Let ~(X)= ~+ xz+x ~z z) X=(~4,~zx3) ; ~(~) let r(n) be the number of integral solutions of the equation QCX) = (1)The following theorem is proved: let n = I, 2, 3, 5, 6 (mod 8) and let r(n, ~) be the number of integral solutions of equation (i) such that Y = ~/~ E ~ where ~ is an arbitrary convex domain with a piecewise smooth boundary on the unit sphere S: Q(Y) = i. Then where D(~) is a measure, normalized by the condition ~(S) = i. A similar result is obtained for the three-dimensional ellipsoid of general form. The mentioned theorem, in combination with the classical Guass--Siegel results on r(n), yields the uniform distribution of the integral points on the three-dimensional sphere (i). i. The questions of the representability of integers by positive definite ternary quadratic forms and the closely related questions on the number of integral points in domains on ellipsoids have been investigated by a series of authors. Ternary quadratic forms of general form have been considered for the first time by Yu. V, Linnik (the discrete ergodic method, going back to his 1939-1940 investigations (see [i]); for applications to ellipsoids, the ergodic method has been developed by A. V. Malyshev [2] and Yu. G. Teterin [3], [4]. Another method of investigation of the ellipsoids belongs to E. P. Golubeva [5], [6]. For their use, both these methods require some assumptions, for example, unproved although weak conjectures on the zeros of Dirichlet L-series, and so far they do not give the final solution of the indicated problems.We consider the most important case, namely the three-dimensional sphere Sph3 ; ~ + ~f + ~ = ~ .

Background Physical inactivity, now the fourth leading cause of death, is a primary element of noncommunicable diseases. Despite a great number of attempts, there is still a lack of effective approaches that can motivate sedentary populations to increase their levels of physical activity over a sustained period. Incentives for exercise can provide an immediate reward for increasing activity levels, but because of limited funding to provide rewards, previous programs using this approach have only shown short-term changes in behavior. Sweatcoin (Sweatco Ltd, UK) is an app-based platform that converts physical movement into virtual currency. The currency can be exchanged for goods and services on their marketplace, providing a continuous incentive to be active. This study investigates the physical activity behavior change observed in Sweatcoin users over a 6-month period of app usage. Objective The aim of this study was to investigate the change in physical activity (measured using daily step count) of a sample of Sweatcoin users, the longevity of the change, and whether this change can be predicted by demographic and other lifestyle variables. Methods Activity data from a sample of 5892 Sweatcoin users were used to analyze daily step count. Activity change was measured in terms of the percentage change in average daily step count for each month after registration, relative to that in the 3 months before using the app. Users were grouped according to having no or negative, moderate, or high activity change. A subset of users completed a questionnaire that allowed differences between groups in terms of activity and demographic status to be investigated using regression analyses. Results Daily step count increased by 19% on average over the 6 months following registration ( P <.001). Of the questionnaire respondents, 728 were valid responses. A multinomial logistic regression identified the key drivers of moderate and high activity behavior change relative to no or negative change based on the defined groupings. There was a clear impact of seasonality, with those registering for the app in winter (odds ratio [OR] 4.67; P =.001) and spring (OR 5.05; P =.001) being more likely to show high positive activity behavior change than those registering in summer. More striking were the results identifying those classified as overweight (measured through body mass index [BMI]; OR 1.83; P =.02) and less active (based on a self-reported scale of physical activity; OR 0.88; P =.048), being most likely to show high levels of physical activity change following registration with the app. Conclusions The results highlight that an incentives-based app can induce significant physical activity behavior change, sustained over a 6-month period. Importantly, the results suggest...

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