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Make sure you understand how to prove that not both ( I ) and ( II ) can hold , and also understand how Tucker 's Existence Theorem is used to prove that at least one of the two statements holds . We conjecture our theorem holds for arbitrary partitions . It can be partially explained by the Schema Theorem ( Holland ) , however , this theorem has been criticized in recent times . " If Kolmogorov 's theorem , or Hecht-Nielsen 's adaptation of it , could be restated in terms of known sigmoid activation functions in the ( single ) hidden and output layers , then Swingler 's alleged upper bound would be correct , but in fact no such restatement of Kolmogorov 's theorem is possible , and Kurkova did not claim to prove any such restatement . Given that there are two of a quite different nature , it now seems extremely likely that this theorem has been proved . The PCP Theorem is one of the deepest and most important theorems in complexity theory . I ’ ve been under the impression that the theorem requires a sort of quantization that doesn ’ t even apply to ordinary gauge theories , but I haven ’ t looked at it — I don ’ t even know the reference — so I ’ d rather not pontificate too much on it . The models all possess the remarkable property that the virial theorem holds locally , from which they earn their name as the hypervirial family . Now the Fundamental Theorem has added another reason for finding antiderivatives : as a tool for calculating ( definite ) integrals . What 's most surprising is the FDH variant , which has a tight security proof despite hashing unrandomized messages ; in the Rabin-Williams context , a minor technical assumption in Coron 's theorem turns out to be a major loophole . In this project we will investigate the combinatorics of the set S of possible secondary struc - tures corresponding to a RNA molecule , and in particular their asymptotics when weighted with the Boltzmann distribution over S. Multivariate Student-T distributions are a family of probability distributions that occur naturally in statistical contexts where the central limit theorem is applied to explain the aggregate distribution of a rescaled sum of in - dependent random vectors and when the variance-covariance matrix of this aggregate distribution has to be estimated from data points . As an application , this discrete intermediate value theorem is shown to provide an elegant proof of an existence theorem in discrete mathematics .