1/13/2024 0 Comments Simple math proofsI think the first person to prove this was G.A. $x$ has order $a$, $y$ has order $b$, and $xy$ has order $c$. There is a theorem in finite group theory, that if $a$, $b$, and $c$ are integers all greater than $1$, there exists a finite group $G$ with elements $x$ and $y$ such that: I decided to make this a community wiki, and I think the usual "one example per answer" guideline makes sense here. I'll post an answer which gives what I would consider to be an example. In summary, I'm looking for results that everyone thought was really hard but which turned out to be almost trivial (or at least natural) when looked at in the right way. it is fine if the machinery is extremely difficult to construct). Finally, I insist that the proofs really be quick (it should be possible to explain it in a few sentences granting the machinery on which it depends) but certainly not necessarily easy (i.e. I would also prefer results which really did have difficult solutions before the quick proofs were found. I am not as interested in problems which motivated the development of complex machinery that eventually solved them, such as the Poincare conjecture in dimension five or higher (which motivated the development of surgery theory) or the Weil conjectures (which motivated the development of l-adic and other cohomology theories). I would like to hear about some examples of problems which were originally solved using arduous direct techniques, but were later found to be corollaries of more sophisticated results. Many problems which first are solved via "direct" methods (long and difficult calculations, tricky estimates, and gritty technical theorems) later turn out to follow beautifully from basic properties of simple devices, though it often takes some work to set up the new machinery. Hence, our technique may be of independent interest.Mathematics is rife with the fruit of abstraction. High density measures correspond to smooth distributions which arise naturally, for instance, in the context of online learning. Our algorithm has a logarithmic runtime over any domain from which we can efficiently sample. We present an algorithm which efficiently approximates the Bregman projection onto the set of high density measures when the Kullback-Leibler divergence is used as a distance function. Bregman projections are widely used in convex optimization and machine learning. The algorithm uses a generalized multiplicative update rule combined with a natural notion of approximate Bregman projection. Then, there is an efficient algorithm which for every input length produces a circuit that computes the function correctly on almost all inputs.Our algorithm significantly simplifies previous proofs of the uniform and the non-uniform hard-core lemma, while matching or improving the previously best known parameters. one that doesn't assign too much weight to any single input) over the inputs produces a circuit such that the circuit computes the boolean function noticeably better than random. Assume there is an efficient algorithm which for every input length and every smooth distribution (i.e. Informally stated, our result is the following: suppose we fix a family of boolean functions. Our result follows from the connection between boosting algorithms and hard-core set constructions discovered by Klivans and Servedio. We give a simple, more efficient and uniform proof of the hard-core lemma, a fundamental result in complexity theory with applications in machine learning and cryptography.
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