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Michael J. Collins

Publications and source records attributed to Michael J. Collins.

8 recordsLinked to original sources
arXiv2026

Computing with Continued Logarithms

Gosper developed an algorithm for performing arithmetic on continued fractions (CFs), and introduced continued logarithms (CLs) as a variant of continued fractions better suited to representing extremely large (or small) numbers. CLs are also well-suited to efficient hardware implementation. Here we present the algorithm for arithmetic on CLs, then extend it to the novel contribution of this paper, an algorithm for computing trigonometric, exponential, and log functions on CLs. These methods can be extended to other transcendental functions. As with the corresponding CF algorithms, computations are entirely in the domain of the CL representation, with no floating-point arithmetic; we read one CL input term at a time, producing the next CL term of the result as soon as it is determined. The CL algorithms are in fact simpler than their CF counterparts. We have implemented these algorithms in Haskell.

math.NT↗
arXiv2025

Transcendental Functions on Continued Fractions

Gosper developed an algorithm for performing arithmetic operations on continued fractions (CFs), getting a CF as the result. Straightforward implementation of the algorithm leads to infinite loops on some inputs. Here we present a modified version of the arithmetic algorithm and prove that it avoids all difficulties with infinite loops. We then combine CF arithmetic with the spigot algorithm to compute the CF expansions of exponential, logarithmic, and trigonometric functions of CFs. We have implemented these algorithms in Haskell.

math.CO↗
arXiv2021

A Biologically Plausible Parser

We describe a parser of English effectuated by biologically plausible neurons and synapses, and implemented through the Assembly Calculus, a recently proposed computational framework for cognitive function. We demonstrate that this device is capable of correctly parsing reasonably nontrivial sentences. While our experiments entail rather simple sentences in English, our results suggest that the parser can be extended beyond what we have implemented, to several directions encompassing much of language. For example, we present a simple Russian version of the parser, and discuss how to handle recursion, embedding, and polysemy.

cs.CL↗
arXiv2018

Equivalence of OEIS A007729 and A174868

We verify the conjecture that the sixth binary partition function is equal (aside from the initial zero term) to the partial sums of the Stern-Brocot sequence.

math.CO↗
arXiv2011

Cost Sharing in the Aspnes Inoculation Model

We consider the use of cost sharing in the Aspnes model of network inoculation, showing that this can improve the cost of the optimal equilibrium by a factor of $O(\sqrt{n})$ in a network of $n$ nodes.

cs.GT↗
arXiv2007

Modular analogues of Jordan's theorem for finite linear groups

In 1878, Jordan showed that a finite subgroup of GL(n,C) contains an abelian normal subgroup whose index is bounded by a function of n alone. Previously, the author has given precise bounds. Here, we consider analogues for finite linear groups over algebraically closed fields of positive characteristic l. A larger normal subgroup must be taken, to eliminate unipotent subgroups and groups of Lie type and characteristic l, and we show that generically the bound is similar to that in characteristic 0 - being (n+1)!, or (n+2)! when l divides (n+2) - given by the faithful representations of minimal degree of the symmetric groups. A complete answer for the optimal bounds is given for all degrees n and every characteristic l.

math.GR↗