BACKGROUND:Non-malignant tumors of the CNS contribute substantially to the morbidity and mortality from CNS tumors. It is critical to understand the epidemiology of non-malignant CNS tumors separately from CNS malignancies to inform resource allocation and policy since treatment and prognosis can differ. High-quality international data on non-malignant CNS tumor burden are needed to accomplish this goal. METHODS:We assessed the cancer registry and vital registration data available to the Global Burden of Disease study by its inclusion of non-malignant CNS tumors, reporting on the availability of data over time and by the World Bank income group. We analyzed preliminary age-standardized incidence rates (ASIRs), age-standardized mortality rates (ASMRs), and proportions of CNS tumors by behavior for adults, children, and all ages. RESULTS:Non-malignant CNS tumors were reported separately in 17.2% (N = 66) of registry reports and in aggregate with malignant CNS tumors in 18.0% (N = 69) of reports. Only 7 low- and middle-income countries (LMICs) had data reporting CNS tumors separately by behavior. Across all ages combined, the median ASIR of non-malignant CNS tumor data was 0.31 (interquartile range: 0.15-0.50) and ASMR was 0.24 (0.10-0.44) per 100,000 in LMICs compared to median ASIR of 3.62 (2.62-4.97) and ASMR of 0.32 (0.16-0.65) in high-income countries (HICs). A larger proportion of incident CNS tumors were reported as non-malignant in HIC data than LMIC data (P < .0001). CONCLUSIONS:Our study alludes to current challenges in understanding global non-malignant CNS tumor burden and a need for increased international data collection. Further research is needed to comprehensively investigate opportunities for future data inclusion.

10573 Background: Liver cancer is a leading cause of global health burden, and was the seventh leading cause of cancer death in 2021. The most common type of liver cancer in adults is hepatocellular carcinoma, which can be due to alcohol, hepatitis B, hepatitis C, non-alcoholic steatohepatitis (NASH), or other causes. In children the most common type of liver cancer is hepatoblastoma. Comprehensive and comparative estimation of liver cancer burden can inform policy decisions and public health interventions to reduce incidence, morbidity, and mortality. This study provides updated liver cancer estimates from 1990 to 2021, and for the first time includes hepatoblastoma. Methods: Using estimation methods from the Global Burden of Diseases, Injuries, and Risk Factors Study 2021 (GBD 2021) we comprehensively estimated total liver cancer incidence, mortality, prevalence, years lived with disability (YLDs), years of life lost (YLLs), and disability adjusted life-years (DALYs). Etiologic proportions from meta-analyses of literature review data were applied to the total. Liver cancers under age 10 were attributed to hepatoblastoma. YLLs were calculated using life expectancy estimates. Prevalence and YLDs were calculated from expected survival, disease sequelae, and disability weights. DALYs were the sum of YLLs and YLDs. Results are presented with 95% Uncertainty Intervals (95% UI). Results: Globally, there were an estimated 529,000 (95% UI 480,000 – 594,000) liver cancer cases and 484,000 (437,000 – 538,000) deaths in 2021, contributing to 12,900,000 (11,600,000 – 14,400,000) DALYs. Since 1990, these represented increases of 114.3% (87.0 – 145.3%) in cases, 102.5% (76.4 – 132.0%) in deaths, and 70.6% (48.7 – 96.8%) in DALYs. For liver cancer etiologies (excluding hepatoblastoma), 37.4% (32.6 – 42.6%) of deaths were due to hepatitis B, 30.3% (26.3 – 34.8%) hepatitis C, 19.1% (15.8 – 22.8%) alcohol, 8.5% (6.9 – 10.3%) NASH, and 4.3% (3.6 – 5.1%) other causes in 2021. Conclusions: These GBD 2021 estimates provide comprehensive estimates of the substantial health burden of liver cancer, highlighting a continued need for public health efforts targeting prevention, vaccination, treatment, or behavioral change.[Table: see text]

This systematic analysis assesses the total and risk-attributable burden of lip and oral cavity cancer and other pharyngeal cancer for 204 countries and territories and by Socio-demographic Index using 2019 Global Burden of Diseases, Injuries, and Risk Factors Study estimates.

Importance:Lip, oral, and pharyngeal cancers are important contributors to cancer burden worldwide, and a comprehensive evaluation of their burden globally, regionally, and nationally is crucial for effective policy planning. Objective:To analyze the total and risk-attributable burden of lip and oral cavity cancer (LOC) and other pharyngeal cancer (OPC) for 204 countries and territories and by Socio-demographic Index (SDI) using 2019 Global Burden of Diseases, Injuries, and Risk Factors (GBD) Study estimates. Evidence Review:The incidence, mortality, and disability-adjusted life years (DALYs) due to LOC and OPC from 1990 to 2019 were estimated using GBD 2019 methods. The GBD 2019 comparative risk assessment framework was used to estimate the proportion of deaths and DALYs for LOC and OPC attributable to smoking, tobacco, and alcohol consumption in 2019. Findings:In 2019, 370 000 (95% uncertainty interval [UI], 338 000-401 000) cases and 199 000 (95% UI, 181 000-217 000) deaths for LOC and 167 000 (95% UI, 153 000-180 000) cases and 114 000 (95% UI, 103 000-126 000) deaths for OPC were estimated to occur globally, contributing 5.5 million (95% UI, 5.0-6.0 million) and 3.2 million (95% UI, 2.9-3.6 million) DALYs, respectively. From 1990 to 2019, low-middle and low SDI regions consistently showed the highest age-standardized mortality rates due to LOC and OPC, while the high SDI strata exhibited age-standardized incidence rates decreasing for LOC and increasing for OPC. Globally in 2019, smoking had the greatest contribution to risk-attributable OPC deaths for both sexes (55.8% [95% UI, 49.2%-62.0%] of all OPC deaths in male individuals and 17.4% [95% UI, 13.8%-21.2%] of all OPC deaths in female individuals). Smoking and alcohol both contributed to substantial LOC deaths globally among male individuals (42.3% [95% UI, 35.2%-48.6%] and 40.2% [95% UI, 33.3%-46.8%] of all risk-attributable cancer deaths, respectively), while chewing tobacco contributed to the greatest attributable LOC deaths among female individuals (27.6% [95% UI, 21.5%-33.8%]), driven by high risk-attributable burden in South and Southeast Asia. Conclusions and Relevance:In this systematic analysis, disparities in LOC and OPC burden existed across the SDI spectrum, and a considerable percentage of burden was attributable to tobacco and alcohol use. These estimates can contribute to an understanding of the distribution and disparities in LOC and OPC burden globally and support cancer control planning efforts.

The divisorial gonality of a graph is the minimum degree of a positive rank divisor on that graph. We introduce the multiplicity-free gonality of a graph, which restricts our consideration to divi-sors that place at most 1 chip on each vertex. We give a sufficient condition in terms of vertex-connectivity for these two versions of gonality to be equal; and we show that no function of gonality can bound multiplicity-free gonality, even for simple graphs. We also prove that multiplicity-free gonality is NP-hard to compute, while still determining it for graph families for which gonality is currently unknown. We also present new gonalities, such as for the wheel graphs.

IMPORTANCE:The Global Burden of Diseases, Injuries, and Risk Factors Study 2019 (GBD 2019) provided systematic estimates of incidence, morbidity, and mortality to inform local and international efforts toward reducing cancer burden. OBJECTIVE:To estimate cancer burden and trends globally for 204 countries and territories and by Sociodemographic Index (SDI) quintiles from 2010 to 2019. EVIDENCE REVIEW:The GBD 2019 estimation methods were used to describe cancer incidence, mortality, years lived with disability, years of life lost, and disability-adjusted life years (DALYs) in 2019 and over the past decade. Estimates are also provided by quintiles of the SDI, a composite measure of educational attainment, income per capita, and total fertility rate for those younger than 25 years. Estimates include 95% uncertainty intervals (UIs). FINDINGS:In 2019, there were an estimated 23.6 million (95% UI, 22.2-24.9 million) new cancer cases (17.2 million when excluding nonmelanoma skin cancer) and 10.0 million (95% UI, 9.36-10.6 million) cancer deaths globally, with an estimated 250 million (235-264 million) DALYs due to cancer. Since 2010, these represented a 26.3% (95% UI, 20.3%-32.3%) increase in new cases, a 20.9% (95% UI, 14.2%-27.6%) increase in deaths, and a 16.0% (95% UI, 9.3%-22.8%) increase in DALYs. Among 22 groups of diseases and injuries in the GBD 2019 study, cancer was second only to cardiovascular diseases for the number of deaths, years of life lost, and DALYs globally in 2019. Cancer burden differed across SDI quintiles. The proportion of years lived with disability that contributed to DALYs increased with SDI, ranging from 1.4% (1.1%-1.8%) in the low SDI quintile to 5.7% (4.2%-7.1%) in the high SDI quintile. While the high SDI quintile had the highest number of new cases in 2019, the middle SDI quintile had the highest number of cancer deaths and DALYs. From 2010 to 2019, the largest percentage increase in the numbers of cases and deaths occurred in the low and low-middle SDI quintiles. CONCLUSIONS AND RELEVANCE:The results of this systematic analysis suggest that the global burden of cancer is substantial and growing, with burden differing by SDI. These results provide comprehensive and comparable estimates that can potentially inform efforts toward equitable cancer control around the world.

Key Points Question What was the burden of cancer globally and across Sociodemographic Index (SDI) groupings in 2019, and how has incidence, morbidity, and mortality changed since 2010? Findings In this systematic analysis, there were 23.6 million new global cancer cases in 2019 (17.2 million when excluding those with nonmelanoma skin cancer), 10.0 million cancer deaths, and an estimated 250 million disability-adjusted life years estimated to be due to cancer; since 2010, these represent increases of 26.3%, 20.9%, and 16.0%, respectively. Absolute cancer burden increased in all SDI quintiles since 2010, but the largest percentage increases occurred in the low and low-middle SDI quintiles. Meanings The study results suggest that increased cancer prevention and control efforts are needed to equitably address the evolving and increasing burden of cancer across the SDI spectrum.

Background Prevention, control, and treatment of respiratory tract cancers are important steps towards achieving target 3.4 of the UN Sustainable Development Goals (SDGs)-a one-third reduction in premature mortality due to non-communicable diseases by 2030. We aimed to provide global, regional, and national estimates of the burden of tracheal, bronchus, and lung cancer and larynx cancer and their attributable risks from 1990 to 2019. Methods Based on the Global Burden of Diseases, Injuries, and Risk Factors Study (GBD) 2019 methodology, we evaluated the incidence, mortality, years lived with disability, years of life lost, and disability-adjusted life-years (DALYs) of respiratory tract cancers (ie, tracheal, bronchus, and lung cancer and larynx cancer). Deaths from tracheal, bronchus, and lung cancer and larynx cancer attributable to each risk factor were estimated on the basis of risk exposure, relative risks, and the theoretical minimum risk exposure level input from 204 countries and territories, stratified by sex and Socio-demographic Index (SDI). Trends were estimated from 1990 to 2019, with an emphasis on the 2010-19 period. Findings Globally, there were 2.26 million (95% uncertainty interval 2.07 to 2.45) new cases of tracheal, bronchus, and lung cancer, and 2.04 million (1.88 to 2.19) deaths and 45.9 million (42.3 to 49.3) DALYs due to tracheal, bronchus, and lung cancer in 2019. There were 209 000 (194 000 to 225 000) new cases of larynx cancer, and 123 000 (115 000 to 133 000) deaths and 3.26 million (3.03 to 3.51) DALYs due to larynx cancer globally in 2019. From 2010 to 2019, the number of new tracheal, bronchus, and lung cancer cases increased by 23.3% (12.9 to 33.6) globally and the number of larynx cancer cases increased by 24.7% (16.0 to 34.1) globally. Global age-standardised incidence rates of tracheal, bronchus, and lung cancer decreased by 7.4% (-16.8 to 1.6) and age-standardised incidence rates of larynx cancer decreased by 3.0% (-10.5 to 5.0) in males over the past decade; however, during the same period, age-standardised incidence rates in females increased by 0.9% (-8.2 to 10.2) for tracheal, bronchus, and lung cancer and decreased by 0.5% (-8.4 to 8.1) for larynx cancer. Furthermore, although age-standardised incidence and death rates declined in both sexes combined from 2010 to 2019 at the global level for tracheal, bronchus, lung and larynx cancers, some locations had rising rates, particularly those on the lower end of the SDI range. Smoking contributed to an estimated 64.2% (61.9-66.4) of all deaths from tracheal, bronchus, and lung cancer and 63.4% (56.3-69.3) of all deaths from larynx cancer in 2019. For males and for both sexes combined, smoking was the leading specific risk factor for age-standardised deaths from tracheal, bronchus, and lung cancer per 100 000 in all SDI quintiles and GBD regions in 2019. However, among females, household air pollution from solid fuels was the leading specific risk factor in the low SDI quintile and in three GBD regions (central, eastern, and western sub-Saharan Africa) in 2019. Interpretation The numbers of incident cases and deaths from tracheal, bronchus, and lung cancer and larynx cancer increased globally during the past decade. Even more concerning, age-standardised incidence and death rates due to tracheal, bronchus, lung cancer and larynx cancer increased in some populations-namely, in the lower SDI quintiles and among females. Preventive measures such as smoking control interventions, air quality management programmes focused on major air pollution sources, and widespread access to clean energy should be prioritised in these settings. Copyright (C) 2021 The Author(s). Published by Elsevier Ltd.

To any graph we associate a sequence of integers called the gonality sequence of the graph, consisting of the minimum degrees of divisors of increasing rank on the graph. This is a tropical analogue of the gonality sequence of an algebraic curve. We study gonality sequences for graphs of low genus, proving that for genus up to $5$, the gonality sequence is determined by the genus and the first gonality. We then prove that any reasonable pair of first two gonalities is achieved by some graph. We also develop a modified version of Dhar's burning algorithm more suited for studying higher gonalities.

We compute the treewidth of a family of graphs we refer to as the glued grids, consisting of the stacked prism graphs and the toroidal grids. Our main technique is constructing strict brambles of large orders. We discuss connections to divisorial graph theory coming from tropical geometry, and use our results to compute the divisorial gonality of these graphs.

Our main theorem in [] contained a mistake regarding the equivalence of two conditions on a graph, which we correct here. Fortunately our main result is not impacted with an additional assumption called the zero-three condition.

I have spent a lot of time thinking this past year and a half about the relationship between asceticism and success. As a mathematics student and a collegiate athlete, I have far too often gotten caught up in the pursuit of objective standards. This chase has left me burnt out and broken. Existential philosophy has been my greatest asset in discerning the true purpose of asceticism. I reflect on this journey and the nature of assessment in this short reflection.

In 2013, Chan classified all metric hyperelliptic graphs, proving that divisorial gonality and geometric gonality are equivalent in the hyperelliptic case. We show that such a classification extends to combinatorial graphs of divisorial gonality three, under certain edge- and vertex-connectivity assumptions. We also give a construction for graphs of divisorial gonality three, and provide conditions for determining when a graph is not of divisorial gonality three.