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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)
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)
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)
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)
The dual of an algebraic curve C in RP 2 defined by the polynomial equation f (x, y, z) = 0 is the locus of points ∂f ∂x (a, b, c) where (a : b : c) ∈ C. The dual can alternatively be defined geometrically as the image under reciprocation of the envelope of tangent lines to the curve. It is known that the dual of an algebraic curve is also an algebraic(More)
We provide an introduction to the statements of the n! and (n + 1) n−1 conjectures of Garsia, Haiman. We introduce several bigraded S n modules (indexed by partitions) and provide evidence to support the fact that their Frobenius series is given by the transformed Macdonald polynomials, thereby showing that the Kostka-Macdonald polynomials count the(More)
Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Symbolic dynamics Shift map a b s t r a c t All continuous endomorphisms f ∞(More)
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