On sets of rational functions which locally represent all of Q\mathbb{Q}

We investigate finite sets of rational functions $\{ f_{1},f_{2}, \dots, f_{r} \}$ defined over some number field $K$ satisfying that any $t_{0} \in K$ is a $K_{p}$-value of one of the functions $f_{i}$ for almost all primes $p$ of $K$. We give strong necessary conditions on the shape of functions appearing in a minimal set with this property, as well as numerous concrete examples showing that these necessary conditions are in a way also close to sufficient. We connect the problem to well-studied concepts such as intersective polynomials and arithmetically exceptional functions

Galois groups of (n0)+(n1)X+…+(n6)X6\binom{n}{0} + \binom{n}{1} X + \ldots + \binom{n}{6} X^6

We show that the Galois group of the polynomial in the title is isomorphic to the full symmetric group on six symbols for all but finitely many $n$. This complements earlier work of Filaseta and Moy, who studied Galois groups of $\binom{n}{0} + \binom{n}{1} X + \ldots + \binom{n}{k} X^k$ for more general pairs $(n,k)$, but had to admit a possibly infinite exceptional set specifically for $k=6$ of at most logarithmic growth in $n$. The proof rests upon invoking Faltings' theorem on a suitable fibration of Galois resolvents to show that this exceptional set is, in fact, finite.Comment: 8 page

On polynomials with roots modulo almost all primes

Call a monic integer polynomial exceptional if it has a root modulo all but a finite number of primes, but does not have an integer root. We classify all irreducible monic integer polynomials $h$ for which there is an irreducible monic quadratic $g$ such that the product $gh$ is exceptional. We construct exceptional polynomials with all factors of the form $X^{p}-b$, $p$ prime and $b$ square free.Comment: 11 page

Large Subsets of Zmn\mathbb{Z}_m^n without Arithmetic Progressions

For integers $m$ and $n$, we study the problem of finding good lower bounds for the size of progression-free sets in $(\mathbb{Z}_{m}^{n},+)$. Let $r_{k}(\mathbb{Z}_{m}^{n})$ denote the maximal size of a subset of $\mathbb{Z}_{m}^{n}$ without arithmetic progressions of length $k$ and let $P^{-}(m)$ denote the least prime factor of $m$. We construct explicit progression-free sets and obtain the following improved lower bounds for $r_{k}(\mathbb{Z}_{m}^{n})$: If $k\geq 5$ is odd and $P^{-}(m)\geq (k+2)/2$, then $$r_k(\mathbb{Z}_m^n) \gg_{m,k} \frac{\bigl\lfloor \frac{k-1}{k+1}m +1\bigr\rfloor^{n}}{n^{\lfloor \frac{k-1}{k+1}m \rfloor/2}}.$$ If $k\geq 4$ is even, $P^{-}(m) \geq k$ and $m \equiv -1 \bmod k$, then $$r_{k}(\mathbb{Z}_{m}^{n}) \gg_{m,k} \frac{\bigl\lfloor \frac{k-2}{k}m + 2\bigr\rfloor^{n}}{n^{\lfloor \frac{k-2}{k}m + 1\rfloor/2}}.$$ Moreover, we give some further improved lower bounds on $r_k(\mathbb{Z}_p^n)$ for primes $p \leq 31$ and progression lengths $4 \leq k \leq 8$.Comment: 10 page

Principal component analysis as an efficient method for capturing multivariate brain signatures of complex disorders—ENIGMA study in people with bipolar disorders and obesity

Multivariate techniques better fit the anatomy of complex neuropsychiatric disorders which are characterized not by alterations in a single region, but rather by variations across distributed brain networks. Here, we used principal component analysis (PCA) to identify patterns of covariance across brain regions and relate them to clinical and demographic variables in a large generalizable dataset of individuals with bipolar disorders and controls. We then compared performance of PCA and clustering on identical sample to identify which methodology was better in capturing links between brain and clinical measures. Using data from the ENIGMA-BD working group, we investigated T1-weighted structural MRI data from 2436 participants with BD and healthy controls, and applied PCA to cortical thickness and surface area measures. We then studied the association of principal components with clinical and demographic variables using mixed regression models. We compared the PCA model with our prior clustering analyses of the same data and also tested it in a replication sample of 327 participants with BD or schizophrenia and healthy controls. The first principal component, which indexed a greater cortical thickness across all 68 cortical regions, was negatively associated with BD, BMI, antipsychotic medications, and age and was positively associated with Li treatment. PCA demonstrated superior goodness of fit to clustering when predicting diagnosis and BMI. Moreover, applying the PCA model to the replication sample yielded significant differences in cortical thickness between healthy controls and individuals with BD or schizophrenia. Cortical thickness in the same widespread regional network as determined by PCA was negatively associated with different clinical and demographic variables, including diagnosis, age, BMI, and treatment with antipsychotic medications or lithium. PCA outperformed clustering and provided an easy-to-use and interpret method to study multivariate associations between brain structure and system-level variables. Practitioner Points: In this study of 2770 Individuals, we confirmed that cortical thickness in widespread regional networks as determined by principal component analysis (PCA) was negatively associated with relevant clinical and demographic variables, including diagnosis, age, BMI, and treatment with antipsychotic medications or lithium. Significant associations of many different system-level variables with the same brain network suggest a lack of one-to-one mapping of individual clinical and demographic factors to specific patterns of brain changes. PCA outperformed clustering analysis in the same data set when predicting group or BMI, providing a superior method for studying multivariate associations between brain structure and system-level variables.</p

Principal component analysis as an efficient method for capturing multivariate brain signatures of complex disorders—ENIGMA study in people with bipolar disorders and obesity

Multivariate techniques better fit the anatomy of complex neuropsychiatric disorders which are characterized not by alterations in a single region, but rather by variations across distributed brain networks. Here, we used principal component analysis (PCA) to identify patterns of covariance across brain regions and relate them to clinical and demographic variables in a large generalizable dataset of individuals with bipolar disorders and controls. We then compared performance of PCA and clustering on identical sample to identify which methodology was better in capturing links between brain and clinical measures. Using data from the ENIGMA-BD working group, we investigated T1-weighted structural MRI data from 2436 participants with BD and healthy controls, and applied PCA to cortical thickness and surface area measures. We then studied the association of principal components with clinical and demographic variables using mixed regression models. We compared the PCA model with our prior clustering analyses of the same data and also tested it in a replication sample of 327 participants with BD or schizophrenia and healthy controls. The first principal component, which indexed a greater cortical thickness across all 68 cortical regions, was negatively associated with BD, BMI, antipsychotic medications, and age and was positively associated with Li treatment. PCA demonstrated superior goodness of fit to clustering when predicting diagnosis and BMI. Moreover, applying the PCA model to the replication sample yielded significant differences in cortical thickness between healthy controls and individuals with BD or schizophrenia. Cortical thickness in the same widespread regional network as determined by PCA was negatively associated with different clinical and demographic variables, including diagnosis, age, BMI, and treatment with antipsychotic medications or lithium. PCA outperformed clustering and provided an easy-to-use and interpret method to study multivariate associations between brain structure and system-level variables. Practitioner Points: In this study of 2770 Individuals, we confirmed that cortical thickness in widespread regional networks as determined by principal component analysis (PCA) was negatively associated with relevant clinical and demographic variables, including diagnosis, age, BMI, and treatment with antipsychotic medications or lithium. Significant associations of many different system-level variables with the same brain network suggest a lack of one-to-one mapping of individual clinical and demographic factors to specific patterns of brain changes. PCA outperformed clustering analysis in the same data set when predicting group or BMI, providing a superior method for studying multivariate associations between brain structure and system-level variables.</p

Projected resurgence of COVID-19 in the United States in July—December 2021 resulting from the increased transmissibility of the Delta variant and faltering vaccination

In Spring 2021, the highly transmissible SARS-CoV-2 Delta variant began to cause increases in cases, hospitalizations, and deaths in parts of the United States. At the time, with slowed vaccination uptake, this novel variant was expected to increase the risk of pandemic resurgence in the US in summer and fall 2021. As part of the COVID-19 Scenario Modeling Hub, an ensemble of nine mechanistic models produced 6-month scenario projections for July–December 2021 for the United States. These projections estimated substantial resurgences of COVID-19 across the US resulting from the more transmissible Delta variant, projected to occur across most of the US, coinciding with school and business reopening. The scenarios revealed that reaching higher vaccine coverage in July–December 2021 reduced the size and duration of the projected resurgence substantially, with the expected impacts was largely concentrated in a subset of states with lower vaccination coverage. Despite accurate projection of COVID-19 surges occurring and timing, the magnitude was substantially underestimated 2021 by the models compared with the of the reported cases, hospitalizations, and deaths occurring during July–December, highlighting the continued challenges to predict the evolving COVID-19 pandemic. Vaccination uptake remains critical to limiting transmission and disease, particularly in states with lower vaccination coverage. Higher vaccination goals at the onset of the surge of the new variant were estimated to avert over 1.5 million cases and 21,000 deaths, although may have had even greater impacts, considering the underestimated resurgence magnitude from the model

Impact of SARS-CoV-2 vaccination of children ages 5–11 years on COVID-19 disease burden and resilience to new variants in the United States, November 2021–March 2022: a multi-model studyResearch in context

Summary: Background: The COVID-19 Scenario Modeling Hub convened nine modeling teams to project the impact of expanding SARS-CoV-2 vaccination to children aged 5–11 years on COVID-19 burden and resilience against variant strains. Methods: Teams contributed state- and national-level weekly projections of cases, hospitalizations, and deaths in the United States from September 12, 2021 to March 12, 2022. Four scenarios covered all combinations of 1) vaccination (or not) of children aged 5–11 years (starting November 1, 2021), and 2) emergence (or not) of a variant more transmissible than the Delta variant (emerging November 15, 2021). Individual team projections were linearly pooled. The effect of childhood vaccination on overall and age-specific outcomes was estimated using meta-analyses. Findings: Assuming that a new variant would not emerge, all-age COVID-19 outcomes were projected to decrease nationally through mid-March 2022. In this setting, vaccination of children 5–11 years old was associated with reductions in projections for all-age cumulative cases (7.2%, mean incidence ratio [IR] 0.928, 95% confidence interval [CI] 0.880–0.977), hospitalizations (8.7%, mean IR 0.913, 95% CI 0.834–0.992), and deaths (9.2%, mean IR 0.908, 95% CI 0.797–1.020) compared with scenarios without childhood vaccination. Vaccine benefits increased for scenarios including a hypothesized more transmissible variant, assuming similar vaccine effectiveness. Projected relative reductions in cumulative outcomes were larger for children than for the entire population. State-level variation was observed. Interpretation: Given the scenario assumptions (defined before the emergence of Omicron), expanding vaccination to children 5–11 years old would provide measurable direct benefits, as well as indirect benefits to the all-age U.S. population, including resilience to more transmissible variants. Funding: Various (see acknowledgments)

Evaluation of the US COVID-19 Scenario Modeling Hub for informing pandemic response under uncertainty

Abstract Our ability to forecast epidemics far into the future is constrained by the many complexities of disease systems. Realistic longer-term projections may, however, be possible under well-defined scenarios that specify the future state of critical epidemic drivers. Since December 2020, the U.S. COVID-19 Scenario Modeling Hub (SMH) has convened multiple modeling teams to make months ahead projections of SARS-CoV-2 burden, totaling nearly 1.8 million national and state-level projections. Here, we find SMH performance varied widely as a function of both scenario validity and model calibration. We show scenarios remained close to reality for 22 weeks on average before the arrival of unanticipated SARS-CoV-2 variants invalidated key assumptions. An ensemble of participating models that preserved variation between models (using the linear opinion pool method) was consistently more reliable than any single model in periods of valid scenario assumptions, while projection interval coverage was near target levels. SMH projections were used to guide pandemic response, illustrating the value of collaborative hubs for longer-term scenario projections