2,667 research outputs found
HR: A System for Machine Discovery in Finite Algebras
We describe the HR concept formation program which invents mathematical definitions and conjectures in finite algebras such as group theory and ring theory. We give the methods behind and the reasons for the concept formation in HR, an evaluation of its performance in its training domain, group theory, and a look at HR in domains other than group theory
The Mass Assembly Histories of Galaxies of Various Morphologies in the GOODS Fields
We present an analysis of the growth of stellar mass with cosmic time
partitioned according to galaxy morphology. Using a well-defined catalog of
2150 galaxies based, in part, on archival data in the GOODS fields, we assign
morphological types in three broad classes (Ellipticals, Spirals,
Peculiar/Irregulars) to a limit of z_AB=22.5 and make the resulting catalog
publicly available. We combine redshift information, optical photometry from
the GOODS catalog and deep K-band imaging to assign stellar masses. We find
little evolution in the form of the galaxy stellar mass function from z~1 to
z=0, especially at the high mass end where our results are most robust.
Although the population of massive galaxies is relatively well established at
z~1, its morphological mix continues to change, with an increasing proportion
of early-type galaxies at later times. By constructing type-dependent stellar
mass functions, we show that in each of three redshift intervals, E/S0's
dominate the higher mass population, while spirals are favored at lower masses.
This transition occurs at a stellar mass of 2--3 times 10^{10} Msun at z~0.3
(similar to local studies) but there is evidence that the relevant mass scale
moves to higher mass at earlier epochs. Such evolution may represent the
morphological extension of the ``downsizing'' phenomenon, in which the most
massive galaxies stop forming stars first, with lower mass galaxies becoming
quiescent later. We infer that more massive galaxies evolve into spheroidal
systems at earlier times, and that this morphological transformation may only
be completed 1--2 Gyr after the galaxies emerge from their active star forming
phase. We discuss several lines of evidence suggesting that merging may play a
key role in generating this pattern of evolution.Comment: 24 pages, 1 table, 8 figures, accepted for publication in Ap
The Use of Classification in Automated Mathematical Concept Formation
Concept formation programs aim to produce a high yield of concepts which are considered interesting. One intelligent way to do this is to base a new concept on one or more concepts which are already known to be interesting. This requires a concrete notion of the `interestingness' of a particular concept. Restricting the concepts formed to mathematical definitions in finite group theory, we derive three measures of the importance of a concept. These measures are based on how much the concept improves a classification of finite groups. Introduction One approach to automatic mathematical concept formation is to perform a heuristic search through a space of sentences which define mathematical concepts. In the space, there will be some sentences which are rubbish, some which are plausible but not very exciting, and some which are important. In order to be able to do an effective search, reducing the number of rubbish sentences, and increasing the yield of important concepts, it is..
A Human-Oriented Term Rewriting System
© Springer Nature Switzerland AG 2019. We introduce a fully automatic system, implemented in the Lean theorem prover, that solves equality problems of everyday mathematics. Our overriding priority in devising the system is that it should construct proofs of equality in a way that is similar to that of humans. A second goal is that the methods it uses should be domain independent. The basic strategy of the system is to operate with a subtask stack: whenever there is no clear way of making progress towards the task at the top of the stack, the program finds a promising subtask, such as rewriting a subterm, and places that at the top of the stack instead. Heuristics guide the choice of promising subtasks and the rewriting process. This makes proofs more human-like by breaking the problem into tasks in the way that a human would. We show that our system can prove equality theorems simply, without having to preselect or orient rewrite rules as in standard theorem provers, and without having to invoke heavy duty tools for performing simple reasoning
The Mass Assembly History of Spheroidal Galaxies: Did Newly-Formed Systems Arise Via Major Mergers?
We examine the properties of a morphologically-selected sample of 0.4<z<1.0
spheroidal galaxies in the GOODS fields in order to ascertain whether their
increase in abundance with time arises primarily from mergers. To address this
question we determine scaling relations between the dynamical mass determined
from stellar velocity dispersions, and the stellar mass determined from optical
and infrared photometry. We exploit these relations across the larger sample
for which we have stellar masses in order to construct the first statistically
robust estimate of the evolving dynamical mass function over 0<z<1. The trends
observed match those seen in the stellar mass functions of Bundy et al. 2005
regarding the top-down growth in the abundance of spheroidal galaxies. By
referencing our dynamical masses to the halo virial mass we compare the growth
rate in the abundance of spheroidals to that predicted by the assembly of dark
matter halos. Our comparisons demonstrate that major mergers do not fully
account for the appearance of new spheroidals since z~1 and that additional
mechanisms, such as morphological transformations, are required to drive the
observed evolution.Comment: Accepted to ApJL; New version corrects the Millennium merger
predictions--further details at
http://www.astro.utoronto.ca/~bundy/millennium
Automatic Invention of Integer Sequences
We report on the application of the HR program (Colton, Bundy, & Walsh 1999) to the problem of automatically inventing integer sequences. Seventeen sequences invented by HR are interesting enough to have been accepted into the Encyclopedia of Integer Sequences (Sloane 2000) and all were supplied with interesting conjectures about their nature, also discovered by HR. By extending HR, we have enabled it to perform a two stage process of invention and investigation. This involves generating both the definition and terms of a new sequence, relating it to sequences already in the Encyclopedia and pruning the output to help identify the most surprising and interesting results
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