820 patients) and a nationwide Danish registry‐based cohort (120,054 patients). RESULTS: Dementia rate was lower both in patients randomized to GLP‐1 RAs versus placebo (hazard ratio [HR]: 0.47 (95% confidence interval [CI]: 0.25–0.86) and in the nationwide cohort (HR: 0.89; 95% CI: 0.86–0.93 with yearly increased exposure to GLP‐1 RAs). DISCUSSION: Treatment with GLP‐1 RAs may provide a new opportunity to reduce the incidence of dementia in patients with type 2 diabetes