Harry J. Altman

activity

20122026

most citedInteger complexity: The integer defect

3 citations · 7 across the 5 of their papers we have counts for

collaborators

11 papers

math.CO2026

A Ring structure on the Class of Combinatorial Games

Harry Altman, Paolo Lipparini

J. Conway defined useful operations on the Class of combinatorial games and also introduced a notion of equivalence between games. Conway showed that, under his equivalence, games…

math.LO2024★ 1 cited

Bounding finite-image sequences of length $ω^{k}$

Harry Altman

Given a well-quasi-order $X$ and an ordinal $α$ , the set $s_{α}^{F} (X)$ of transfinite sequences on $X$ with length less than $α$ and with finite image is also a well-quasi-order, as p…

math.NT2021

Integer complexity: Stability and self-similarity

Harry Altman, Juan Arias de Reyna

Define $∣∣ n ∣∣$ to be the complexity of $n$ , the smallest number of ones needed to write $n$ using an arbitrary combination of addition and multiplication. The set $D$ of…

math.LO2019

Maximum linearizations of lower sets in $N^{m}$ with application to monomial ideals

Harry Altman, Andreas Weiermann

We compute the type (maximum linearization) of the well partial order of bounded lower sets in $N^{m}$ , ordered under inclusion, and find it is $ω^{ω^{m - 1}}$ . Moreover we co…

math.NT2018★ 3 cited

Integer complexity: The integer defect

Harry Altman

Define $∥ n ∥$ to be the complexity of $n$ , the smallest number of ones needed to write $n$ using an arbitrary combination of addition and multiplication. John Selfridge showed tha…

math.NT2016★ 3 cited

Integer complexity: algorithms and computational results

Harry Altman

Define $∥ n ∥$ to be the complexity of $n$ , the smallest number of ones needed to write $n$ using an arbitrary combination of addition and multiplication. Define $n$ to be stable i…

math.NT2016

Integer Complexity: Representing Numbers of Bounded Defect

Harry Altman

Define $∥ n ∥$ to be the complexity of $n$ , the smallest number of ones needed to write $n$ using an arbitrary combination of addition and multiplication. John Selfridge showed tha…

math.LO2015

Intermediate arithmetic operations on ordinal numbers

Harry Altman

There are two well-known ways of doing arithmetic with ordinal numbers: the "ordinary" addition, multiplication, and exponentiation, which are defined by transfinite iteration; and…

math.NT2014

Internal Structure of Addition Chains: Well-Ordering

Harry Altman

An addition chain for $n$ is defined to be a sequence $(a_{0}, a_{1}, \dots, a_{r})$ such that $a_{0} = 1$ , $a_{r} = n$ , and, for any $1 \leq k \leq r$ , there exist $0 \leq i, j < k$ such that $a_k = a_i…

math.NT2013

Integer Complexity and Well-Ordering

Harry Altman

Define $∥ n ∥$ to be the complexity of $n$ , the smallest number of ones needed to write $n$ using an arbitrary combination of addition and multiplication. John Selfridge showed tha…

math.NT2012

Numbers with Integer Complexity Close to the Lower Bound

Harry Altman, Joshua Zelinsky

Define $∣ n ∣$ to be the complexity of $n$ , the smallest number of 1's needed to write $n$ using an arbitrary combination of addition and multiplication. John Selfridge showed that $…

A Ring structure on the Class of Combinatorial Games

Bounding finite-image sequences of length ωk

Integer complexity: Stability and self-similarity

Maximum linearizations of lower sets in Nm with application to monomial ideals

Integer complexity: The integer defect

Integer complexity: algorithms and computational results

Integer Complexity: Representing Numbers of Bounded Defect

Intermediate arithmetic operations on ordinal numbers

Internal Structure of Addition Chains: Well-Ordering

Integer Complexity and Well-Ordering

Numbers with Integer Complexity Close to the Lower Bound

Bounding finite-image sequences of length $ω^{k}$

Maximum linearizations of lower sets in $N^{m}$ with application to monomial ideals