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There are many examples of countries that have converged with developed countries which validate the catch-up theory. [5] Based on case studies on Japan, Mexico and other countries, Nakaoka studied social capabilities for industrialization and clarified the features of human and social attitudes in the catching-up process of Japan in the Meiji period (1868-1912).
The different notions of convergence capture different properties about the sequence, with some notions of convergence being stronger than others. For example, convergence in distribution tells us about the limit distribution of a sequence of random variables. This is a weaker notion than convergence in probability, which tells us about the ...
Symbolic convergence theory (SCT) is a communication theory developed by Ernest Bormann proposing that the holding of fantasies in common transforms collections of individuals into cohesive groups. SCT offers an explanation for the appearance of a group's cohesiveness, consisting of shared emotions, motives, and meanings.
For (,) a measurable space, a sequence μ n is said to converge setwise to a limit μ if = ()for every set .. Typical arrow notations are and .. For example, as a consequence of the Riemann–Lebesgue lemma, the sequence μ n of measures on the interval [−1, 1] given by μ n (dx) = (1 + sin(nx))dx converges setwise to Lebesgue measure, but it does not converge in total variation.
The following result is a generalisation of the monotone convergence of non negative sums theorem above to the measure theoretic setting. It is a cornerstone of measure and integration theory with many applications and has Fatou's lemma and the dominated convergence theorem as direct consequence.
The Riemann hypothesis catastrophe thought experiment provides one example of instrumental convergence. Marvin Minsky, the co-founder of MIT's AI laboratory, suggested that an artificial intelligence designed to solve the Riemann hypothesis might decide to take over all of Earth's resources to build supercomputers to help achieve its goal. [2]
In measure theory, Lebesgue's dominated convergence theorem ... For example, order the rationals ... One corollary to the dominated convergence theorem is the bounded ...
A fundamental example, Donsker's theorem, is convergence of rescaled random walk to Brownian motion. The mathematical theory, combining probability and functional analysis, was first developed in the 1950s by Skorokhod and Prokhorov, but was regarded as a specialized advanced topic.