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The need to describe abrupt changes or response of nonlinear systems to impulsive stimuli is ubiquitous in applications. Also the informal use of infinitesimal and infinite quantities is still a method used to construct idealized but tractable models within the famous J. von Neumann reasonably wide area of applicability. We review the theory of generalized smooth functions as a candidate to address both these needs: a rigorous but simple language of infinitesimal and infinite quantities, and the possibility to deal with continuous and generalized function as if they were smooth maps: with pointwise values, free composition and hence nonlinear operations, all the classical theorems of calculus, a good integration theory, and new existence results for differential equations. We exemplify the applications of this theory through several models of singular dynamical systems: deduction of the heat and wave equations extended to generalized functions, a singular variable length pendulum wrapping on a parallelepiped, the oscillation of a pendulum damped by different media, a nonlinear stress–strain model of steel, singular Lagrangians as used in optics, and some examples from quantum mechanics.

We prove Banach, Newton-Raphson and Brouwer fixed point theorems in the framework of generalized smooth functions, a minimal extension of Colombeau's theory (and hence of classical distribution theory) which makes it possible to model nonlinear singular problems, while at the same time sharing a number of fundamental properties with ordinary smooth functions, such as the closure with respect to composition and several non trivial classical theorems of the calculus. The proved results allows one to deal with equations of the form F(x)=0, where F is a generalized smooth function, in particular, a Sobolev-Schwartz distribution. We consider examples with singularities that are not included in the classical version of these theorems.

Alcohol use is causally linked to multiple cancers. We present global, regional, and national estimates of alcohol-attributable cancer burden in 2020 to inform alcohol policy and cancer control across different settings globally. In this population-based study, population attributable fractions (PAFs) calculated using a theoretical minimum-risk exposure of lifetime abstention and 2010 alcohol consumption estimates from the Global Information System on Alcohol and Health (assuming a 10-year latency period between alcohol consumption and cancer diagnosis), combined with corresponding relative risk estimates from systematic literature reviews as part of the WCRF Continuous Update Project, were applied to cancer incidence data from GLOBOCAN 2020 to estimate new cancer cases attributable to alcohol. We also calculated the contribution of moderate (<20 g per day), risky (20–60 g per day), and heavy (>60 g per day) drinking to the total alcohol-attributable cancer burden, as well as the contribution by 10 g per day increment (up to a maximum of 150 g). 95% uncertainty intervals (UIs) were estimated using a Monte Carlo-like approach. Globally, an estimated 741 300 (95% UI 558 500–951 200), or 4·1% (3·1–5·3), of all new cases of cancer in 2020 were attributable to alcohol consumption. Males accounted for 568 700 (76·7%; 95% UI 422 500–731 100) of total alcohol-attributable cancer cases, and cancers of the oesophagus (189 700 cases [110 900–274 600]), liver (154 700 cases [43 700–281 500]), and breast (98 300 cases [68 200–130 500]) contributed the most cases. PAFs were lowest in northern Africa (0·3% [95% UI 0·1–3·3]) and western Asia (0·7% [0·5–1·2]), and highest in eastern Asia (5·7% [3·6–7·9]) and central and eastern Europe (5·6% [4·6–6·6]). The largest burden of alcohol-attributable cancers was represented by heavy drinking (346 400 [46·7%; 95% UI 227 900–489 400] cases) and risky drinking (291 800 [39·4%; 227 700–333 100] cases), whereas moderate drinking contributed 103 100 (13·9%; 82 600–207 200) cases, and drinking up to 10 g per day contributed 41 300 (35 400–145 800) cases. Our findings highlight the need for effective policy and interventions to increase awareness of cancer risks associated with alcohol use and decrease overall alcohol consumption to prevent the burden of alcohol-attributable cancers. None.

A summary evaluation of the 2015 American Cancer Society (ACS) challenge goal showed that overall US mortality from all cancers combined declined 26% over the period from 1990 to 2015. Recent research suggests that US cancer mortality can still be lowered considerably by applying known interventions broadly and equitably. The ACS Board of Directors, therefore, commissioned ACS researchers to determine challenge goals for reductions in cancer mortality by 2035. A statistical model was used to estimate the average annual percent decline in overall cancer death rates among the US general population and among college‐educated Americans during the most recent period. Then, the average annual percent decline in the overall cancer death rates of college graduates was applied to the death rates in the general population to project future rates in the United States beginning in 2020. If overall cancer death rates from 2020 through 2035 nationally decline at the pace of those of college graduates, then death rates in 2035 in the United States will drop by 38.3% from the 2015 level and by 54.4% from the 1990 level. On the basis of these results, the ACS 2035 challenge goal was set as a 40% reduction from the 2015 level. Achieving this goal could lead to approximately 1.3 million fewer cancer deaths than would have occurred from 2020 through 2035 and 122,500 fewer cancer deaths in 2035 alone. The results also show that reducing the prevalence of risk factors and achieving optimal adherence to evidence‐based screening guidelines by 2025 could lead to a 33.5% reduction in the overall cancer death rate by 2035, attaining 85% of the challenge goal.