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All continuous endomorphisms f ∞ of the shift dynamical system S on the 2-adic integers Z 2 are induced by some f : B n → {0, 1}, where n is a positive integer, B n is the set of n-blocks over {0, 1}, and f ∞ (x) = y 0 y 1 y 2. .. We prove that D, V • D, S, and V • S are conjugate to S and are the only continuous endomorphisms of S whose parity vector(More)
Given a partition λ of n, a k-minor of λ is a partition of n − k whose Young diagram fits inside that of λ. We find an explicit function g(n) such that any partition of n can be reconstructed from its set of k-minors if and only if k ≤ g(n). In particular, partitions of n ≥ k 2 + 2k are uniquely determined by their sets of k-minors. This result completely(More)
Let R(w; q) be Dyson's generating function for partition ranks. For roots of unity ζ = 1, it is known that R(ζ; q) and R(ζ; 1/q) are given by harmonic Maass forms, Eichler integrals, and modular units. We show that modular forms arise from G(w; q), the generating function for ranks of partitions into distinct parts, in a similar way. If D(w; q) := (1 +(More)
Let Q(n) denote the number of partitions of n into distinct parts. We show that Dyson's rank provides a combinatorial interpretation of the well-known fact that Q(n) is almost always divisible by 4. This interpretation gives rise to a new false theta function identity that reveals surprising analytic properties of one of Ramanujan's mock theta functions,(More)
The 3x + 1 Conjecture asserts that the T-orbit of every positive integer contains 1, where T maps x → x/2 for x even and x → (3x + 1)/2 for x odd. A set S of positive integers is sufficient if the orbit of each positive integer intersects the orbit of some member of S. In [9] it was shown that every arithmetic sequence is sufficient. In this paper we(More)
Let A be a finite alphabet and let L ⊂ (A *) n be an n-variable language over A. We say that L is regular if it is the language accepted by a synchronous n-tape finite state automaton, it is quasi-regular if it is accepted by an asynchronous n-tape automaton, and it is weakly regular if it is accepted by a non-deterministic asynchronous n-tape automaton. We(More)
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