The Sitnikov problem is a restricted three body problem where the eccentricity of the primaries acts as a parameter. We find families of symmetric periodic solutions bifurcating from the equilibrium at the center of mass.
ImportanceNeuroimaging studies have documented racial and ethnic disparities in brain health in old age. It remains unclear whether these disparities are apparent in midlife.ObjectiveTo assess racial and ethnic disparities in magnetic resonance imaging (MRI) markers of cerebrovascular disease and neurodegeneration in midlife and late life.Design, Setting, and ParticipantsData from 2 community-based cohort studies, Washington Heights–Inwood Columbia Aging Project (WHICAP) and the Offspring Study of Racial and Ethnic Disparities in Alzheimer Disease (Offspring), were used. Enrollment took place from March 2011 and June 2017, in WHICAP and Offspring, respectively, to January 2021. Of the 822 Offspring and 1254 WHICAP participants approached for MRI scanning, 285 and 176 refused participation in MRI scanning, 36 and 76 were excluded for contraindications/ineligibility, and 4 and 32 were excluded for missing key variables, respectively.Main Outcomes and MeasuresCortical thickness in Alzheimer disease–related regions, white matter hyperintensity (WMH) volume.ResultsThe final sample included 1467 participants. Offspring participants (497 [33.9%]) had a mean (SD) age of 55 (10.7) years, had a mean (SD) of 13 (3.5) years of education, and included 117 Black individuals (23.5%), 348 Latinx individuals (70%), 32 White individuals (6.4%), and 324 women (65.2%). WHICAP participants (970 [66.1%]) had a mean (SD) age of 75 (6.5) years, had a mean (SD) of 12 (4.7) years of education, and included 338 Black individuals (34.8%), 389 Latinx individuals (40.1%), 243 White individuals (25.1%), and 589 women (65.2%). Racial and ethnic disparities in cerebrovascular disease were observed in both midlife (Black-White: B = 0.357; 95% CI, 0.708-0.007; P = .046) and late life (Black-Latinx: B = 0.149, 95% CI, 0.068-0.231; P < .001; Black-White: B = 0.166; 95% CI, 0.254-0.077; P < .001), while disparities in cortical thickness were evident in late life only (Black-Latinx: B = −0.037; 95% CI, −0.055 to −0.019; P < .001; Black-White: B = −0.064; 95% CI −0.044 to −0.084; P < .001). Overall, Black-White disparities were larger than Latinx-White disparities for cortical thickness and WMH volume. Brain aging, or the association of age with MRI measures, was greater in late life compared with midlife for Latinx (cortical thickness: B = 0.006; 95% CI, 0.004-0.008; P < .001; WMH volume: B = −0.010; 95% CI, −0.018 to −0.001; P = .03) and White (cortical thickness: B = 0.005; 95% CI, 0.002-0.008; P = .001; WMH volume: B = −0.021; 95% CI −0.043 to 0.002; P = .07) participants but not Black participants (cortical thickness: B = 0.001; 95% CI, −0.002 to 0.004; P =.64; WMH volume: B = 0.003; 95% CI, −0.010 to 0.017; P = .61), who evidenced a similarly strong association between age and MRI measures in midlife and late life.Conclusions and RelevanceIn this study, racial and ethnic disparities in small vessel cerebrovascular disease were apparent in midlife. In Latinx and White adults, brain aging was more pronounced in late life than midlife, whereas Black adults showed accelerated pattern of brain aging beginning in midlife.
This paper studies a special restricted (N + 1)-body problem which can be reduced to the Sitnikov problem with an appropriate positive parameter. According to the number of bodies we prove the existence (or nonexistence) of a finite (or infinite) number of symmetric families of periodic solutions. These solutions bifurcate from the equilibrium at the center of mass of the system. Introduction.In celestial mechanics there is a restricted 3-body problem known as the Sitnikov problem. In this problem we have two material points (bodies) P 1 , P 2 (called primaries) with equal mass m 1 = m 2 moving in the plane x, y around the origin (center of mass) describing elliptic orbits of the 2-body problem and a body P 0 of infinitesimal mass that moves along the z-axis. The Sitnikov problem deals with the study of the orbits of P 0 . If we choose m 1 = m 2 = 1/2, the gravitational constant G = 1, and the period of the orbit described by the primaries is equal to 2π, then the equation of motion of P 0 is
We provide sufficient conditions for the existence and stability of periodic solutions of the second-order non-autonomous differential equation of the Nathanson's model x + x + aẋ − b(v 0 + δv(ωt)) 2 (1 − x) 2 = 0, and of the comb-drive finger model x + x + aẋ − 4b(v 0 + δv(ωt)) 2 x (1 − x 2) 2 = 0, where x ∈ R, c, β, v 0 and δ are positive parameters, v(ωt) is a 2π/ω-periodic function. The results are obtained using the averaging theory.
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