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This paper proves the existence and uniqueness of a time-invariant measure for the 2D Navier-Stokes equations on the sphere under a random kick-force and a time-periodic deterministic force. Several examples of deterministic force satisfying the necessary conditions for a unique invariant measure to exist are given. The support of the measure is examined and given explicitly for several cases.
In the late 19th century arguments explaining incest avoidance were framed separately by Edward Tylor and Edward Westermarck. Tylor offered an environmental theory asserting that people have to marry outside of their own kin and communities or die out from the detrimental effects of isolation. Westermarck turned to Darwin’s theory to explain that harmful inbreeding had been selected against in the human genome. By the late 20th and early 21st centuries explanations of human behaviors have become increasingly encompassed by natural selection theory. The debate concerning the productiveness of evolutionary biology for explaining complex human behaviors is highly contentious and continues unabated. Although human evolutionists repeatedly say that environment is important for understanding human behavior they often do not develop this part of the equation. Behind the prestige of evolutionary biology selection models of human behavior have passed into popular science and the public psyche. Often heard today from a wide range of highly visible media sources is an assortment of topics on human behaviors which are framed by Darwinian assumptions. Contemplations about incest and inbreeding avoidance fall into this category and are presented by Darwinian social science as the best case example demonstrating evolutionary suppositions about human behavior. In the article that follows these issues are framed and examined. The argument is offered that evolutionary approaches are not always the most compelling and that convincing environmental explanations are overlooked.