Multiset automata are a class of automata for which the symbols can be read in any order and obtain the same result. We investigate weighted multiset automata and show how to construct them from weighted regular expressions. We present training methods to learn the weights for weighted regular expressions and for general multiset automata from data. Finally, we examine situations in which inside weights can be computed more efficiently.
Abstract. Following Bhargava and Hanke's celebrated 290-theorem, we prove a universality theorem for all positive-definite integer-valued quadratic forms that represent all positive integers coprime to 3. In particular, if a positive-definite quadratic form represents all positive integers coprime to 3 and ≤ 290, then it represents all positive integers coprime to 3. We use similar methods to those used by Rouse to prove (assuming GRH) that a positive-definite quadratic form representing every odd integer between 1 and 451 represents all positive odd integers.
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