The Theory Behind TheoryMine

Abstract. We describe the technology behind the TheoryMine novelty gift company, which sells the rights to name novel mathematical theorems. A tower of four computer systems is used to generate recursive theories, then to speculate conjectures in those theories and then to prove these conjectures. All stages of the process are entirely automatic. The process guarantees large numbers of sound, novel theorems of some intrinsic merit.

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

Case-Analysis for Rippling and Inductive Proof

Rippling is a heuristic used to guide rewriting and is typically used for inductive theorem proving. We introduce a method to support case-analysis within rippling. Like earlier work, this allows goals containing if-statements to be proved automatically. The new contribution is that our method also supports case-analysis on datatypes. By locating the case-analysis as a step within rippling we also maintain the termination. The work has been implemented in IsaPlanner and used to extend the existing inductive proof method. We evaluate this extended prover on a large set of examples from Isabelle’s theory library and from the inductive theorem proving literature. We find that this leads to a significant improvement in the coverage of inductive theorem proving. The main limitations of the extended prover are identified, highlight the need for advances in the treatment of assumptions during rippling and when conjecturing lemmas

A Proof Strategy Language and Proof Script Generation for Isabelle/HOL

We introduce a language, PSL, designed to capture high level proof strategies in Isabelle/HOL. Given a strategy and a proof obligation, PSL's runtime system generates and combines various tactics to explore a large search space with low memory usage. Upon success, PSL generates an efficient proof script, which bypasses a large part of the proof search. We also present PSL's monadic interpreter to show that the underlying idea of PSL is transferable to other ITPs.Comment: This paper has been submitted to CADE2

The Sloan Lens ACS Survey. IX. Colors, Lensing and Stellar Masses of Early-type Galaxies

We present the current photometric dataset for the Sloan Lens ACS (SLACS) Survey, including HST photometry from ACS, WFPC2, and NICMOS. These data have enabled the confirmation of an additional 15 grade `A' (certain) lens systems, bringing the number of SLACS grade `A' lenses to 85; including 13 grade `B' (likely) systems, SLACS has identified nearly 100 lenses and lens candidates. Approximately 80% of the grade `A' systems have elliptical morphologies while ~10% show spiral structure; the remaining lenses have lenticular morphologies. Spectroscopic redshifts for the lens and source are available for every system, making SLACS the largest homogeneous dataset of galaxy-scale lenses to date. We have developed a novel Bayesian stellar population analysis code to determine robust stellar masses with accurate error estimates. We apply this code to deep, high-resolution HST imaging and determine stellar masses with typical statistical errors of 0.1 dex; we find that these stellar masses are unbiased compared to estimates obtained using SDSS photometry, provided that informative priors are used. The stellar masses range from 10^10.5 to 10^11.8 M$_\odot$ and the typical stellar mass fraction within the Einstein radius is 0.4, assuming a Chabrier IMF. The ensemble properties of the SLACS lens galaxies, e.g. stellar masses and projected ellipticities, appear to be indistinguishable from other SDSS galaxies with similar stellar velocity dispersions. This further supports that SLACS lenses are representative of the overall population of massive early-type galaxies with M* >~ 10^11 M$_\odot$, and are therefore an ideal dataset to investigate the kpc-scale distribution of luminous and dark matter in galaxies out to z ~ 0.5.Comment: 20 pages, 18 figures, 5 tables, published in Ap

Nonuniform high-gamma (60-500 Hz) power changes dissociate cognitive task and anatomy in human cortex

High-gamma-band (\u3e60 Hz) power changes in cortical electrophysiology are a reliable indicator of focal, event-related cortical activity. Despite discoveries of oscillatory subthreshold and synchronous suprathreshold activity at the cellular level, there is an increasingly popular view that high-gamma-band amplitude changes recorded from cellular ensembles are the result of asynchronous firing activity that yields wideband and uniform power increases. Others have demonstrated independence of power changes in the low- and high-gamma bands, but to date, no studies have shown evidence of any such independence above 60 Hz. Based on nonuniformities in time-frequency analyses of electrocorticographic (ECoG) signals, we hypothesized that induced high-gamma-band (60-500 Hz) power changes are more heterogeneous than currently understood. Using single-word repetition tasks in six human subjects, we showed that functional responsiveness of different ECoG high-gamma sub-bands can discriminate cognitive task (e.g., hearing, reading, speaking) and cortical locations. Power changes in these sub-bands of the high-gamma range are consistently present within single trials and have statistically different time courses within the trial structure. Moreover, when consolidated across all subjects within three task-relevant anatomic regions (sensorimotor, Broca\u27s area, and superior temporal gyrus), these behavior- and location-dependent power changes evidenced nonuniform trends across the population. Together, the independence and nonuniformity of power changes across a broad range of frequencies suggest that a new approach to evaluating high-gamma-band cortical activity is necessary. These findings show that in addition to time and location, frequency is another fundamental dimension of high-gamma dynamics

The development and validation of an age-structured model for the evaluation of disease control strategies for intestinal helminths

Epidemiological modelling can be a useful tool for the evaluation of parasite control strategies. An age-structured epidemiological model of intestinal helminth dynamics is developed. This model includes the explicit representation of changing worm distributions between hosts as a result of treatment, and estimates the morbidity due to heavy infections. The model is used to evaluate the effectiveness of different programmes of age-targeted community chemotherapy in reducing the amount of morbidity due to helminth infection. The magnitude of age-related heterogeneities is found to be very important in determining the results of age-targeted treatment programmes. The model was verified using field data from control programmes for Ascaris lumbricoides and Trichuris trichiura, and was found to provide accurate predictions of prevalence and mean intensities of infection during and following different control regime

Identification and characterisation of enteroaggregative Escherichia coli subtypes associated with human disease

Enteroaggregative E. coli (EAEC) are a major cause of diarrhoea worldwide. Due to their heterogeneity and carriage in healthy individuals, identification of diagnostic virulence markers for pathogenic strains has been difficult. In this study, we have determined phenotypic and genotypic differences between EAEC strains of sequence types (STs) epidemiologically associated with asymptomatic carriage (ST31) and diarrhoeal disease (ST40). ST40 strains demonstrated significantly enhanced intestinal adherence, biofilm formation, and pro-inflammatory interleukin-8 secretion compared with ST31 isolates. This was independent of whether strains were derived from diarrhoea patients or healthy controls. Whole genome sequencing revealed differences in putative virulence genes encoding aggregative adherence fimbriae, E. coli common pilus, flagellin and EAEC heat-stable enterotoxin 1. Our results indicate that ST40 strains have a higher intrinsic potential of human pathogenesis due to a specific combination of virulence-related factors which promote host cell colonization and inflammation. These findings may contribute to the development of genotypic and/or phenotypic markers for EAEC strains of high virulence

Mock observations with the Millennium Simulation: cosmological downsizing and intermediate-redshift observations

Only by incorporating various forms of feedback can theories of galaxy formation reproduce the present-day luminosity function of galaxies. It has also been argued that such feedback processes might explain the counterintuitive behaviour of 'downsizing' witnessed since redshifts z ≃ 1–2. To examine this question, observations spanning 0.4 < z < 1.4 from the Deep Extragalactic Evolutionary Probe (DEEP)2/Palomar survey are compared with a suite of equivalent mock observations derived from the Millennium Simulation, populated with galaxies using the galform code. Although the model successfully reproduces the observed total mass function and the general trend of 'downsizing', it fails to accurately reproduce the colour distribution and type-dependent mass functions at all redshifts probed. This failure is shared by other semi-analytical models which collectively appear to 'over-quench' star formation in intermediate-mass systems. These mock lightcones are also a valuable tool for investigating the reliability of the observational results in terms of cosmic variance. Using variance estimates derived from the lightcones, we confirm the significance of the decline since z ∼ 1 in the observed number density of massive blue galaxies which, we argue, provides the bulk of the associated growth in the red sequence. We also assess the limitations arising from cosmic variance in terms of our ability to observe mass-dependent growth since z ∼ 1

Adaptive Optics Imaging of QSOs with Double-Peaked Narrow Lines: Are they Dual AGNs?

Active galaxies hosting two accreting and merging super-massive black holes (SMBHs) -- dual Active Galactic Nuclei (AGN) -- are predicted by many current and popular models of black hole-galaxy co-evolution. We present here the results of a program that has identified a set of probable dual AGN candidates based on near Infra-red (NIR) Laser Guide-Star Adaptive Optics (LGS AO) imaging with the Keck II telescope. These candidates are selected from a complete sample of radio-quiet Quasi-stellar Objects (QSOs) drawn from the Sloan Digital Sky Survey (SDSS), which show double-peaked narrow AGN emission lines. Of the twelve AGNs imaged, we find six with double galaxy structure, of which four are in galaxy mergers. We measure the ionization of the two velocity components in the narrow AGN lines to test the hypothesis that both velocity components come from an active nucleus. The combination of a well-defined parent sample and high-quality imaging allows us to place constraints on the fraction of SDSS QSOs that host dual accreting black holes separated on kiloparsec (kpc) scales: ~0.3%-0.65%. We derive from this fraction the time spent in a QSO phase during a typical merger and find a value that is much lower than estimates that arise from QSO space densities and galaxy merger statistics. We discuss possible reasons for this difference. Finally, we compare the SMBH mass distributions of single and dual AGN and find little difference between the two within the limited statistics of our program, hinting that most SMBH growth happens in the later stages of a merger process.Comment: 9 pages, 4 figures, 1 table; accepted to the Astrophysical Journa