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So, a permutation is a special arrangement of symbols. A “superpermutation,” in a sense, is a special arrangement of permutations: It is a sequence of symbols that contains every possible permutation of those symbols. For example, stringing together all six permutations of {A, B, C} gives us ABCACBBACBCACABCBA, a superpermutation ...Read More

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"A little high school geometry can help you understand the basic math behind movie recommendation engines."

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I remember reading these papers when they were first widely available. I am no great mathematician (and certainly not adequate to adjudicate the controversy in Corollary 3.12), but I remember having a sense of how the proof might work.
In essence, Mochizuki is trying to arrive at a more abstract understanding of numbers, whereby manipulating the fundamental attributes of numbers changes the relationships inherent in basic operators. The way I conceptualize it is roughly analogous to the following: 1 + 1 = 2 is true, but only within certain ...Read More

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"Digital security depends on the difficulty of factoring large numbers. A new proof shows why one method for breaking digital encryption won’t work."

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"A founder of modern geometric analysis who produced “some of the most dramatic advances in mathematics in the last 40 years,” Uhlenbeck is the first woman to be"

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"In requiring theories to be beautiful and natural, we may be pre-rejecting reality."