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In the 1820s, Americans found conditions in Texas to be much to their liking. The Mexican government encouraged immigration and strove to ease the process for Americans, allowing them a great degree of freedom in choosing the location of settlements and the political organization as well.
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They had cultural ties with them; British news/propaganda also swayed the American public opinion in favor of the Allies.
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During his trial for suspicion of heresy, Galileo chose his words carefully. It was only after the trial, angered by his conviction no doubt, that he was said to have muttered to the inquisitors, “Eppur si muove”(“And yet it moves)”, as if to say that they may have won this battle, but in the end, truth would win out.
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Comparative advantage. It is being able to produce goods by using fewer resources, at a lower opportunity cost, that gives countries a comparative advantage. The gradient of a PPF reflects the opportunity cost of production. Increasing the production of one good means that less of another can be produced.