We introduce Elgot categories, a sort of distributive monoidal category with additional structure in which the partial recursive functions are representable. Moreover, we construct an initial Elgot category, the morph...
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Hofstadter’s G function is recursively defined via G(0) = 0 and then G(n) = n−G(G(n−1)). Following Hofstadter, a family (Fk) of similar functions is obtained by varying the number k of nested recursive calls in this ...
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We give Euler-like recursive formulas for the t-colored partition function when t = 2 or t = 3, as well as for all t-regular partition functions. In particular, we derive an infinite family of "triangular number&...
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In this paper, we show that if (Un)n≥1 is any nondegenerate linearly recurrent sequence of integers whose general term is up to sign not a polynomial in n, then the inequality φ(|Un|) ≥ |Uφ(n)| holds on a set of p...
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In this paper, we define a novel recursive Heaviside step sequence function and demonstrate its applicability to modeling human mental states such as thought processes, memory recall, and forgetfulness. By extending t...
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Concerning classical computational models able to express all the Primitive recursive functions (PRF), there are interesting results regarding limits on their algorithmic expressiveness or, equivalently, efficiency, n...
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This paper proposes an asymptotic theory for online inference of the stochastic gradient descent (SGD) iterates with dropout regularization in linear regression. Specifically, we establish the geometric-moment contrac...
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In the past few years, a successful line of research has lead to lower bounds for several fundamental local graph problems in the distributed setting. These results were obtained via a technique called round eliminati...
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In the past few years, a successful line of research has lead to lower bounds for several fundamental local graph problems in the distributed setting. These results were obtained via a technique called round elimination. On a high level, the round elimination technique can be seen as a recursive application of a function that takes as input a problem Π and outputs a problem Π′ that is one round easier than Π. Applying this function recursively to concrete problems of interest can be highly nontrivial, which is one of the reasons that has made the technique difficult to approach. The contribution of our paper is threefold. Firstly, we develop a new and fully automatic method for finding so-called fixed point relaxations under round elimination. The detection of a non-0-round solvable fixed point relaxation of a problem Π immediately implies lower bounds of Ω(log∆ n) and Ω(log∆ log n) rounds for deterministic and randomized algorithms for Π, respectively. Secondly, we show that this automatic method is indeed useful, by obtaining lower bounds for defective coloring problems. More precisely, as an application of our procedure, we show that the problem of coloring the nodes of a graph with 3 colors and defect at most (∆ − 3)/2 requires Ω(log∆ n) rounds for deterministic algorithms and Ω(log∆ log n) rounds for randomized ones. Additionally, we provide a simplified proof for an existing defective coloring lower bound. We note that lower bounds for coloring problems are notoriously challenging to obtain, both in general, and via the round elimination technique. Both the first and (indirectly) the second contribution build on our third contribution—a new and conceptually simple way to compute the one-round easier problem Π′ in the round elimination framework. This new procedure provides a clear and easy recipe for applying round elimination, thereby making a substantial step towards the greater goal of having a fully automatic procedure for obtaining lower bounds in the dis
We introduce the Tor groups (formula presented) for a loopless matroid M as a way to study the extra relations occurring in the linear ideal of the Feichtner–Yuzvinsky presentation of the Chow ring A•(M). This extend...
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